The following describes the semantics of operations defined in the
XlaBuilder
interface. Typically, these operations map onetoone to operations defined in
the RPC interface in
xla_data.proto
.
A note on nomenclature: the generalized data type XLA deals with is an Ndimensional array holding elements of some uniform type (such as 32bit float). Throughout the documentation, array is used to denote an arbitrarydimensional array. For convenience, special cases have more specific and familiar names; for example a vector is a 1dimensional array and a matrix is a 2dimensional array.
AfterAll
See also
XlaBuilder::AfterAll
.
AfterAll takes a variadic number of tokens and produces a single token. Tokens
are primitive types which can be threaded between sideeffecting operations to
enforce ordering. AfterAll
can be used as a join of tokens for ordering a
operation after a set operations.
AfterAll(operands)
Arguments  Type  Semantics 

operands 
XlaOp 
variadic number of tokens 
AllToAll
See also
XlaBuilder::AllToAll
.
Alltoall is a collective operation that sends data from all cores to all cores. It has two phases:
 the scatter phase. On each core, the operand is split into
split_count
number of blocks along thesplit_dimensions
, and the blocks are scattered to all cores, e.g., the ith block is send to the ith core.  the gather phase. Each core concatenates the received blocks along the
concat_dimension
.
The participating cores can be configured by:
replica_groups
: each ReplicaGroup contains a list of replica id. If empty, all replicas belong to one group in the order of 0  (n1). Alltoall will be applied within subgroups in the specified order. For example, replica groups = { {1,2,3},{4,5,0}} means, an Alltoall will be applied within replica 1, 2, 3, and in the gather phase, the received blocks will be concatenated in the order of 1, 2, 3; another Alltoall will be applied within replica 4, 5, 0, and the concatenation order is 4, 5, 0.
Prerequisites:
 The dimension size of the operand on the split_dimension is divisible by split_count.
 The operand's shape is not tuple.
AllToAll(operand, split_dimension, concat_dimension, split_count,
replica_groups)
Arguments  Type  Semantics 

operand 
XlaOp 
n dimensional input array 
split_dimension

int64

A value in the interval [0,
n) that names the dimension
along which the operand is
split 
concat_dimension

int64

a value in the interval [0,
n) that names the dimension
along which the split blocks
are concatenated 
split_count

int64

the number of cores that
participate this operation. If
replica_groups is empty, this
should be the number of
replicas; otherwise, this
should be equal to the number
of replicas in each group. 
replica_groups

ReplicaGroup vector

each group contains a list of replica id. 
Below shows an example of Alltoall.
XlaBuilder b("alltoall");
auto x = Parameter(&b, 0, ShapeUtil::MakeShape(F32, {4, 16}), "x");
AllToAll(x, /*split_dimension=*/1, /*concat_dimension=*/0, /*split_count=*/4);
In this example, there are 4 cores participating the Alltoall. On each core, the operand is split into 4 parts along dimension 0, so each part has shape f32[4,4]. The 4 parts are scattered to all cores. Then each core concatenates the received parts along dimension 1, in the order or core 04. So the output on each core has shape f32[16,4].
BatchNormGrad
See also
XlaBuilder::BatchNormGrad
and the original batch normalization paper
for a detailed description of the algorithm.
Calculates gradients of batch norm.
BatchNormGrad(operand, scale, mean, variance, grad_output, epsilon, feature_index)
Arguments  Type  Semantics 

operand

XlaOp

n dimensional array to be normalized (x) 
scale

XlaOp

1 dimensional array (\(\gamma\)) 
mean 
XlaOp 
1 dimensional array (\(\mu\)) 
variance

XlaOp

1 dimensional array (\(\sigma^2\)) 
grad_output

XlaOp

Gradients passed to
BatchNormTraining
(\( \nabla y\)) 
epsilon 
float 
Epsilon value (\(\epsilon\)) 
feature_index

int64

Index to feature dimension in
operand 
For each feature in the feature dimension (feature_index
is the index for the
feature dimension in operand
), the operation calculates the gradients with
respect to operand
, offset
and scale
across all the other dimensions. The
feature_index
must be a valid index for the feature dimension in operand
.
The three gradients are defined by the following formulas (assuming a
4dimensional array as operand
and with feature dimension index
m
and spatial sizes w
and h
):
\[ \begin{split} c_l&= \frac{1}{mwh}\sum_{i=1}^m\sum_{j=1}^w\sum_{k=1}^h \left( \nabla y_{ijkl} \frac{x_{ijkl}  \mu_l}{\sigma^2_l+\epsilon} \right) \\\\ \nabla x_{ijkl} &= \frac{\gamma_{l}}{\sqrt{\sigma^2_{l}+\epsilon}} \left( \nabla y_{ijkl}  \mathrm{mean}(\nabla y)  c_l (x_{ijkl}  \mu_{l}) \right) \\\\ \nabla \gamma_l &= \sum_{i=1}^m\sum_{j=1}^w\sum_{k=1}^h \left( \nabla y_{ijkl} \frac{x_{ijkl}  \mu_l}{\sqrt{\sigma^2_{l}+\epsilon}} \right) \\\\\ \nabla \beta_l &= \sum_{i=1}^m\sum_{j=1}^w\sum_{k=1}^h \nabla y_{ijkl} \end{split} \]
The inputs mean
and variance
represent moments value
across batch and spatial dimensions.
The output type is a tuple of three handles:
Outputs  Type  Semantics 

grad_operand

XlaOp

gradient with respect to input
operand (\( \nabla x\)) 
grad_scale

XlaOp

gradient with respect to input
scale (\( \nabla \gamma\)) 
grad_offset

XlaOp

gradient with respect to input
offset (\( \nabla \beta\)) 
BatchNormInference
See also
XlaBuilder::BatchNormInference
and the original batch normalization paper
for a detailed description of the algorithm.
Normalizes an array across batch and spatial dimensions.
BatchNormInference(operand, scale, offset, mean, variance, epsilon, feature_index)
Arguments  Type  Semantics 

operand 
XlaOp 
n dimensional array to be normalized 
scale 
XlaOp 
1 dimensional array 
offset 
XlaOp 
1 dimensional array 
mean 
XlaOp 
1 dimensional array 
variance 
XlaOp 
1 dimensional array 
epsilon 
float 
Epsilon value 
feature_index 
int64 
Index to feature dimension in operand 
For each feature in the feature dimension (feature_index
is the index for the
feature dimension in operand
), the operation calculates the mean and variance
across all the other dimensions and uses the mean and variance to normalize each
element in operand
. The feature_index
must be a valid index for the feature
dimension in operand
.
BatchNormInference
is equivalent to calling BatchNormTraining
without
computing mean
and variance
for each batch. It uses the input mean
and
variance
instead as estimated values. The purpose of this op is to reduce
latency in inference, hence the name BatchNormInference
.
The output is an ndimensional, normalized array with the same shape as input
operand
.
BatchNormTraining
See also
XlaBuilder::BatchNormTraining
and the original batch normalization paper
for a detailed description of the algorithm.
Normalizes an array across batch and spatial dimensions.
BatchNormTraining(operand, scale, offset, epsilon, feature_index)
Arguments  Type  Semantics 

operand 
XlaOp 
n dimensional array to be normalized (x) 
scale 
XlaOp 
1 dimensional array (\(\gamma\)) 
offset 
XlaOp 
1 dimensional array (\(\beta\)) 
epsilon 
float 
Epsilon value (\(\epsilon\)) 
feature_index 
int64 
Index to feature dimension in operand 
For each feature in the feature dimension (feature_index
is the index for the
feature dimension in operand
), the operation calculates the mean and variance
across all the other dimensions and uses the mean and variance to normalize each
element in operand
. The feature_index
must be a valid index for the feature
dimension in operand
.
The algorithm goes as follows for each batch in operand
\(x\) that
contains m
elements with w
and h
as the size of spatial dimensions
(assuming operand
is an 4 dimensional array):
Calculates batch mean \(\mu_l\) for each feature
l
in feature dimension: \(\mu_l=\frac{1}{mwh}\sum_{i=1}^m\sum_{j=1}^w\sum_{k=1}^h x_{ijkl}\)Calculates batch variance \(\sigma^2_l\): \(\sigma^2_l=\frac{1}{mwh}\sum_{i=1}^m\sum_{j=1}^w\sum_{k=1}^h (x_{ijkl}  \mu_l)^2\)
Normalizes, scales and shifts: \(y_{ijkl}=\frac{\gamma_l(x_{ijkl}\mu_l)}{\sqrt[2]{\sigma^2_l+\epsilon}}+\beta_l\)
The epsilon value, usually a small number, is added to avoid dividebyzero errors.
The output type is a tuple of three XlaOp
s:
Outputs  Type  Semantics 

output

XlaOp

n dimensional array with the same
shape as input operand (y) 
batch_mean 
XlaOp 
1 dimensional array (\(\mu\)) 
batch_var 
XlaOp 
1 dimensional array (\(\sigma^2\)) 
The batch_mean
and batch_var
are moments calculated across the batch and
spatial dimensions using the formulas above.
BitcastConvertType
See also
XlaBuilder::BitcastConvertType
.
Similar to a tf.bitcast
in TensorFlow, performs an elementwise bitcast
operation from a data shape to a target shape. The dimensions must match, and
the conversion is an elementwise one; e.g. s32
elements become f32
elements
via bitcast routine. Bitcast is implemented as a lowlevel cast, so machines
with different floatingpoint representations will give different results.
BitcastConvertType(operand, new_element_type)
Arguments  Type  Semantics 

operand 
XlaOp 
array of type T with dims D 
new_element_type 
PrimitiveType 
type U 
The dimensions of the operand and the target shape must match. The bitwidth of the source and destination element types must be equal. The source and destination element types must not be tuples.
Broadcast
See also
XlaBuilder::Broadcast
.
Adds dimensions to an array by duplicating the data in the array.
Broadcast(operand, broadcast_sizes)
Arguments  Type  Semantics 

operand 
XlaOp 
The array to duplicate 
broadcast_sizes 
ArraySlice<int64> 
The sizes of the new dimensions 
The new dimensions are inserted on the left, i.e. if broadcast_sizes
has
values {a0, ..., aN}
and the operand shape has dimensions {b0, ..., bM}
then
the shape of the output has dimensions {a0, ..., aN, b0, ..., bM}
.
The new dimensions index into copies of the operand, i.e.
output[i0, ..., iN, j0, ..., jM] = operand[j0, ..., jM]
For example, if operand
is a scalar f32
with value 2.0f
, and
broadcast_sizes
is {2, 3}
, then the result will be an array with shape
f32[2, 3]
and all the values in the result will be 2.0f
.
BroadcastInDim
See also
XlaBuilder::BroadcastInDim
.
Expands the size and rank of an array by duplicating the data in the array.
BroadcastInDim(operand, out_dim_size, broadcast_dimensions)
Arguments  Type  Semantics 

operand 
XlaOp 
The array to duplicate 
out_dim_size

ArraySlice<int64>

The sizes of the dimensions of the target shape 
broadcast_dimensions

ArraySlice<int64>

Which dimension in the target shape each dimension of the operand shape corresponds to 
Similar to Broadcast, but allows adding dimensions anywhere and expanding existing dimensions with size 1.
The operand
is broadcast to the shape described by out_dim_size
.
broadcast_dimensions
maps the dimensions of operand
to the dimensions of the
target shape, i.e. the i'th dimension of the operand is mapped to the
broadcast_dimension[i]'th dimension of the output shape. The dimensions of
operand
must have size 1 or be the same size as the dimension in in the output
shape they are mapped to. The remaining dimensions are filled with dimensions of
size 1. Degeneratedimension broadcasting then broadcasts along these degenerate
dimensions to reach the output shape. The semantics are described in detail on
the broadcasting page.
Call
See also
XlaBuilder::Call
.
Invokes a computation with the given arguments.
Call(computation, args...)
Arguments  Type  Semantics 

computation

XlaComputation

computation of type T_0, T_1, ...,
T_N > S with N parameters of
arbitrary type 
args 
sequence of N XlaOp s 
N arguments of arbitrary type 
The arity and types of the args
must match the parameters of the
computation
. It is allowed to have no args
.
Cholesky
See also
XlaBuilder::Cholesky
.
Computes the Cholesky decomposition of a batch of symmetric (Hermitian) positive definite matrices.
Cholesky(a, lower)
Arguments  Type  Semantics 

a 
XlaOp 
a rank > 2 array of a complex or floatingpoint type. 
lower 
bool 
whether to use the upper or lower triangle of a . 
If lower
is true
, computes lowertriangular matrices l
such that
lower
is false
, computes uppertriangular matrices u
such
that Input data is read only from the lower/upper triangle of a
, depending on the
value of lower
. Values from the other triangle are ignored. Output data is
returned in the same triangle; the values in the other triangle are
implementationdefined and may be anything.
If the rank of a
is greater than 2, a
is treated as a batch of matrices,
where all except the minor 2 dimensions are batch dimensions.
If a
is not symmetric (Hermitian) positive definite, the result is
implementationdefined.
Clamp
See also
XlaBuilder::Clamp
.
Clamps an operand to within the range between a minimum and maximum value.
Clamp(min, operand, max)
Arguments  Type  Semantics 

min 
XlaOp 
array of type T 
operand 
XlaOp 
array of type T 
max 
XlaOp 
array of type T 
Given an operand and minimum and maximum values, returns the operand if it is in
the range between the minimum and maximum, else returns the minimum value if the
operand is below this range or the maximum value if the operand is above this
range. That is, clamp(a, x, b) = min(max(a, x), b)
.
All three arrays must be the same shape. Alternatively, as a restricted form of
broadcasting, min
and/or max
can be a scalar of type T
.
Example with scalar min
and max
:
let operand: s32[3] = {1, 5, 9};
let min: s32 = 0;
let max: s32 = 6;
==>
Clamp(min, operand, max) = s32[3]{0, 5, 6};
Collapse
See also
XlaBuilder::Collapse
and the tf.reshape
operation.
Collapses dimensions of an array into one dimension.
Collapse(operand, dimensions)
Arguments  Type  Semantics 

operand 
XlaOp 
array of type T 
dimensions 
int64 vector 
inorder, consecutive subset of T's dimensions. 
Collapse replaces the given subset of the operand's dimensions by a single
dimension. The input arguments are an arbitrary array of type T and a
compiletimeconstant vector of dimension indices. The dimension indices must be
an inorder (low to high dimension numbers), consecutive subset of T's
dimensions. Thus, {0, 1, 2}, {0, 1}, or {1, 2} are all valid dimension sets, but
{1, 0} or {0, 2} are not. They are replaced by a single new dimension, in the
same position in the dimension sequence as those they replace, with the new
dimension size equal to the product of original dimension sizes. The lowest
dimension number in dimensions
is the slowest varying dimension (most major)
in the loop nest which collapses these dimension, and the highest dimension
number is fastest varying (most minor). See the tf.reshape
operator
if more general collapse ordering is needed.
For example, let v be an array of 24 elements:
let v = f32[4x2x3] { { {10, 11, 12}, {15, 16, 17}},
{ {20, 21, 22}, {25, 26, 27}},
{ {30, 31, 32}, {35, 36, 37}},
{ {40, 41, 42}, {45, 46, 47}}};
// Collapse to a single dimension, leaving one dimension.
let v012 = Collapse(v, {0,1,2});
then v012 == f32[24] {10, 11, 12, 15, 16, 17,
20, 21, 22, 25, 26, 27,
30, 31, 32, 35, 36, 37,
40, 41, 42, 45, 46, 47};
// Collapse the two lower dimensions, leaving two dimensions.
let v01 = Collapse(v, {0,1});
then v01 == f32[4x6] { {10, 11, 12, 15, 16, 17},
{20, 21, 22, 25, 26, 27},
{30, 31, 32, 35, 36, 37},
{40, 41, 42, 45, 46, 47}};
// Collapse the two higher dimensions, leaving two dimensions.
let v12 = Collapse(v, {1,2});
then v12 == f32[8x3] { {10, 11, 12},
{15, 16, 17},
{20, 21, 22},
{25, 26, 27},
{30, 31, 32},
{35, 36, 37},
{40, 41, 42},
{45, 46, 47}};
CollectivePermute
See also
XlaBuilder::CollectivePermute
.
CollectivePermute is a collective operation that sends and receives data cross replicas.
CollectivePermute(operand, source_target_pairs)
Arguments  Type  Semantics 

operand 
XlaOp 
n dimensional input array 
source_target_pairs

<int64, int64> vector

A list of (source_replica_id, target_replica_id) pairs. For each pair, the operand is sent from source replica to target replica. 
Note that there are the following restrictions on the source_target_pair
:
 Any two pairs should not have the same target replica id, and they should not have the same source replica id.
 If a replica id is not a target in any pair, then the output on that replica is a tensor consists of 0(s) with the same shape as the input.
Concatenate
See also
XlaBuilder::ConcatInDim
.
Concatenate composes an array from multiple array operands. The array is of the same rank as each of the input array operands (which must be of the same rank as each other) and contains the arguments in the order that they were specified.
Concatenate(operands..., dimension)
Arguments  Type  Semantics 

operands

sequence of N XlaOp

N arrays of type T with dimensions [L0, L1, ...]. Requires N >= 1. 
dimension

int64

A value in the interval [0, N) that
names the dimension to be concatenated
between the operands . 
With the exception of dimension
all dimensions must be the same. This is
because XLA does not support "ragged" arrays. Also note that rank0 values
cannot be concatenated (as it's impossible to name the dimension along which the
concatenation occurs).
1dimensional example:
Concat({ {2, 3}, {4, 5}, {6, 7}}, 0)
>>> {2, 3, 4, 5, 6, 7}
2dimensional example:
let a = {
{1, 2},
{3, 4},
{5, 6},
};
let b = {
{7, 8},
};
Concat({a, b}, 0)
>>> {
{1, 2},
{3, 4},
{5, 6},
{7, 8},
}
Diagram:
Conditional
See also
XlaBuilder::Conditional
.
Conditional(pred, true_operand, true_computation, false_operand,
false_computation)
Arguments  Type  Semantics 

pred 
XlaOp 
Scalar of type PRED 
true_operand 
XlaOp 
Argument of type \( T_0 \) 
true_computation 
XlaComputation 
XlaComputation of type \( T_0 \to S\) 
false_operand 
XlaOp 
Argument of type \( T_1 \) 
false_computation 
XlaComputation 
XlaComputation of type \( T_1 \to S \) 
Executes true_computation
if pred
is true
, false_computation
if pred
is false
, and returns the result.
The true_computation
must take in a single argument of type
true_operand
which must be of the same type. The
false_computation
must take in a single argument of type false_operand
which must be of the same type. The type of the
returned value of true_computation
and false_computation
must be the same.
Note that only one of true_computation
and false_computation
will be
executed depending on the value of pred
.
Conditional(branch_index, branch_computations, branch_operands)
Arguments  Type  Semantics 

branch_index 
XlaOp 
Scalar of type S32 
branch_computations

sequence of N
XlaComputation

XlaComputations of type \(
T_0 \to S , T_1 \to S , ...,
T_{N1} \to S \) 
branch_operands

sequence of N XlaOp

Arguments of type \( T_0 ,
T_1 , ..., T_{N1} \) 
Executes branch_computations[branch_index]
, and returns the result. If
branch_index
is an S32
which is < 0 or >= N, then branch_computations[N1]
is executed as the default branch.
Each branch_computations[b]
must take in a single argument of type T_b
and
will be invoked with branch_operands[b]
which must be of the same type. The
type of the returned value of each branch_computations[b]
must be the same.
Note that only one of the branch_computations
will be executed depending on
the value of branch_index
.
Conv (convolution)
See also
XlaBuilder::Conv
.
As ConvWithGeneralPadding, but the padding is specified in a shorthand way as
either SAME or VALID. SAME padding pads the input (lhs
) with zeroes so that
the output has the same shape as the input when not taking striding into
account. VALID padding simply means no padding.
ConvWithGeneralPadding (convolution)
See also
XlaBuilder::ConvWithGeneralPadding
.
Computes a convolution of the kind used in neural networks. Here, a convolution can be thought of as a ndimensional window moving across a ndimensional base area and a computation is performed for each possible position of the window.
Arguments  Type  Semantics 

lhs 
XlaOp 
rank n+2 array of inputs 
rhs

XlaOp

rank n+2 array of kernel weights 
window_strides

ArraySlice<int64>

nd array of kernel strides 
padding

ArraySlice< pair<int64,
int64>> 
nd array of (low, high) padding 
lhs_dilation

ArraySlice<int64>

nd lhs dilation factor array 
rhs_dilation

ArraySlice<int64>

nd rhs dilation factor array 
feature_group_count

int64  the number of feature groups 
batch_group_count

int64  the number of batch groups 
Let n be the number of spatial dimensions. The lhs
argument is a rank n+2
array describing the base area. This is called the input, even though of course
the rhs is also an input. In a neural network, these are the input activations.
The n+2 dimensions are, in this order:
batch
: Each coordinate in this dimension represents an independent input for which convolution is carried out.z/depth/features
: Each (y,x) position in the base area has a vector associated to it, which goes into this dimension.spatial_dims
: Describes then
spatial dimensions that define the base area that the window moves across.
The rhs
argument is a rank n+2 array describing the convolutional
filter/kernel/window. The dimensions are, in this order:
outputz
: Thez
dimension of the output.inputz
: The size of this dimension timesfeature_group_count
should equal the size of thez
dimension in lhs.spatial_dims
: Describes then
spatial dimensions that define the nd window that moves across the base area.
The window_strides
argument specifies the stride of the convolutional window
in the spatial dimensions. For example, if the stride in the first spatial
dimension is 3, then the window can only be placed at coordinates where the
first spatial index is divisible by 3.
The padding
argument specifies the amount of zero padding to be applied to the
base area. The amount of padding can be negative  the absolute value of
negative padding indicates the number of elements to remove from the specified
dimension before doing the convolution. padding[0]
specifies the padding for
dimension y
and padding[1]
specifies the padding for dimension x
. Each
pair has the low padding as the first element and the high padding as the second
element. The low padding is applied in the direction of lower indices while the
high padding is applied in the direction of higher indices. For example, if
padding[1]
is (2,3)
then there will be a padding by 2 zeroes on the left and
by 3 zeroes on the right in the second spatial dimension. Using padding is
equivalent to inserting those same zero values into the input (lhs
) before
doing the convolution.
The lhs_dilation
and rhs_dilation
arguments specify the dilation factor to
be applied to the lhs and rhs, respectively, in each spatial dimension. If the
dilation factor in a spatial dimension is d, then d1 holes are implicitly
placed between each of the entries in that dimension, increasing the size of the
array. The holes are filled with a noop value, which for convolution means
zeroes.
Dilation of the rhs is also called atrous convolution. For more details, see
tf.nn.atrous_conv2d
. Dilation of the lhs is also called transposed
convolution. For more details, see tf.nn.conv2d_transpose
.
The feature_group_count
argument (default value 1) can be used for grouped
convolutions. feature_group_count
needs to be a divisor of both the input and
the output feature dimension. If feature_group_count
is greater than 1, it
means that conceptually the input and output feature dimension and the rhs
output feature dimension are split evenly into feature_group_count
many
groups, each group consisting of a consecutive subsequence of features. The
input feature dimension of rhs
needs to be equal to the lhs
input feature
dimension divided by feature_group_count
(so it already has the size of a
group of input features). The ith groups are used together to compute
feature_group_count
many separate convolutions. The results of these
convolutions are concatenated together in the output feature dimension.
For depthwise convolution the feature_group_count
argument would be set to the
input feature dimension, and the filter would be reshaped from
[filter_height, filter_width, in_channels, channel_multiplier]
to
[filter_height, filter_width, 1, in_channels * channel_multiplier]
. For more
details, see tf.nn.depthwise_conv2d
.
The batch_group_count
(default value 1) argument can be used for depthwise
filters during backpropagation. batch_group_count
needs to be a divisor of the
size of the lhs
(input) batch dimension. If batch_group_count
is greater
than 1, it means that the output batch dimension should be of size
batch_group_size
where batch_group_size = input batch / batch_group_count
.
For convolutions with batch_group_count
greater than 1, the input batch size
must evenly divide into batch_group_size and output feature size, which implies
that the output feature size must be equal to batch_group_count. Conceptually,
this can be achieved by performing the usual convolution, and then scraping
batch_group_size
number of elements on the diagonal of the matrix formed by
output batch and output feature.
The output shape has these dimensions, in this order:
batch
: The size of this dimension timesbatch_group_count
should equal the size of thebatch
dimension in lhs.z
: Same size asoutputz
on the kernel (rhs
).spatial_dims
: One value for each valid placement of the convolutional window.
The valid placements of the convolutional window are determined by the strides and the size of the base area after padding.
To describe what a convolution does, consider a 2d convolution, and pick some
fixed batch
, z
, y
, x
coordinates in the output. Then (y,x)
is a
position of a corner of the window within the base area (e.g. the upper left
corner, depending on how you interpret the spatial dimensions). We now have a 2d
window, taken from the base area, where each 2d point is associated to a 1d
vector, so we get a 3d box. From the convolutional kernel, since we fixed the
output coordinate z
, we also have a 3d box. The two boxes have the same
dimensions, so we can take the sum of the elementwise products between the two
boxes (similar to a dot product). That is the output value.
Note that if outputz
is e.g., 5, then each position of the window produces 5
values in the output into the z
dimension of the output. These values differ
in what part of the convolutional kernel is used  there is a separate 3d box of
values used for each outputz
coordinate. So you could think of it as 5
separate convolutions with a different filter for each of them.
Here is pseudocode for a 2d convolution with padding and striding:
for (b, oz, oy, ox) { // output coordinates
value = 0;
for (iz, ky, kx) { // kernel coordinates and input z
iy = oy*stride_y + ky  pad_low_y;
ix = ox*stride_x + kx  pad_low_x;
if ((iy, ix) inside the base area considered without padding) {
value += input(b, iz, iy, ix) * kernel(oz, iz, ky, kx);
}
}
output(b, oz, oy, ox) = value;
}
ConvertElementType
See also
XlaBuilder::ConvertElementType
.
Similar to an elementwise static_cast
in C++, performs an elementwise
conversion operation from a data shape to a target shape. The dimensions must
match, and the conversion is an elementwise one; e.g. s32
elements become
f32
elements via an s32
tof32
conversion routine.
ConvertElementType(operand, new_element_type)
Arguments  Type  Semantics 

operand 
XlaOp 
array of type T with dims D 
new_element_type 
PrimitiveType 
type U 
The dimensions of the operand and the target shape must match. The source and destination element types must not be tuples.
A conversion such as T=s32
to U=f32
will perform a normalizing inttofloat
conversion routine such as roundtonearesteven.
let a: s32[3] = {0, 1, 2};
let b: f32[3] = convert(a, f32);
then b == f32[3]{0.0, 1.0, 2.0}
CrossReplicaSum
See also
XlaBuilder::CrossReplicaSum
.
Computes a sum across replicas.
CrossReplicaSum(operand)
Arguments  Type  Semantics 

operand 
XlaOp 
Array to sum across replicas. 
replica_group_ids 
int64 vector 
Group ID for each replica. 
The output shape is the same as the input shape. For example, if there are two
replicas and the operand has the value (1.0, 2.5)
and (3.0, 5.25)
respectively on the two replicas, then the output value from this op will be
(4.0, 7.75)
on both replicas.
replica_group_ids
identifies the group ID of each replica. The group ID must
either be empty (all replicas belong to a single group), or contain the same
number of elements as the number of replicas. For example, if
replica_group_ids
= {0, 1, 2, 3, 0, 1, 2, 3} has eight replicas, there are
four subgroups of replica IDs: {0, 4}, {1, 5}, {2, 6}, and {3, 7}. The size of
each subgroup must be identical, so, for example, using:
replica_group_ids
= {0, 1, 2, 0} for four replicas is invalid.
Computing the result of CrossReplicaSum requires having one input from each replica, so if one replica executes a CrossReplicaSum node more times than another, then the former replica will wait forever. Since the replicas are all running the same program, there are not a lot of ways for that to happen, but it is possible when a while loop's condition depends on data from infeed and the data that is infed causes the while loop to iterate more times on one replica than another.
CustomCall
See also
XlaBuilder::CustomCall
.
Call a userprovided function within a computation.
CustomCall(target_name, args..., shape)
Arguments  Type  Semantics 

target_name

string

Name of the function. A call instruction will be emitted which targets this symbol name. 
args

sequence of N XlaOp s

N arguments of arbitrary type, which will be passed to the function. 
shape 
Shape 
Output shape of the function 
The function signature is the same, regardless of the arity or type of args:
extern "C" void target_name(void* out, void** in);
For example, if CustomCall is used as follows:
let x = f32[2] {1,2};
let y = f32[2x3] { {10, 20, 30}, {40, 50, 60}};
CustomCall("myfunc", {x, y}, f32[3x3])
Here is an example of an implementation of myfunc
:
extern "C" void myfunc(void* out, void** in) {
float (&x)[2] = *static_cast<float(*)[2]>(in[0]);
float (&y)[2][3] = *static_cast<float(*)[2][3]>(in[1]);
EXPECT_EQ(1, x[0]);
EXPECT_EQ(2, x[1]);
EXPECT_EQ(10, y[0][0]);
EXPECT_EQ(20, y[0][1]);
EXPECT_EQ(30, y[0][2]);
EXPECT_EQ(40, y[1][0]);
EXPECT_EQ(50, y[1][1]);
EXPECT_EQ(60, y[1][2]);
float (&z)[3][3] = *static_cast<float(*)[3][3]>(out);
z[0][0] = x[1] + y[1][0];
// ...
}
The userprovided function must not have sideeffects and its execution must be idempotent.
Dot
See also
XlaBuilder::Dot
.
Dot(lhs, rhs)
Arguments  Type  Semantics 

lhs 
XlaOp 
array of type T 
rhs 
XlaOp 
array of type T 
The exact semantics of this operation depend on the ranks of the operands:
Input  Output  Semantics 

vector [n] dot vector
[n] 
scalar  vector dot product 
matrix [m x k] dot
vector [k] 
vector [m]  matrixvector multiplication 
matrix [m x k] dot
matrix [k x n] 
matrix [m x n]  matrixmatrix multiplication 
The operation performs sum of products over the second dimension of lhs
(or
the first if it has rank 1) and the first dimension of rhs
. These are the
"contracted" dimensions. The contracted dimensions of lhs
and rhs
must be of
the same size. In practice, it can be used to perform dot products between
vectors, vector/matrix multiplications or matrix/matrix multiplications.
DotGeneral
See also
XlaBuilder::DotGeneral
.
DotGeneral(lhs, rhs, dimension_numbers)
Arguments  Type  Semantics 

lhs 
XlaOp 
array of type T 
rhs 
XlaOp 
array of type T 
dimension_numbers 
DotDimensionNumbers 
array of type T 
As Dot, but allows contracting and batch dimension numbers to be specified for both the 'lhs' and 'rhs'.
DotDimensionNumbers Fields  Type  Semantics 

'lhs_contracting_dimensions'  repeated int64  'lhs' contracting dimension numbers 
'rhs_contracting_dimensions'  repeated int64  'rhs' contracting dimension numbers 
'lhs_batch_dimensions'  repeated int64  'lhs' batch dimension numbers 
'rhs_batch_dimensions'  repeated int64  'rhs' batch dimension numbers 
DotGeneral performs the sum of products over contracting dimensions specified in 'dimension_numbers'.
Associated contracting dimension numbers from the 'lhs' and 'rhs' do not need to be the same and but must have the same dimension sizes.
Example with contracting dimension numbers:
lhs = { {1.0, 2.0, 3.0},
{4.0, 5.0, 6.0} }
rhs = { {1.0, 1.0, 1.0},
{2.0, 2.0, 2.0} }
DotDimensionNumbers dnums;
dnums.add_lhs_contracting_dimensions(1);
dnums.add_rhs_contracting_dimensions(1);
DotGeneral(lhs, rhs, dnums) > { {6.0, 12.0},
{15.0, 30.0} }
Associated batch dimension numbers from the 'lhs' and 'rhs' must have the same dimension sizes.
Example with batch dimension numbers (batch size 2, 2x2 matrices):
lhs = { { {1.0, 2.0},
{3.0, 4.0} },
{ {5.0, 6.0},
{7.0, 8.0} } }
rhs = { { {1.0, 0.0},
{0.0, 1.0} },
{ {1.0, 0.0},
{0.0, 1.0} } }
DotDimensionNumbers dnums;
dnums.add_lhs_contracting_dimensions(2);
dnums.add_rhs_contracting_dimensions(1);
dnums.add_lhs_batch_dimensions(0);
dnums.add_rhs_batch_dimensions(0);
DotGeneral(lhs, rhs, dnums) > { { {1.0, 2.0},
{3.0, 4.0} },
{ {5.0, 6.0},
{7.0, 8.0} } }
Input  Output  Semantics 

[b0, m, k] dot [b0, k, n] 
[b0, m, n]  batch matmul 
[b0, b1, m, k] dot [b0, b1, k, n] 
[b0, b1, m, n]  batch matmul 
It follows that the resulting dimension number starts with the batch dimension, then the 'lhs' noncontracting/nonbatch dimension, and finally the 'rhs' noncontracting/nonbatch dimension.
DynamicSlice
See also
XlaBuilder::DynamicSlice
.
DynamicSlice extracts a subarray from the input array at dynamic
start_indices
. The size of the slice in each dimension is passed in
size_indices
, which specify the end point of exclusive slice intervals in each
dimension: [start, start + size). The shape of start_indices
must be rank ==
1, with dimension size equal to the rank of operand
.
DynamicSlice(operand, start_indices, size_indices)
Arguments  Type  Semantics 

operand 
XlaOp 
N dimensional array of type T 
start_indices

sequence of N XlaOp

List of N scalar integers containing the starting indices of the slice for each dimension. Value must be greater than or equal to zero. 
size_indices

ArraySlice<int64>

List of N integers containing the slice size for each dimension. Each value must be strictly greater than zero, and start + size must be less than or equal to the size of the dimension to avoid wrapping modulo dimension size. 
The effective slice indices are computed by applying the following
transformation for each index i
in [1, N)
before performing the slice:
start_indices[i] = clamp(start_indices[i], 0, operand.dimension_size[i]  size_indices[i])
This ensures that the extracted slice is always inbounds with respect to the operand array. If the slice is inbounds before the transformation is applied, the transformation has no effect.
1dimensional example:
let a = {0.0, 1.0, 2.0, 3.0, 4.0}
let s = {2}
DynamicSlice(a, s, {2}) produces:
{2.0, 3.0}
2dimensional example:
let b =
{ {0.0, 1.0, 2.0},
{3.0, 4.0, 5.0},
{6.0, 7.0, 8.0},
{9.0, 10.0, 11.0} }
let s = {2, 1}
DynamicSlice(b, s, {2, 2}) produces:
{ { 7.0, 8.0},
{10.0, 11.0} }
DynamicUpdateSlice
See also
XlaBuilder::DynamicUpdateSlice
.
DynamicUpdateSlice generates a result which is the value of the input array
operand
, with a slice update
overwritten at start_indices
.
The shape of update
determines the shape of the subarray of the result which
is updated.
The shape of start_indices
must be rank == 1, with dimension size equal to
the rank of operand
.
DynamicUpdateSlice(operand, update, start_indices)
Arguments  Type  Semantics 

operand 
XlaOp 
N dimensional array of type T 
update

XlaOp

N dimensional array of type T containing the slice update. Each dimension of update shape must be strictly greater than zero, and start + update must be less than or equal to the operand size for each dimension to avoid generating outofbounds update indices. 
start_indices

sequence of N XlaOp

List of N scalar integers containing the starting indices of the slice for each dimension. Value must be greater than or equal to zero. 
The effective slice indices are computed by applying the following
transformation for each index i
in [1, N)
before performing the slice:
start_indices[i] = clamp(start_indices[i], 0, operand.dimension_size[i]  update.dimension_size[i])
This ensures that the updated slice is always inbounds with respect to the operand array. If the slice is inbounds before the transformation is applied, the transformation has no effect.
1dimensional example:
let a = {0.0, 1.0, 2.0, 3.0, 4.0}
let u = {5.0, 6.0}
let s = {2}
DynamicUpdateSlice(a, u, s) produces:
{0.0, 1.0, 5.0, 6.0, 4.0}
2dimensional example:
let b =
{ {0.0, 1.0, 2.0},
{3.0, 4.0, 5.0},
{6.0, 7.0, 8.0},
{9.0, 10.0, 11.0} }
let u =
{ {12.0, 13.0},
{14.0, 15.0},
{16.0, 17.0} }
let s = {1, 1}
DynamicUpdateSlice(b, u, s) produces:
{ {0.0, 1.0, 2.0},
{3.0, 12.0, 13.0},
{6.0, 14.0, 15.0},
{9.0, 16.0, 17.0} }
Elementwise binary arithmetic operations
See also
XlaBuilder::Add
.
A set of elementwise binary arithmetic operations is supported.
Op(lhs, rhs)
Where Op
is one of Add
(addition), Sub
(subtraction), Mul
(multiplication), Div
(division), Rem
(remainder), Max
(maximum), Min
(minimum), LogicalAnd
(logical AND), or LogicalOr
(logical OR).
Arguments  Type  Semantics 

lhs 
XlaOp 
lefthandside operand: array of type T 
rhs 
XlaOp 
righthandside operand: array of type T 
The arguments' shapes have to be either similar or compatible. See the broadcasting documentation about what it means for shapes to be compatible. The result of an operation has a shape which is the result of broadcasting the two input arrays. In this variant, operations between arrays of different ranks are not supported, unless one of the operands is a scalar.
When Op
is Rem
, the sign of the result is taken from the dividend, and the
absolute value of the result is always less than the divisor's absolute value.
Integer division overflow (signed/unsigned division/remainder by zero or signed
division/remainder of INT_SMIN
with 1
) produces an implementation defined
value.
An alternative variant with differentrank broadcasting support exists for these operations:
Op(lhs, rhs, broadcast_dimensions)
Where Op
is the same as above. This variant of the operation should be used
for arithmetic operations between arrays of different ranks (such as adding a
matrix to a vector).
The additional broadcast_dimensions
operand is a slice of integers used to
expand the rank of the lowerrank operand up to the rank of the higherrank
operand. broadcast_dimensions
maps the dimensions of the lowerrank shape to
the dimensions of the higherrank shape. The unmapped dimensions of the expanded
shape are filled with dimensions of size one. Degeneratedimension broadcasting
then broadcasts the shapes along these degenerate dimensions to equalize the
shapes of both operands. The semantics are described in detail on the
broadcasting page.
Elementwise comparison operations
See also
XlaBuilder::Eq
.
A set of standard elementwise binary comparison operations is supported. Note that standard IEEE 754 floatingpoint comparison semantics apply when comparing floatingpoint types.
Op(lhs, rhs)
Where Op
is one of Eq
(equalto), Ne
(not equalto), Ge
(greaterorequalthan), Gt
(greaterthan), Le
(lessorequalthan), Lt
(lessthan).
Arguments  Type  Semantics 

lhs 
XlaOp 
lefthandside operand: array of type T 
rhs 
XlaOp 
righthandside operand: array of type T 
The arguments' shapes have to be either similar or compatible. See the
broadcasting documentation about what it means for shapes to
be compatible. The result of an operation has a shape which is the result of
broadcasting the two input arrays with the element type PRED
. In this variant,
operations between arrays of different ranks are not supported, unless one of
the operands is a scalar.
An alternative variant with differentrank broadcasting support exists for these operations:
Op(lhs, rhs, broadcast_dimensions)
Where Op
is the same as above. This variant of the operation should be used
for comparison operations between arrays of different ranks (such as adding a
matrix to a vector).
The additional broadcast_dimensions
operand is a slice of integers specifying
the dimensions to use for broadcasting the operands. The semantics are described
in detail on the broadcasting page.
Elementwise unary functions
XlaBuilder supports these elementwise unary functions:
Abs(operand)
Elementwise abs x > x
.
Ceil(operand)
Elementwise ceil x > ⌈x⌉
.
Cos(operand)
Elementwise cosine x > cos(x)
.
Exp(operand)
Elementwise natural exponential x > e^x
.
Floor(operand)
Elementwise floor x > ⌊x⌋
.
IsFinite(operand)
Tests whether each element of operand
is finite,
i.e., is not positive or negative infinity, and is not NaN
. Returns an array
of PRED
values with the same shape as the input, where each element is true
if and only if the corresponding input element is finite.
Log(operand)
Elementwise natural logarithm x > ln(x)
.
LogicalNot(operand)
Elementwise logical not x > !(x)
.
PopulationCount(operand)
Computes the number of bits set in each
element of operand
.
Neg(operand)
Elementwise negation x > x
.
Sign(operand)
Elementwise sign operation x > sgn(x)
where
using the comparison operator of the element type of operand
.
Tanh(operand)
Elementwise hyperbolic tangent x > tanh(x)
.
Arguments  Type  Semantics 

operand 
XlaOp 
The operand to the function 
The function is applied to each element in the operand
array, resulting in an
array with the same shape. It is allowed for operand
to be a scalar (rank 0).
Fft
The XLA FFT operation implements the forward and inverse Fourier Transforms for real and complex inputs/outputs. Multidimensional FFTs on up to 3 axes are supported, except on TPU, where only a single axis is supported (please file a github issue if you require higher order).
See also
XlaBuilder::Fft
.
Arguments  Type  Semantics 

operand

XlaOp

The array we are Fourier transforming. 
fft_type 
FftType 
See the table below. 
fft_length

ArraySlice<int64>

The timedomain lengths
of the axes being
transformed. This is
needed in particular for
IRFFT to rightsize the
innermost axis, since
RFFT(fft_length=[16])
has the same output
shape as
RFFT(fft_length=[17]) . 
FftType 
Semantics 

FFT 
Forward complextocomplex FFT. Shape is unchanged. 
IFFT 
Inverse complextocomplex FFT. Shape is unchanged. 
RFFT

Forward realtocomplex FFT. Shape of the innermost axis is
reduced to fft_length[1] // 2 + 1 if fft_length[1] is a
nonzero value, omitting the reversed conjugate part of the
transformed signal beyond the Nyquist frequency. 
IRFFT

Inverse realtocomplex FFT (i.e. takes complex, returns real).
Shape of the innermost axis is expanded to fft_length[1] if
fft_length[1] is a nonzero value, inferring the part of the
transformed signal beyond the Nyquist frequency from the reverse
conjugate of the 1 to fft_length[1] // 2 + 1 entries. 
Multidimensional FFT
When more than 1 fft_length
is provided, this is equivalent to applying a
cascade of FFT operations to each of the innermost axes. Note that for the
real>complex and complex>real cases, the innermost axis transform is
(effectively) performed first (RFFT; last for IRFFT), which is why the innermost
axis is the one which changes size. Other axis transforms will then be
complex>complex.
Implementation details
CPU FFT is backed by Eigen's TensorFFT. GPU FFT uses cuFFT.
Gather
The XLA gather operation stitches together several slices (each slice at a potentially different runtime offset) of an input array.
General Semantics
See also
XlaBuilder::Gather
.
For a more intuitive description, see the "Informal Description" section below.
gather(operand, start_indices, offset_dims, collapsed_slice_dims, slice_sizes, start_index_map)
Arguments  Type  Semantics 

operand

XlaOp

The array we’re gathering from. 
start_indices

XlaOp

Array containing the starting indices of the slices we gather. 
index_vector_dim

int64

The dimension in
start_indices that
"contains" the starting
indices. See below for a
detailed description. 
offset_dims

ArraySlice<int64>

The set of dimensions in the output shape that offset into an array sliced from operand. 
slice_sizes

ArraySlice<int64>

slice_sizes[i] is the
bounds for the slice on
dimension i . 
collapsed_slice_dims

ArraySlice<int64>

The set of dimensions in each slice that are collapsed away. These dimensions must have size 1. 
start_index_map

ArraySlice<int64>

A map that describes how to
map indices in
start_indices to legal
indices into operand. 
indices_are_sorted

bool

Whether the indices are guaranteed to be sorted by the caller. 
For convenience, we label dimensions in the output array not in offset_dims
as batch_dims
.
The output is an array of rank batch_dims.size
+ offset_dims.size
.
The operand.rank
must equal the sum of offset_dims.size
and
collapsed_slice_dims
. Also, slice_sizes.size
has to be equal to
operand.rank
.
If index_vector_dim
is equal to start_indices.rank
we implicitly consider
start_indices
to have a trailing 1
dimension (i.e. if start_indices
was of
shape [6,7]
and index_vector_dim
is 2
then we implicitly consider the
shape of start_indices
to be [6,7,1]
).
The bounds for the output array along dimension i
is computed as follows:
If
i
is present inbatch_dims
(i.e. is equal tobatch_dims[k]
for somek
) then we pick the corresponding dimension bounds out ofstart_indices.shape
, skippingindex_vector_dim
(i.e. pickstart_indices.shape.dims
[k
] ifk
<index_vector_dim
andstart_indices.shape.dims
[k
+1
] otherwise).If
i
is present inoffset_dims
(i.e. equal tooffset_dims
[k
] for somek
) then we pick the corresponding bound out ofslice_sizes
after accounting forcollapsed_slice_dims
(i.e. we pickadjusted_slice_sizes
[k
] whereadjusted_slice_sizes
isslice_sizes
with the bounds at indicescollapsed_slice_dims
removed).
Formally, the operand index In
corresponding to a given output index Out
is
calculated as follows:
Let
G
= {Out
[k
] fork
inbatch_dims
}. UseG
to slice out a vectorS
such thatS
[i
] =start_indices
[Combine(G
,i
)] where Combine(A, b) inserts b at positionindex_vector_dim
into A. Note that this is well defined even ifG
is empty  ifG
is empty thenS
=start_indices
.Create a starting index,
S
_{in}, intooperand
usingS
by scatteringS
usingstart_index_map
. More precisely:S
_{in}[start_index_map
[k
]] =S
[k
] ifk
<start_index_map.size
.S
_{in}[_
] =0
otherwise.
Create an index
O
_{in} intooperand
by scattering the indices at the offset dimensions inOut
according to thecollapsed_slice_dims
set. More precisely:O
_{in}[remapped_offset_dims
(k
)] =Out
[offset_dims
[k
]] ifk
<offset_dims.size
(remapped_offset_dims
is defined below).O
_{in}[_
] =0
otherwise.
In
isO
_{in} +S
_{in} where + is elementwise addition.
remapped_offset_dims
is a monotonic function with domain [0
, offset.size
)
and range [0
, operand.rank
) \ collapsed_slice_dims
. So if, e.g.,
offset.size
is 4
, operand.rank
is 6
and collapsed_slice_dims
is {0
,
2
} then remapped_offset_dims
is {0
→1
, 1
→3
, 2
→4
, 3
→5
}.
If indices_are_sorted
is set to true then XLA can assume that start_indices
are sorted (in ascending start_index_map
order) by the user. If they are not
then the semantics is implementation defined.
Informal Description and Examples
Informally, every index Out
in the output array corresponds to an element E
in the operand array, computed as follows:
We use the batch dimensions in
Out
to look up a starting index fromstart_indices
.We use
start_index_map
to map the starting index (whose size may be less than operand.rank) to a "full" starting index into theoperand
.We dynamicslice out a slice with size
slice_sizes
using the full starting index.We reshape the slice by collapsing the
collapsed_slice_dims
dimensions. Since all collapsed slice dimensions must have a bound of 1, this reshape is always legal.We use the offset dimensions in
Out
to index into this slice to get the input element,E
, corresponding to output indexOut
.
index_vector_dim
is set to start_indices.rank
 1
in all of the examples
that follow. More interesting values for index_vector_dim
do not change the
operation fundamentally, but make the visual representation more cumbersome.
To get an intuition on how all of the above fits together, let's look at an
example that gathers 5 slices of shape [8,6]
from a [16,11]
array. The
position of a slice into the [16,11]
array can be represented as an index
vector of shape S64[2]
, so the set of 5 positions can be represented as a
S64[5,2]
array.
The behavior of the gather operation can then be depicted as an index
transformation that takes [G
,O
_{0},O
_{1}], an index in
the output shape, and maps it to an element in the input array in the following
way:
We first select an (X
,Y
) vector from the gather indices array using G
.
The element in the output array at index
[G
,O
_{0},O
_{1}] is then the element in the input
array at index [X
+O
_{0},Y
+O
_{1}].
slice_sizes
is [8,6]
, which decides the range of O_{0} and
O_{1}, and this in turn decides the bounds of the slice.
This gather operation acts as a batch dynamic slice with G
as the batch
dimension.
The gather indices may be multidimensional. For instance, a more general
version of the example above using a "gather indices" array of shape [4,5,2]
would translate indices like this:
Again, this acts as a batch dynamic slice G
_{0} and
G
_{1} as the batch dimensions. The slice size is still [8,6]
.
The gather operation in XLA generalizes the informal semantics outlined above in the following ways:
We can configure which dimensions in the output shape are the offset dimensions (dimensions containing
O
_{0},O
_{1} in the last example). The output batch dimensions (dimensions containingG
_{0},G
_{1} in the last example) are defined to be the output dimensions that are not offset dimensions.The number of output offset dimensions explicitly present in the output shape may be smaller than the input rank. These "missing" dimensions, which are listed explicitly as
collapsed_slice_dims
, must have a slice size of1
. Since they have a slice size of1
the only valid index for them is0
and eliding them does not introduce ambiguity.The slice extracted from the "Gather Indices" array ((
X
,Y
) in the last example) may have fewer elements than the input array rank, and an explicit mapping dictates how the index should be expanded to have the same rank as the input.
As a final example, we use (2) and (3) to implement tf.gather_nd
:
G
_{0} and G
_{1} are used to slice out a starting index
from the gather indices array as usual, except the starting index has only one
element, X
. Similarly, there is only one output offset index with the value
O
_{0}. However, before being used as indices into the input array,
these are expanded in accordance to "Gather Index Mapping" (start_index_map
in
the formal description) and "Offset Mapping" (remapped_offset_dims
in the
formal description) into [X
,0
] and [0
,O
_{0}] respectively,
adding up to [X
,O
_{0}]. In other words, the output index
[G
_{0},G
_{1},O
_{0}] maps to the input index
[GatherIndices
[G
_{0},G
_{1},0
],X
] which gives us
the semantics for tf.gather_nd
.
slice_sizes
for this case is [1,11]
. Intuitively this means that every
index X
in the gather indices array picks an entire row and the result is the
concatenation of all these rows.
GetDimensionSize
See also
XlaBuilder::GetDimensionSize
.
Returns the size of the given dimension of the operand. The operand must be array shaped.
GetDimensionSize(operand, dimension)
Arguments  Type  Semantics 

operand 
XlaOp 
n dimensional input array 
dimension

int64

A value in the interval [0, n) that specifies the
dimension 
GetTupleElement
See also
XlaBuilder::GetTupleElement
.
Indexes into a tuple with a compiletimeconstant value.
The value must be a compiletimeconstant so that shape inference can determine the type of the resulting value.
This is analogous to std::get<int N>(t)
in C++. Conceptually:
let v: f32[10] = f32[10]{0, 1, 2, 3, 4, 5, 6, 7, 8, 9};
let s: s32 = 5;
let t: (f32[10], s32) = tuple(v, s);
let element_1: s32 = gettupleelement(t, 1); // Inferred shape matches s32.
See also tf.tuple
.
Infeed
See also
XlaBuilder::Infeed
.
Infeed(shape)
Argument  Type  Semantics 

shape

Shape

Shape of the data read from the Infeed interface. The layout field of the shape must be set to match the layout of the data sent to the device; otherwise its behavior is undefined. 
Reads a single data item from the implicit Infeed streaming interface of the
device, interpreting the data as the given shape and its layout, and returns a
XlaOp
of the data. Multiple Infeed operations are allowed in a
computation, but there must be a total order among the Infeed operations. For
example, two Infeeds in the code below have a total order since there is a
dependency between the while loops.
result1 = while (condition, init = init_value) {
Infeed(shape)
}
result2 = while (condition, init = result1) {
Infeed(shape)
}
Nested tuple shapes are not supported. For an empty tuple shape, the Infeed operation is effectively a noop and proceeds without reading any data from the Infeed of the device.
Iota
Iota()
Builds a constant literal on device rather than a potentially large host
transfer. Creates a rank 1 array of values starting at zero and incrementing by
one. For floatingpoint types, the produced array is equivalent to
ConvertElementType(Iota(...))
where the Iota
is of integral type and the
conversion is to the floatingpoint type.
Arguments  Type  Semantics 

type 
PrimitiveType 
type U 
size 
int64 
The number of elements in the array. 
iota_dimension 
int64 
The dimension to increment along. 
Map
See also
XlaBuilder::Map
.
Map(operands..., computation)
Arguments  Type  Semantics 

operands 
sequence of N XlaOp s 
N arrays of types T0..T{N1} 
computation

XlaComputation

computation of type T_0, T_1,
..., T_{N + M 1} > S with N
parameters of type T and M of
arbitrary type 
dimensions 
int64 array 
array of map dimensions 
Applies a scalar function over the given operands
arrays, producing an array
of the same dimensions where each element is the result of the mapped function
applied to the corresponding elements in the input arrays.
The mapped function is an arbitrary computation with the restriction that it has
N inputs of scalar type T
and a single output with type S
. The output has
the same dimensions as the operands except that the element type T is replaced
with S.
For example: Map(op1, op2, op3, computation, par1)
maps elem_out <
computation(elem1, elem2, elem3, par1)
at each (multidimensional) index in the
input arrays to produce the output array.
Pad
See also
XlaBuilder::Pad
.
Pad(operand, padding_value, padding_config)
Arguments  Type  Semantics 

operand 
XlaOp 
array of type T 
padding_value

XlaOp

scalar of type T to fill in the added
padding 
padding_config

PaddingConfig

padding amount on both edges (low, high) and between the elements of each dimension 
Expands the given operand
array by padding around the array as well as between
the elements of the array with the given padding_value
. padding_config
specifies the amount of edge padding and the interior padding for each
dimension.
PaddingConfig
is a repeated field of PaddingConfigDimension
, which contains
three fields for each dimension: edge_padding_low
, edge_padding_high
, and
interior_padding
.
edge_padding_low
and edge_padding_high
specify the amount of padding added
at the lowend (next to index 0) and the highend (next to the highest index) of
each dimension respectively. The amount of edge padding can be negative  the
absolute value of negative padding indicates the number of elements to remove
from the specified dimension.
interior_padding
specifies the amount of padding added between any two
elements in each dimension; it may not be negative. Interior padding occurs
logically before edge padding, so in the case of negative edge padding, elements
are removed from the interiorpadded operand.
This operation is a noop if the edge padding pairs are all (0, 0) and the
interior padding values are all 0. The figure below shows examples of different
edge_padding
and interior_padding
values for a twodimensional array.
Recv
See also
XlaBuilder::Recv
.
Recv(shape, channel_handle)
Arguments  Type  Semantics 

shape 
Shape 
shape of the data to receive 
channel_handle 
ChannelHandle 
unique identifier for each send/recv pair 
Receives data of the given shape from a Send
instruction in another
computation that shares the same channel handle. Returns a
XlaOp for the received data.
The client API of Recv
operation represents synchronous communication.
However, the instruction is internally decomposed into 2 HLO instructions
(Recv
and RecvDone
) to enable asynchronous data transfers. See also
HloInstruction::CreateRecv
and HloInstruction::CreateRecvDone
.
Recv(const Shape& shape, int64 channel_id)
Allocates resources required to receive data from a Send
instruction with the
same channel_id. Returns a context for the allocated resources, which is used
by a following RecvDone
instruction to wait for the completion of the data
transfer. The context is a tuple of {receive buffer (shape), request identifier
(U32)} and it can only be used by a RecvDone
instruction.
RecvDone(HloInstruction context)
Given a context created by a Recv
instruction, waits for the data transfer to
complete and returns the received data.
Reduce
See also
XlaBuilder::Reduce
.
Applies a reduction function to one or more arrays in parallel.
Reduce(operands..., init_values..., computation, dimensions)
Arguments  Type  Semantics 

operands 
Sequence of N XlaOp 
N arrays of types T_0, ..., T_N . 
init_values 
Sequence of N XlaOp 
N scalars of types T_0, ..., T_N . 
computation 
XlaComputation 
computation of type T_0, ..., T_N, T_0, ..., T_N > Collate(T_0, ..., T_N) . 
dimensions 
int64 array 
unordered array of dimensions to reduce. 
Where:
 N is required to be greater or equal to 1.
 All input arrays must have the same dimensions.
 If
N = 1
,Collate(T)
isT
.  If
N > 1
,Collate(T_0, ..., T_N)
is a tuple ofN
elements of typeT
.
The output of the op is Collate(Q_0, ..., Q_N)
where Q_i
is an array of type
T_i
, the dimensions of which are described below.
This operation reduces one or more dimensions of each input array into scalars.
The rank of each returned array is rank(operand)  len(dimensions)
. The
initial value used for every reduction is init_value
, and it may be inserted
anywhere during computation by the backend. In most cases, init_value
is an
identity of the reduction function (for example, 0
for addition). The applied
computation
is always passed the init_value
on the lefthand side.
The evaluation order of the reduction function is arbitrary and may be nondeterministic. Therefore, the reduction function should not be overly sensitive to reassociation.
Some reduction functions like addition are not strictly associative for floats. However, if the range of the data is limited, floatingpoint addition is close enough to being associative for most practical uses. It is possible to conceive of some completely nonassociative reductions, however, and these will produce incorrect or unpredictable results in XLA.
As an example, when reducing across one dimension in a single 1D array with
values [10, 11, 12, 13]
, with reduction function f
(this is computation
)
then that could be computed as
f(10, f(11, f(12, f(init_value, 13)))
but there are also many other possibilities, e.g.
f(init_value, f(f(10, f(init_value, 11)), f(f(init_value, 12), f(init_value, 13))))
The following is a rough pseudocode example of how reduction could be implemented, using summation as the reduction computation with an initial value of 0.
result_shape < remove all dims in dimensions from operand_shape
# Iterate over all elements in result_shape. The number of r's here is equal
# to the rank of the result
for r0 in range(result_shape[0]), r1 in range(result_shape[1]), ...:
# Initialize this result element
result[r0, r1...] < 0
# Iterate over all the reduction dimensions
for d0 in range(dimensions[0]), d1 in range(dimensions[1]), ...:
# Increment the result element with the value of the operand's element.
# The index of the operand's element is constructed from all ri's and di's
# in the right order (by construction ri's and di's together index over the
# whole operand shape).
result[r0, r1...] += operand[ri... di]
Here's an example of reducing a 2D array (matrix). The shape has rank 2, dimension 0 of size 2 and dimension 1 of size 3:
Results of reducing dimensions 0 or 1 with an "add" function:
Note that both reduction results are 1D arrays. The diagram shows one as column and another as row just for visual convenience.
For a more complex example, here is a 3D array. Its rank is 3, dimension 0 of size 4, dimension 1 of size 2 and dimension 2 of size 3. For simplicity, the values 1 to 6 are replicated across dimension 0.
Similarly to the 2D example, we can reduce just one dimension. If we reduce dimension 0, for example, we get a rank2 array where all values across dimension 0 were folded into a scalar:
 4 8 12 
 16 20 24 
If we reduce dimension 2, we also get a rank2 array where all values across dimension 2 were folded into a scalar:
 6 15 
 6 15 
 6 15 
 6 15 
Note that the relative order between the remaining dimensions in the input is preserved in the output, but some dimensions may get assigned new numbers (since the rank changes).
We can also reduce multiple dimensions. Addreducing dimensions 0 and 1 produces
the 1D array [20, 28, 36]
.
Reducing the 3D array over all its dimensions produces the scalar 84
.
Variadic Reduce
When N > 1
, reduce function application is slightly more complex, as it is
applied simultaneously to all inputs. The operands are supplied to the
computation in the following order:
 Running reduced value for the first operand
 ...
 Running reduced value for the N'th operand
 Input value for the first operand
 ...
 Input value for the N'th operand
For example, consider the following reduction function, which can be used to compute the max and the argmax of a 1D array in parallel:
f: (Float, Int, Float, Int) > Float, Int
f(max, argmax, value, index):
if value >= max:
return (value, index)
else:
return (max, argmax)
For 1D Input arrays V = Float[N], K = Int[N]
, and init values
I_V = Float, I_K = Int
, the result f_(N1)
of reducing across the only
input dimension is equivalent to the following recursive application:
f_0 = f(I_V, I_K, V_0, K_0)
f_1 = f(f_0.first, f_0.second, V_1, K_1)
...
f_(N1) = f(f_(N2).first, f_(N2).second, V_(N1), K_(N1))
Applying this reduction to an array of values, and an array of sequential indices (i.e. iota), will coiterate over the arrays, and return a tuple containing the maximal value and the matching index.
ReducePrecision
See also
XlaBuilder::ReducePrecision
.
Models the effect of converting floatingpoint values to a lowerprecision format (such as IEEEFP16) and back to the original format. The number of exponent and mantissa bits in the lowerprecision format can be specified arbitrarily, although all bit sizes may not be supported on all hardware implementations.
ReducePrecision(operand, mantissa_bits, exponent_bits)
Arguments  Type  Semantics 

operand 
XlaOp 
array of floatingpoint type T . 
exponent_bits 
int32 
number of exponent bits in lowerprecision format 
mantissa_bits 
int32 
number of mantissa bits in lowerprecision format 
The result is an array of type T
. The input values are rounded to the nearest
value representable with the given number of mantissa bits (using "ties to even"
semantics), and any values that exceed the range specified by the number of
exponent bits are clamped to positive or negative infinity. NaN
values are
retained, although they may be converted to canonical NaN
values.
The lowerprecision format must have at least one exponent bit (in order to
distinguish a zero value from an infinity, since both have a zero mantissa), and
must have a nonnegative number of mantissa bits. The number of exponent or
mantissa bits may exceed the corresponding value for type T
; the corresponding
portion of the conversion is then simply a noop.
ReduceWindow
See also
XlaBuilder::ReduceWindow
.
Applies a reduction function to all elements in each window of the input
multidimensional array, producing an output multidimensional array with the
same number of elements as the number of valid positions of the window. A
pooling layer can be expressed as a ReduceWindow
. Similar to
Reduce
, the applied computation
is always passed the init_value
on the lefthand side.
ReduceWindow(operand, init_value, computation, window_dimensions,
window_strides, padding)
Arguments  Type  Semantics 

operand

XlaOp

N dimensional array containing elements of type T. This is the base area on which the window is placed. 
init_value

XlaOp

Starting value for the reduction. See Reduce for details. 
computation

XlaComputation

Reduction function of type T, T
> T , to apply to all elements
in each window 
window_dimensions

ArraySlice<int64>

array of integers for window dimension values 
window_strides

ArraySlice<int64>

array of integers for window stride values 
base_dilations

ArraySlice<int64>

array of integers for base dilation values 
window_dilations

ArraySlice<int64>

array of integers for window dilation values 
padding

Padding

padding type for window (Padding::kSame, which pads so as to have the same output shape as input if the stride is 1, or Padding::kValid, which uses no no padding and "stops" the window once it no longer fits) 
Below code and figure shows an example of using ReduceWindow
. Input is a
matrix of size [4x6] and both window_dimensions and window_stride_dimensions are
[2x3].
// Create a computation for the reduction (maximum).
XlaComputation max;
{
XlaBuilder builder(client_, "max");
auto y = builder.Parameter(0, ShapeUtil::MakeShape(F32, {}), "y");
auto x = builder.Parameter(1, ShapeUtil::MakeShape(F32, {}), "x");
builder.Max(y, x);
max = builder.Build().ConsumeValueOrDie();
}
// Create a ReduceWindow computation with the max reduction computation.
XlaBuilder builder(client_, "reduce_window_2x3");
auto shape = ShapeUtil::MakeShape(F32, {4, 6});
auto input = builder.Parameter(0, shape, "input");
builder.ReduceWindow(
input,
/*init_val=*/builder.ConstantLiteral(LiteralUtil::MinValue(F32)),
*max,
/*window_dimensions=*/{2, 3},
/*window_stride_dimensions=*/{2, 3},
Padding::kValid);
Stride of 1 in a dimension specifies that the position of a window in the dimension is 1 element away from its adjacent window. In order to specify that no windows overlap with each other, window_stride_dimensions should be equal to window_dimensions. The figure below illustrates the use of two different stride values. Padding is applied to each dimension of the input and the calculations are the same as though the input came in with the dimensions it has after padding.
For a nontrivial padding example, consider computing reducewindow minimum
(initial value is MAX_FLOAT
) with dimension 3
and stride 2
over the input
array [10000, 1000, 100, 10, 1]
. Padding kValid
computes minimums over two
valid windows: [10000, 1000, 100]
and [100, 10, 1]
, resulting in the
output [100, 1]
. Padding kSame
first pads the array so that the shape after
the reducewindow would be the same as input for stride one by adding initial
elements on both sides, getting [MAX_VALUE, 10000, 1000, 100, 10, 1,
MAX_VALUE]
. Running reducewindow over the padded array operates on three
windows [MAX_VALUE, 10000, 1000]
, [1000, 100, 10]
, [10, 1, MAX_VALUE]
, and
yields [1000, 10, 1]
.
The evaluation order of the reduction function is arbitrary and may be
nondeterministic. Therefore, the reduction function should not be overly
sensitive to reassociation. See the discussion about associativity in the
context of Reduce
for more details.
ReplicaId
See also
XlaBuilder::ReplicaId
.
Returns the unique ID (U32 scalar) of the replica.
ReplicaId()
The unique ID of each replica is an unsigned integer in the interval [0, N)
,
where N
is the number of replicas. Since all the replicas are running the same
program, a ReplicaId()
call in the program will return a different value on
each replica.
Reshape
See also
XlaBuilder::Reshape
and the Collapse
operation.
Reshapes the dimensions of an array into a new configuration.
Reshape(operand, new_sizes)
Reshape(operand, dimensions, new_sizes)
Arguments  Type  Semantics 

operand 
XlaOp 
array of type T 
dimensions 
int64 vector 
order in which dimensions are collapsed 
new_sizes 
int64 vector 
vector of sizes of new dimensions 
Conceptually, reshape first flattens an array into a onedimensional vector of
data values, and then refines this vector into a new shape. The input arguments
are an arbitrary array of type T, a compiletimeconstant vector of dimension
indices, and a compiletimeconstant vector of dimension sizes for the result.
The values in the dimension
vector, if given, must be a permutation of all of
T's dimensions; the default if not given is {0, ..., rank  1}
. The order of
the dimensions in dimensions
is from slowestvarying dimension (most major) to
fastestvarying dimension (most minor) in the loop nest which collapses the
input array into a single dimension. The new_sizes
vector determines the size
of the output array. The value at index 0 in new_sizes
is the size of
dimension 0, the value at index 1 is the size of dimension 1, and so on. The
product of the new_size
dimensions must equal the product of the operand's
dimension sizes. When refining the collapsed array into the multidimensional
array defined by new_sizes
, the dimensions in new_sizes
are ordered from
slowest varying (most major) and to fastest varying (most minor).
For example, let v be an array of 24 elements:
let v = f32[4x2x3] { { {10, 11, 12}, {15, 16, 17}},
{ {20, 21, 22}, {25, 26, 27}},
{ {30, 31, 32}, {35, 36, 37}},
{ {40, 41, 42}, {45, 46, 47}}};
Inorder collapse:
let v012_24 = Reshape(v, {0,1,2}, {24});
then v012_24 == f32[24] {10, 11, 12, 15, 16, 17, 20, 21, 22, 25, 26, 27,
30, 31, 32, 35, 36, 37, 40, 41, 42, 45, 46, 47};
let v012_83 = Reshape(v, {0,1,2}, {8,3});
then v012_83 == f32[8x3] { {10, 11, 12}, {15, 16, 17},
{20, 21, 22}, {25, 26, 27},
{30, 31, 32}, {35, 36, 37},
{40, 41, 42}, {45, 46, 47}};
Outoforder collapse:
let v021_24 = Reshape(v, {1,2,0}, {24});
then v012_24 == f32[24] {10, 20, 30, 40, 11, 21, 31, 41, 12, 22, 32, 42,
15, 25, 35, 45, 16, 26, 36, 46, 17, 27, 37, 47};
let v021_83 = Reshape(v, {1,2,0}, {8,3});
then v021_83 == f32[8x3] { {10, 20, 30}, {40, 11, 21},
{31, 41, 12}, {22, 32, 42},
{15, 25, 35}, {45, 16, 26},
{36, 46, 17}, {27, 37, 47}};
let v021_262 = Reshape(v, {1,2,0}, {2,6,2});
then v021_262 == f32[2x6x2] { { {10, 20}, {30, 40},
{11, 21}, {31, 41},
{12, 22}, {32, 42}},
{ {15, 25}, {35, 45},
{16, 26}, {36, 46},
{17, 27}, {37, 47}}};
As a special case, reshape can transform a singleelement array to a scalar and vice versa. For example,
Reshape(f32[1x1] { {5}}, {0,1}, {}) == 5;
Reshape(5, {}, {1,1}) == f32[1x1] { {5}};
Rev (reverse)
See also
XlaBuilder::Rev
.
Rev(operand, dimensions)
Arguments  Type  Semantics 

operand 
XlaOp 
array of type T 
dimensions 
ArraySlice<int64> 
dimensions to reverse 
Reverses the order of elements in the operand
array along the specified
dimensions
, generating an output array of the same shape. Each element of the
operand array at a multidimensional index is stored into the output array at a
transformed index. The multidimensional index is transformed by reversing the
index in each dimension to be reversed (i.e., if a dimension of size N is one of
the reversing dimensions, its index i is transformed into N  1  i).
One use for the Rev
operation is to reverse the convolution weight array along
the two window dimensions during the gradient computation in neural networks.
RngNormal
See also
XlaBuilder::RngNormal
.
Constructs an output of a given shape with random numbers generated following the
RngNormal(mu, sigma, shape)
Arguments  Type  Semantics 

mu

XlaOp

Scalar of type T specifying mean of generated numbers 
sigma

XlaOp

Scalar of type T specifying standard deviation of generated numbers 
shape 
Shape 
Output shape of type T 
RngUniform
See also
XlaBuilder::RngUniform
.
Constructs an output of a given shape with random numbers generated following the uniform distribution over the interval
RngUniform(a, b, shape)
Arguments  Type  Semantics 

a

XlaOp

Scalar of type T specifying lower limit of interval 
b

XlaOp

Scalar of type T specifying upper limit of interval 
shape 
Shape 
Output shape of type T 
Scatter
The XLA scatter operation generates a result which is the value of the input
array operand
, with several slices (at indices specified by scatter_indices
)
updated with the values in updates
using update_computation
.
See also
XlaBuilder::Scatter
.
scatter(operand, scatter_indices, updates, update_computation, index_vector_dim, update_window_dims, inserted_window_dims, scatter_dims_to_operand_dims)
Arguments  Type  Semantics 

operand 
XlaOp 
Array to be scattered into. 
scatter_indices 
XlaOp 
Array containing the starting indices of the slices that must be scattered to. 
updates 
XlaOp 
Array containing the values that must be used for scattering. 
update_computation 
XlaComputation 
Computation to be used for combining the existing values in the input array and the updates during scatter. This computation should be of type (T, T) > T . 
index_vector_dim 
int64 
The dimension in scatter_indices that contains the starting indices. 
update_window_dims 
ArraySlice<int64> 
The set of dimensions in updates shape that are window dimensions. 
inserted_window_dims 
ArraySlice<int64> 
The set of window dimensions that must be inserted into updates shape. 
scatter_dims_to_operand_dims 
ArraySlice<int64> 
A dimensions map from the scatter indices to the operand index space. This array is interpreted as mapping i to scatter_dims_to_operand_dims[i] . It has to be onetoone and total. 
indices_are_sorted 
bool 
Whether the indices are guaranteed to be sorted by the caller. 
If index_vector_dim
is equal to scatter_indices.rank
we implicitly consider
scatter_indices
to have a trailing 1
dimension.
We define update_scatter_dims
of type ArraySlice<int64>
as the set of
dimensions in updates
shape that are not in update_window_dims
, in ascending
order.
The arguments of scatter should follow these constraints:
updates
array must be of rankupdate_window_dims.size + scatter_indices.rank  1
.Bounds of dimension
i
inupdates
must conform to the following: If
i
is present inupdate_window_dims
(i.e. equal toupdate_window_dims
[k
] for somek
), then the bound of dimensioni
inupdates
must not exceed the corresponding bound ofoperand
after accounting for theinserted_window_dims
(i.e.adjusted_window_bounds
[k
], whereadjusted_window_bounds
contains the bounds ofoperand
with the bounds at indicesinserted_window_dims
removed).  If
i
is present inupdate_scatter_dims
(i.e. equal toupdate_scatter_dims
[k
] for somek
), then the bound of dimensioni
inupdates
must be equal to the corresponding bound ofscatter_indices
, skippingindex_vector_dim
(i.e.scatter_indices.shape.dims
[k
], ifk
<index_vector_dim
andscatter_indices.shape.dims
[k+1
] otherwise).
 If
update_window_dims
must be in ascending order, not have any repeating dimension numbers, and be in the range[0, updates.rank)
.inserted_window_dims
must be in ascending order, not have any repeating dimension numbers, and be in the range[0, operand.rank)
.operand.rank
must equal the sum ofupdate_window_dims.size
andinserted_window_dims.size
.scatter_dims_to_operand_dims.size
must be equal toscatter_indices
[index_vector_dim
], and its values must be in the range[0, operand.rank)
.
For a given index U
in the updates
array, the corresponding index I
in the
operand
array into which this update has to be applied is computed as follows:
 Let
G
= {U
[k
] fork
inupdate_scatter_dims
}. UseG
to look up an index vectorS
in thescatter_indices
array such thatS
[i
] =scatter_indices
[Combine(G
,i
)] where Combine(A, b) inserts b at positionsindex_vector_dim
into A.  Create an index
S
_{in} intooperand
usingS
by scatteringS
using thescatter_dims_to_operand_dims
map. More formally:S
_{in}[scatter_dims_to_operand_dims
[k
]] =S
[k
] ifk
<scatter_dims_to_operand_dims.size
.S
_{in}[_
] =0
otherwise.
 Create an index
W
_{in} intooperand
by scattering the indices atupdate_window_dims
inU
according toinserted_window_dims
. More formally:W
_{in}[window_dims_to_operand_dims
(k
)] =U
[k
] ifk
is inupdate_window_dims
, wherewindow_dims_to_operand_dims
is the monotonic function with domain [0
,update_window_dims.size
) and range [0
,operand.rank
) \inserted_window_dims
. (For example, ifupdate_window_dims.size
is4
,operand.rank
is6
, andinserted_window_dims
is {0
,2
} thenwindow_dims_to_operand_dims
is {0
→1
,1
→3
,2
→4
,3
→5
}).W
_{in}[_
] =0
otherwise.
I
isW
_{in} +S
_{in} where + is elementwise addition.
In summary, the scatter operation can be defined as follows.
 Initialize
output
withoperand
, i.e. for all indicesO
in theoperand
array:
output
[O
] =operand
[O
]  For every index
U
in theupdates
array and the corresponding indexO
in theoperand
array, ifO
is a valid index foroutput
:
output
[O
] =update_computation
(output
[O
],updates
[U
])
The order in which updates are applied is nondeterministic. So, when multiple
indices in updates
refer to the same index in operand
, the corresponding
value in output
will be nondeterministic.
Note that the first parameter that is passed into the update_computation
will
always be the current value from the output
array and the second parameter
will always be the value from the updates
array. This is important
specifically for cases when the update_computation
is not commutative.
If indices_are_sorted
is set to true then XLA can assume that start_indices
are sorted (in ascending start_index_map
order) by the user. If they are not
then the semantics is implementation defined.
Informally, the scatter op can be viewed as an inverse of the gather op, i.e. the scatter op updates the elements in the input that are extracted by the corresponding gather op.
For a detailed informal description and examples, refer to the
"Informal Description" section under Gather
.
Select
See also
XlaBuilder::Select
.
Constructs an output array from elements of two input arrays, based on the values of a predicate array.
Select(pred, on_true, on_false)
Arguments  Type  Semantics 

pred 
XlaOp 
array of type PRED 
on_true 
XlaOp 
array of type T 
on_false 
XlaOp 
array of type T 
The arrays on_true
and on_false
must have the same shape. This is also the
shape of the output array. The array pred
must have the same dimensionality as
on_true
and on_false
, with the PRED
element type.
For each element P
of pred
, the corresponding element of the output array is
taken from on_true
if the value of P
is true
, and from on_false
if the
value of P
is false
. As a restricted form of broadcasting,
pred
can be a scalar of type PRED
. In this case, the output array is taken
wholly from on_true
if pred
is true
, and from on_false
if pred
is false
.
Example with nonscalar pred
:
let pred: PRED[4] = {true, false, false, true};
let v1: s32[4] = {1, 2, 3, 4};
let v2: s32[4] = {100, 200, 300, 400};
==>
Select(pred, v1, v2) = s32[4]{1, 200, 300, 4};
Example with scalar pred
:
let pred: PRED = true;
let v1: s32[4] = {1, 2, 3, 4};
let v2: s32[4] = {100, 200, 300, 400};
==>
Select(pred, v1, v2) = s32[4]{1, 2, 3, 4};
Selections between tuples are supported. Tuples are considered to be scalar
types for this purpose. If on_true
and on_false
are tuples (which must have
the same shape!) then pred
has to be a scalar of type PRED
.
SelectAndScatter
See also
XlaBuilder::SelectAndScatter
.
This operation can be considered as a composite operation that first computes
ReduceWindow
on the operand
array to select an element from each window, and
then scatters the source
array to the indices of the selected elements to
construct an output array with the same shape as the operand array. The binary
select
function is used to select an element from each window by applying it
across each window, and it is called with the property that the first
parameter's index vector is lexicographically less than the second parameter's
index vector. The select
function returns true
if the first parameter is
selected and returns false
if the second parameter is selected, and the
function must hold transitivity (i.e., if select(a, b)
and select(b, c)
are
true
, then select(a, c)
is also true
) so that the selected element does
not depend on the order of the elements traversed for a given window.
The function scatter
is applied at each selected index in the output array. It
takes two scalar parameters:
 Current value at the selected index in the output array
 The scatter value from
source
that applies to the selected index
It combines the two parameters and returns a scalar value that's used to update
the value at the selected index in the output array. Initially, all indices of
the output array are set to init_value
.
The output array has the same shape as the operand
array and the source
array must have the same shape as the result of applying a ReduceWindow
operation on the operand
array. SelectAndScatter
can be used to
backpropagate the gradient values for a pooling layer in a neural network.
SelectAndScatter(operand, select, window_dimensions, window_strides,
padding, source, init_value, scatter)
Arguments  Type  Semantics 

operand

XlaOp

array of type T over which the windows slide 
select

XlaComputation

binary computation of type T, T
> PRED , to apply to all
elements in each window; returns
true if the first parameter is
selected and returns false if
the second parameter is selected 
window_dimensions

ArraySlice<int64>

array of integers for window dimension values 
window_strides

ArraySlice<int64>

array of integers for window stride values 
padding

Padding

padding type for window (Padding::kSame or Padding::kValid) 
source

XlaOp

array of type T with the values to scatter 
init_value

XlaOp

scalar value of type T for the initial value of the output array 
scatter

XlaComputation

binary computation of type T, T
> T , to apply each scatter
source element with its
destination element 
The figure below shows examples of using SelectAndScatter
, with the select
function computing the maximal value among its parameters. Note that when the
windows overlap, as in the figure (2) below, an index of the operand
array may
be selected multiple times by different windows. In the figure, the element of
value 9 is selected by both of the top windows (blue and red) and the binary
addition scatter
function produces the output element of value 8 (2 + 6).
The evaluation order of the scatter
function is arbitrary and may be
nondeterministic. Therefore, the scatter
function should not be overly
sensitive to reassociation. See the discussion about associativity in the
context of Reduce
for more details.
Send
See also
XlaBuilder::Send
.
Send(operand, channel_handle)
Arguments  Type  Semantics 

operand 
XlaOp 
data to send (array of type T) 
channel_handle 
ChannelHandle 
unique identifier for each send/recv pair 
Sends the given operand data to a Recv
instruction in another computation
that shares the same channel handle. Does not return any data.
Similar to the Recv
operation, the client API of Send
operation represents
synchronous communication, and is internally decomposed into 2 HLO instructions
(Send
and SendDone
) to enable asynchronous data transfers. See also
HloInstruction::CreateSend
and HloInstruction::CreateSendDone
.
Send(HloInstruction operand, int64 channel_id)
Initiates an asynchronous transfer of the operand to the resources allocated by
the Recv
instruction with the same channel id. Returns a context, which is
used by a following SendDone
instruction to wait for the completion of the
data transfer. The context is a tuple of {operand (shape), request identifier
(U32)} and it can only be used by a SendDone
instruction.
SendDone(HloInstruction context)
Given a context created by a Send
instruction, waits for the data transfer to
complete. The instruction does not return any data.
Scheduling of channel instructions
The execution order of the 4 instructions for each channel (Recv
, RecvDone
,
Send
, SendDone
) is as below.
Recv
happens beforeSend
Send
happens beforeRecvDone
Recv
happens beforeRecvDone
Send
happens beforeSendDone
When the backend compilers generate a linear schedule for each computation that communicates via channel instructions, there must not be cycles across the computations. For example, below schedules lead to deadlocks.
Slice
See also
XlaBuilder::Slice
.
Slicing extracts a subarray from the input array. The subarray is of the same rank as the input and contains the values inside a bounding box within the input array where the dimensions and indices of the bounding box are given as arguments to the slice operation.
Slice(operand, start_indices, limit_indices)
Arguments  Type  Semantics 

operand 
XlaOp 
N dimensional array of type T 
start_indices

ArraySlice<int64>

List of N integers containing the starting indices of the slice for each dimension. Values must be greater than or equal to zero. 
limit_indices

ArraySlice<int64>

List of N integers containing the
ending indices (exclusive) for the
slice for each dimension. Each value
must be greater than or equal to the
respective start_indices value for
the dimension and less than or equal
to the size of the dimension. 
strides

ArraySlice<int64>

List of N integers that decides the
input stride of the slice. The slice
picks every strides[d] element in
dimension d . 
1dimensional example:
let a = {0.0, 1.0, 2.0, 3.0, 4.0}
Slice(a, {2}, {4}) produces:
{2.0, 3.0}
2dimensional example:
let b =
{ {0.0, 1.0, 2.0},
{3.0, 4.0, 5.0},
{6.0, 7.0, 8.0},
{9.0, 10.0, 11.0} }
Slice(b, {2, 1}, {4, 3}) produces:
{ { 7.0, 8.0},
{10.0, 11.0} }
Sort
See also
XlaBuilder::Sort
.
Sort(operands, comparator, dimension, is_stable)
Arguments  Type  Semantics 

operands 
ArraySlice<XlaOp> 
The operands to sort. 
comparator 
XlaComputation 
The comparator computation to use. 
dimension 
int64 
The dimension along which to sort. 
is_stable 
bool 
Whether stable sorting should be used. 
If only one operand is provided:
If the operand is a rank1 tensor (an array), the result is a sorted array. If you want to sort the array into ascending order, the comparator should perform a lessthan comparison. Formally, after the array is sorted, it holds for all index positions
i, j
withi < j
that eithercomparator(value[i], value[j]) = comparator(value[j], value[i]) = false
orcomparator(value[i], value[j]) = true
.If the operand has higher rank, the operand is sorted along the provided dimension. For example, for a rank2 tensor (a matrix), a dimension value of
0
will independently sort every column, and a dimension value of1
will independently sort each row. If no dimension number is provided, then the last dimension is chosen by default. For the dimension which is sorted, the same sorting order applies as in the rank1 case.
If n > 1
operands are provided:
All
n
operands must be tensors with the same dimensions. The element types of the tensors may be different.All operands are sorted together, not individually. Conceptually the operands are treated as a tuple. When checking whether the elements of each operand at index positions
i
andj
need to be swapped, the comparator is called with2 * n
scalar parameters, where parameter2 * k
corresponds to the value at positioni
from thekth
operand, and parameter2 * k + 1
corresponds to the value at positionj
from thekth
operand. Usually, the comparator would thus compare parameters2 * k
and2 * k + 1
with each other and possibly use other parameter pairs as tie breakers.The result is a tuple that consists of the operands in sorted order (along the provided dimension, as above). The
ith
operand of the tuple corresponds to theith
operand of Sort.
For example, if there are three operands operand0 = [3, 1]
,
operand1 = [42, 50]
, operand2 = [3.0, 1.1]
, and the comparator compares
only the values of operand0
with lessthan, then the output of the sort is the
tuple ([1, 3], [50, 42], [1.1, 3.0])
.
If is_stable
is set to true, the sort is guaranteed to be stable, that is, if
there are elements which are considered to be equal by the comparator, the
relative order of the equal values is preserved. By default, is_stable
is set
to false.
Transpose
See also the tf.reshape
operation.
Transpose(operand)
Arguments  Type  Semantics 

operand 
XlaOp 
The operand to transpose. 
permutation 
ArraySlice<int64> 
How to permute the dimensions. 
Permutes the operand dimensions with the given permutation, so
∀ i . 0 ≤ i < rank ⇒ input_dimensions[permutation[i]] = output_dimensions[i]
.
This is the same as Reshape(operand, permutation, Permute(permutation, operand.shape.dimensions)).
TriangularSolve
See also
XlaBuilder::TriangularSolve
.
Solves systems of linear equations with lower or upper triangular coefficient
matrices by forward or backsubstitution. Broadcasting along leading
dimensions, this routine solves one of the matrix systems op(a) * x =
b
, or x * op(a) = b
, for the variable x
, given a
and b
, where op(a)
is
either op(a) = a
, or op(a) = Transpose(a)
, or op(a) = Conj(Transpose(a))
.
TriangularSolve(a, b, left_side, lower, unit_diagonal, transpose_a)
Arguments  Type  Semantics 

a

XlaOp

a rank > 2 array of a complex or
floatingpoint type with shape [..., M,
M] . 
b

XlaOp

a rank > 2 array of the same type with shape
[..., M, K] if left_side is true, [...,
K, M] otherwise. 
left_side

bool

indicates whether to solve a system of the
form op(a) * x = b (true ) or x *
op(a) = b (false ). 
lower

bool

whether to use the upper or lower triangle
of a . 
unit_diagonal

bool

if true , the diagonal elements of a are
assumed to be 1 and not accessed. 
transpose_a

Transpose

whether to use a as is, transpose it or
take its conjugate transpose. 
Input data is read only from the lower/upper triangle of a
, depending on the
value of lower
. Values from the other triangle are ignored. Output data is
returned in the same triangle; the values in the other triangle are
implementationdefined and may be anything.
If the rank of a
and b
are greater than 2, they are treated as batches of
matrices, where all except the minor 2 dimensions are batch dimensions. a
and
b
must have equal batch dimensions.
Tuple
See also
XlaBuilder::Tuple
.
A tuple containing a variable number of data handles, each of which has its own shape.
This is analogous to std::tuple
in C++. Conceptually:
let v: f32[10] = f32[10]{0, 1, 2, 3, 4, 5, 6, 7, 8, 9};
let s: s32 = 5;
let t: (f32[10], s32) = tuple(v, s);
Tuples can be deconstructed (accessed) via the GetTupleElement
operation.
While
See also
XlaBuilder::While
.
While(condition, body, init)
Arguments  Type  Semantics 

condition

XlaComputation

XlaComputation of type T > PRED which
defines the termination condition of the
loop. 
body

XlaComputation

XlaComputation of type T > T which
defines the body of the loop. 
init

T

Initial value for the parameter of
condition and body . 
Sequentially executes the body
until the condition
fails. This is similar to
a typical while loop in many other languages except for the differences and
restrictions listed below.
 A
While
node returns a value of typeT
, which is the result from the last execution of thebody
.  The shape of the type
T
is statically determined and must be the same across all iterations.
The T parameters of the computations are initialized with the init
value in
the first iteration and are automatically updated to the new result from body
in each subsequent iteration.
One main use case of the While
node is to implement the repeated execution of
training in neural networks. Simplified pseudocode is shown below with a graph
that represents the computation. The code can be found in
while_test.cc
.
The type T
in this example is a Tuple
consisting of an int32
for the
iteration count and a vector[10]
for the accumulator. For 1000 iterations, the
loop keeps adding a constant vector to the accumulator.
// Pseudocode for the computation.
init = {0, zero_vector[10]} // Tuple of int32 and float[10].
result = init;
while (result(0) < 1000) {
iteration = result(0) + 1;
new_vector = result(1) + constant_vector[10];
result = {iteration, new_vector};
}