# tfp.experimental.substrates.numpy.math.psd_kernels.SchurComplement

The SchurComplement kernel.

Inherits From: `PositiveSemidefiniteKernel`

Given a block matrix `M = [[A, B], [C, D]]`, the Schur complement of D in M is written `M / D = A - B @ Inverse(D) @ C`.

This class represents a PositiveSemidefiniteKernel whose behavior is as follows. We compute a matrix, analogous to `D` in the above definition, by calling `base_kernel.matrix(fixed_inputs, fixed_inputs)`. Then given new input locations `x` and `y`, we can construct the remaining pieces of `M` above, and compute the Schur complement of `D` in `M` (see Mathematical Details, below).

Notably, this kernel uses a bijector (Invert(CholeskyOuterProduct)), as an intermediary for the requisite matrix solve, which means we get a caching benefit after the first use.

### Mathematical Details

Suppose we have a kernel `k` and some fixed collection of inputs `Z = [z0, z1, ..., zN]`. Given new inputs `x` and `y`, we can form a block matrix

``````   M = [
[k(x, y), k(x, z0), ..., k(x, zN)],
[k(z0, y), k(z0, z0), ..., k(z0, zN)],
...,
[k(zN, y), k(z0, zN), ..., k(zN, zN)],
]
``````

We might write this, so as to emphasize the block structure,

``````   M = [
[xy, xZ],
[yZ^T, ZZ],
],

xy = [k(x, y)]
xZ = [k(x, z0), ..., k(x, zN)]
yZ = [k(y, z0), ..., k(y, zN)]
ZZ = "the matrix of k(zi, zj)'s"
``````

Then we have the definition of this kernel's apply method:

`schur_comp.apply(x, y) = xy - xZ @ ZZ^{-1} @ yZ^T`

and similarly, if x and y are collections of inputs.

As with other PSDKernels, the `apply` method acts as a (possibly vectorized) scalar function of 2 inputs. Given a single `x` and `y`, `apply` will yield a scalar output. Given two (equal size!) collections `X` and `Y`, it will yield another (equal size!) collection of scalar outputs.

### Examples

Here's a simple example usage, with no particular motivation.

``````from tensorflow_probability.math import psd_kernels

# 3 points in 1-dimensional space (shape [3, 1]).
z = [[0.], [3.], [4.]]

schur_kernel = psd_kernels.SchurComplement(
base_kernel=base_kernel,
fixed_inputs=z)

# Two individual 1-d points
x = [1.]
y = [2.]
print(schur_kernel.apply(x, y))
# ==> k(x, y) - k(x, z) @ Inverse(k(z, z)) @ k(z, y)
``````

A more motivating application of this kernel is in constructing a Gaussian process that is conditioned on some observed data.

``````from tensorflow_probability import distributions as tfd
from tensorflow_probability.math import psd_kernels

observation_index_points = np.random.uniform(-1., 1., [50, 1])
observations = np.sin(2 * np.pi * observation_index_points[..., 0])

posterior_kernel = psd_kernels.SchurComplement(
base_kernel=base_kernel,
fixed_inputs=observation_index_points)

# Assume we use a zero prior mean, and compute the posterior mean.
def posterior_mean_fn(x):
k_x_obs_linop = tf.linalg.LinearOperatorFullMatrix(
base_kernel.matrix(x, observation_index_points))
chol_linop = tf.linalg.LinearOperatorLowerTriangular(
posterior_kernel.divisor_matrix_cholesky())

return k_x_obs_linop.matvec(
chol_linop.solvevec(
chol_linop.solvevec(observations),

# Construct the GP posterior distribution at some new points.
gp_posterior = tfp.distributions.GaussianProcess(
index_points=np.linspace(-1., 1., 100)[..., np.newaxis],
kernel=posterior_kernel,
mean_fn=posterior_mean_fn)

# Draw 5 samples on the above 100-point grid
samples = gp_posterior.sample(5)
``````

`base_kernel` A `PositiveSemidefiniteKernel` instance, the kernel used to build the block matrices of which this kernel computes the Schur complement.
`fixed_inputs` A Tensor, representing a collection of inputs. The Schur complement that this kernel computes comes from a block matrix, whose bottom-right corner is derived from ```base_kernel.matrix(fixed_inputs, fixed_inputs)```, and whose top-right and bottom-left pieces are constructed by computing the base_kernel at pairs of input locations together with these `fixed_inputs`. `fixed_inputs` is allowed to be an empty collection (either `None` or having a zero shape entry), in which case the kernel falls back to the trivial application of `base_kernel` to inputs. See class-level docstring for more details on the exact computation this does; `fixed_inputs` correspond to the `Z` structure discussed there. `fixed_inputs` is assumed to have shape ```[b1, ..., bB, N, f1, ..., fF]``` where the `b`'s are batch shape entries, the `f`'s are feature_shape entries, and `N` is the number of fixed inputs. Use of this kernel entails a 1-time O(N^3) cost of computing the Cholesky decomposition of the k(Z, Z) matrix. The batch shape elements of `fixed_inputs` must be broadcast compatible with `base_kernel.batch_shape`.
`diag_shift` A floating point scalar to be added to the diagonal of the divisor_matrix before computing its Cholesky.
`validate_args` If `True`, parameters are checked for validity despite possibly degrading runtime performance. Default value: `False`
`name` Python `str` name prefixed to Ops created by this class. Default value: `"SchurComplement"`

`base_kernel`

`batch_shape` The batch_shape property of a PositiveSemidefiniteKernel.

This property describes the fully broadcast shape of all kernel parameters. For example, consider an ExponentiatedQuadratic kernel, which is parameterized by an amplitude and length_scale:

``````exp_quad(x, x') := amplitude * exp(||x - x'||**2 / length_scale**2)
``````

The batch_shape of such a kernel is derived from broadcasting the shapes of `amplitude` and `length_scale`. E.g., if their shapes were

``````amplitude.shape = [2, 1, 1]
length_scale.shape = [1, 4, 3]
``````

then `exp_quad`'s batch_shape would be `[2, 4, 3]`.

Note that this property defers to the private _batch_shape method, which concrete implementation sub-classes are obliged to provide.

`cholesky_bijector`

`diag_shift`

`dtype` DType over which the kernel operates.
`feature_ndims` The number of feature dimensions.

Kernel functions generally act on pairs of inputs from some space like

``````R^(d1 x ... x dD)
``````

or, in words: rank-`D` real-valued tensors of shape `[d1, ..., dD]`. Inputs can be vectors in some `R^N`, but are not restricted to be. Indeed, one might consider kernels over matrices, tensors, or even more general spaces, like strings or graphs.

`fixed_inputs`

`name` Name prepended to all ops created by this class.
`trainable_variables`

`validate_args` Python `bool` indicating possibly expensive checks are enabled.
`variables`

## Methods

### `apply`

View source

Apply the kernel function pairs of inputs.

Args
`x1` `Tensor` input to the kernel, of shape `B1 + E1 + F`, where `B1` and `E1` may be empty (ie, no batch/example dims, resp.) and `F` (the feature shape) must have rank equal to the kernel's `feature_ndims` property. Batch shape must broadcast with the batch shape of `x2` and with the kernel's batch shape. Example shape must broadcast with example shape of `x2`. `x1` and `x2` must have the same number of example dims (ie, same rank).
`x2` `Tensor` input to the kernel, of shape `B2 + E2 + F`, where `B2` and `E2` may be empty (ie, no batch/example dims, resp.) and `F` (the feature shape) must have rank equal to the kernel's `feature_ndims` property. Batch shape must broadcast with the batch shape of `x2` and with the kernel's batch shape. Example shape must broadcast with example shape of `x2`. `x1` and `x2` must have the same number of example
`example_ndims` A python integer, the number of example dims in the inputs. In essence, this parameter controls how broadcasting of the kernel's batch shape with input batch shapes works. The kernel batch shape will be broadcast against everything to the left of the combined example and feature dimensions in the input shapes.
`name` name to give to the op

Returns
`Tensor` containing the results of applying the kernel function to inputs `x1` and `x2`. If the kernel parameters' batch shape is `Bk` then the shape of the `Tensor` resulting from this method call is `broadcast(Bk, B1, B2) + broadcast(E1, E2)`.

Given an index set `S`, a kernel function is mathematically defined as a real- or complex-valued function on `S` satisfying the positive semi-definiteness constraint:

``````sum_i sum_j (c[i]*) c[j] k(x[i], x[j]) >= 0
``````

for any finite collections `{x[1], ..., x[N]}` in `S` and `{c[1], ..., c[N]}` in the reals (or the complex plane). '*' is the complex conjugate, in the complex case.

This method most closely resembles the function described in the mathematical definition of a kernel. Given a PositiveSemidefiniteKernel `k` with scalar parameters and inputs `x` and `y` in `S`, `apply(x, y)` yields a single scalar value.

#### Examples

``````import tensorflow_probability as tfp; tfp = tfp.experimental.substrates.numpy

# Suppose `SomeKernel` acts on vectors (rank-1 tensors)
scalar_kernel = tfp.math.psd_kernels.SomeKernel(param=.5)
scalar_kernel.batch_shape
# ==> []

# `x` and `y` are batches of five 3-D vectors:
x = np.ones([5, 3], np.float32)
y = np.ones([5, 3], np.float32)
scalar_kernel.apply(x, y).shape
# ==> [5]
``````

The above output is the result of vectorized computation of the five values

``````[k(x[0], y[0]), k(x[1], y[1]), ..., k(x[4], y[4])]
``````

Now we can consider a kernel with batched parameters:

``````batch_kernel = tfp.math.psd_kernels.SomeKernel(param=[.2, .5])
batch_kernel.batch_shape
# ==> [2]
batch_kernel.apply(x, y).shape
# ==> Error! [2] and [5] can't broadcast.
``````

The parameter batch shape of `[2]` and the input batch shape of `[5]` can't be broadcast together. We can fix this in either of two ways:

1. Give the parameter a shape of `[2, 1]` which will correctly broadcast with `[5]` to yield `[2, 5]`:
``````batch_kernel = tfp.math.psd_kernels.SomeKernel(
param=[[.2], [.5]])
batch_kernel.batch_shape
# ==> [2, 1]
batch_kernel.apply(x, y).shape
# ==> [2, 5]
``````
1. By specifying `example_ndims`, which tells the kernel to treat the `5` in the input shape as part of the "example shape", and "pushing" the kernel batch shape to the left:
``````batch_kernel = tfp.math.psd_kernels.SomeKernel(param=[.2, .5])
batch_kernel.batch_shape
# ==> [2]
batch_kernel.apply(x, y, example_ndims=1).shape
# ==> [2, 5]

<h3 id="batch_shape_tensor"><code>batch_shape_tensor</code></h3>

<a target="_blank" href="https://github.com/tensorflow/probability/blob/v0.11.1/tensorflow_probability/python/math/psd_kernels/_numpy/positive_semidefinite_kernel.py#L306-L318">View source</a>

<code>batch_shape_tensor()
</code></pre>

The batch_shape property of a PositiveSemidefiniteKernel as a `Tensor`.

<!-- Tabular view -->
<table class="responsive fixed orange">
<colgroup><col width="214px"><col></colgroup>
<tr><th colspan="2">Returns</th></tr>
<tr class="alt">
<td colspan="2">
`Tensor` which evaluates to a vector of integers which are the
fully-broadcast shapes of the kernel parameters.
</td>
</tr>

</table>

<h3 id="divisor_matrix"><code>divisor_matrix</code></h3>

<a target="_blank" href="https://github.com/tensorflow/probability/blob/v0.11.1/tensorflow_probability/python/math/psd_kernels/_numpy/schur_complement.py#L360-L361">View source</a>

<code>divisor_matrix()
</code></pre>

<h3 id="divisor_matrix_cholesky"><code>divisor_matrix_cholesky</code></h3>

<a target="_blank" href="https://github.com/tensorflow/probability/blob/v0.11.1/tensorflow_probability/python/math/psd_kernels/_numpy/schur_complement.py#L367-L368">View source</a>

<code>divisor_matrix_cholesky(
fixed_inputs=None
)
</code></pre>

<h3 id="matrix"><code>matrix</code></h3>

<a target="_blank" href="https://github.com/tensorflow/probability/blob/v0.11.1/tensorflow_probability/python/math/psd_kernels/_numpy/positive_semidefinite_kernel.py#L506-L671">View source</a>

<code>matrix(
x1, x2, name='matrix'
)
</code></pre>

Construct (batched) matrices from (batches of) collections of inputs.

<!-- Tabular view -->
<table class="responsive fixed orange">
<colgroup><col width="214px"><col></colgroup>
<tr><th colspan="2">Args</th></tr>

<tr>
<td>
`x1`
</td>
<td>
`Tensor` input to the first positional parameter of the kernel, of
shape `B1 + [e1] + F`, where `B1` may be empty (ie, no batch dims,
resp.), `e1` is a single integer (ie, `x1` has example ndims exactly 1),
and `F` (the feature shape) must have rank equal to the kernel's
`feature_ndims` property. Batch shape must broadcast with the batch
shape of `x2` and with the kernel's batch shape.
</td>
</tr><tr>
<td>
`x2`
</td>
<td>
`Tensor` input to the second positional parameter of the kernel,
shape `B2 + [e2] + F`, where `B2` may be empty (ie, no batch dims,
resp.), `e2` is a single integer (ie, `x2` has example ndims exactly 1),
and `F` (the feature shape) must have rank equal to the kernel's
`feature_ndims` property. Batch shape must broadcast with the batch
shape of `x1` and with the kernel's batch shape.
</td>
</tr><tr>
<td>
`name`
</td>
<td>
name to give to the op
</td>
</tr>
</table>

<!-- Tabular view -->
<table class="responsive fixed orange">
<colgroup><col width="214px"><col></colgroup>
<tr><th colspan="2">Returns</th></tr>
<tr class="alt">
<td colspan="2">
`Tensor` containing the matrix (possibly batched) of kernel applications
to pairs from inputs `x1` and `x2`. If the kernel parameters' batch shape
is `Bk` then the shape of the `Tensor` resulting from this method call is
`broadcast(Bk, B1, B2) + [e1, e2]` (note this differs from `apply`: the
example dimensions are concatenated, whereas in `apply` the example dims
</td>
</tr>

</table>

Given inputs `x1` and `x2` of shapes

```none
[b1, ..., bB, e1, f1, ..., fF]
``````

and

``````[c1, ..., cC, e2, f1, ..., fF]
``````

This method computes the batch of `e1 x e2` matrices resulting from applying the kernel function to all pairs of inputs from `x1` and `x2`. The shape of the batch of matrices is the result of broadcasting the batch shapes of `x1`, `x2`, and the kernel parameters (see examples below). As such, it's required that these shapes all be broadcast compatible. However, the kernel parameter batch shapes need not broadcast against the 'example shapes' (`e1` and `e2` above).

When the two inputs are the (batches of) identical collections, the resulting matrix is the so-called Gram (or Gramian) matrix (https://en.wikipedia.org/wiki/Gramian_matrix).

#### Examples

First, consider a kernel with a single scalar parameter.

``````import tensorflow_probability as tfp; tfp = tfp.experimental.substrates.numpy

scalar_kernel = tfp.math.psd_kernels.SomeKernel(param=.5)
scalar_kernel.batch_shape
# ==> []

# Our inputs are two lists of 3-D vectors
x = np.ones([5, 3], np.float32)
y = np.ones([4, 3], np.float32)
scalar_kernel.matrix(x, y).shape
# ==> [5, 4]
``````

The result comes from applying the kernel to the entries in `x` and `y` pairwise, across all pairs:

``````  | k(x[0], y[0])    k(x[0], y[1])  ...  k(x[0], y[3]) |
| k(x[1], y[0])    k(x[1], y[1])  ...  k(x[1], y[3]) |
|      ...              ...                 ...      |
| k(x[4], y[0])    k(x[4], y[1])  ...  k(x[4], y[3]) |
``````

Now consider a kernel with batched parameters with the same inputs

``````batch_kernel = tfp.math.psd_kernels.SomeKernel(param=[1., .5])
batch_kernel.batch_shape
# ==> [2]

batch_kernel.matrix(x, y).shape
# ==> [2, 5, 4]
``````

This results in a batch of 2 matrices, one computed from the kernel with `param = 1.` and the other with `param = .5`.

We also support batching of the inputs. First, let's look at that with the scalar kernel again.

``````# Batch of 10 lists of 5 vectors of dimension 3
x = np.ones([10, 5, 3], np.float32)

# Batch of 10 lists of 4 vectors of dimension 3
y = np.ones([10, 4, 3], np.float32)

scalar_kernel.matrix(x, y).shape
# ==> [10, 5, 4]
``````

The result is a batch of 10 matrices built from the batch of 10 lists of input vectors. These batch shapes have to be broadcastable. The following will not work:

``````x = np.ones([10, 5, 3], np.float32)
y = np.ones([20, 4, 3], np.float32)
scalar_kernel.matrix(x, y).shape
# ==> Error! [10] and [20] can't broadcast.
``````

Now let's consider batches of inputs in conjunction with batches of kernel parameters. We require that the input batch shapes be broadcastable with the kernel parameter batch shapes, otherwise we get an error:

``````x = np.ones([10, 5, 3], np.float32)
y = np.ones([10, 4, 3], np.float32)

batch_kernel = tfp.math.psd_kernels.SomeKernel(params=[1., .5])
batch_kernel.batch_shape
# ==> [2]
batch_kernel.matrix(x, y).shape
# ==> Error! [2] and [10] can't broadcast.
``````

The fix is to make the kernel parameter shape broadcastable with `[10]` (or reshape the inputs to be broadcastable!):

``````x = np.ones([10, 5, 3], np.float32)
y = np.ones([10, 4, 3], np.float32)

batch_kernel = tfp.math.psd_kernels.SomeKernel(
params=[[1.], [.5]])
batch_kernel.batch_shape
# ==> [2, 1]
batch_kernel.matrix(x, y).shape
# ==> [2, 10, 5, 4]

# Or, make the inputs broadcastable:
x = np.ones([10, 1, 5, 3], np.float32)
y = np.ones([10, 1, 4, 3], np.float32)

batch_kernel = tfp.math.psd_kernels.SomeKernel(
params=[1., .5])
batch_kernel.batch_shape
# ==> [2]
batch_kernel.matrix(x, y).shape
# ==> [10, 2, 5, 4]
``````

Here, we have the result of applying the kernel, with 2 different parameters, to each of a batch of 10 pairs of input lists.

### `tensor`

View source

Construct (batched) tensors from (batches of) collections of inputs.

Args
`x1` `Tensor` input to the first positional parameter of the kernel, of shape `B1 + E1 + F`, where `B1` and `E1` arbitrary shapes which may be empty (ie, no batch/example dims, resp.), and `F` (the feature shape) must have rank equal to the kernel's `feature_ndims` property. Batch shape must broadcast with the batch shape of `x2` and with the kernel's batch shape.
`x2` `Tensor` input to the second positional parameter of the kernel, shape `B2 + E2 + F`, where `B2` and `E2` arbitrary shapes which may be empty (ie, no batch/example dims, resp.), and `F` (the feature shape) must have rank equal to the kernel's `feature_ndims` property. Batch shape must broadcast with the batch shape of `x1` and with the kernel's batch shape.
`x1_example_ndims` A python integer greater than or equal to 0, the number of example dims in the first input. This affects both the alignment of batch shapes and the shape of the final output of the function. Everything left of the feature shape and the example dims in `x1` is considered "batch shape", and must broadcast as specified above.
`x2_example_ndims` A python integer greater than or equal to 0, the number of example dims in the second input. This affects both the alignment of batch shapes and the shape of the final output of the function. Everything left of the feature shape and the example dims in `x1` is considered "batch shape", and must broadcast as specified above.
`name` name to give to the op

Returns
`Tensor` containing (possibly batched) kernel applications to pairs from inputs `x1` and `x2`. If the kernel parameters' batch shape is `Bk` then the shape of the `Tensor` resulting from this method call is `broadcast(Bk, B1, B2) + E1 + E2`. Note this differs from `apply`: the example dimensions are concatenated, whereas in `apply` the example dims are broadcast together. It also differs from `matrix`: the example shapes are arbitrary here, and the result accrues a rank equal to the sum of the ranks of the input example shapes.

#### Examples

First, consider a kernel with a single scalar parameter.

``````import tensorflow_probability as tfp; tfp = tfp.experimental.substrates.numpy

scalar_kernel = tfp.math.psd_kernels.SomeKernel(param=.5)
scalar_kernel.batch_shape
# ==> []

# Our inputs are two rank-2 collections of 3-D vectors
x = np.ones([5, 6, 3], np.float32)
y = np.ones([7, 8, 3], np.float32)
scalar_kernel.tensor(x, y, x1_example_ndims=2, x2_example_ndims=2).shape
# ==> [5, 6, 7, 8]

# Empty example shapes work too!
x = np.ones([3], np.float32)
y = np.ones([5, 3], np.float32)
scalar_kernel.tensor(x, y, x1_example_ndims=0, x2_example_ndims=1).shape
# ==> [5]
``````

The result comes from applying the kernel to the entries in `x` and `y` pairwise, across all pairs:

``````  | k(x[0], y[0])    k(x[0], y[1])  ...  k(x[0], y[3]) |
| k(x[1], y[0])    k(x[1], y[1])  ...  k(x[1], y[3]) |
|      ...              ...                 ...      |
| k(x[4], y[0])    k(x[4], y[1])  ...  k(x[4], y[3]) |
``````

Now consider a kernel with batched parameters.

``````batch_kernel = tfp.math.psd_kernels.SomeKernel(param=[1., .5])
batch_kernel.batch_shape
# ==> [2]

# Inputs are two rank-2 collections of 3-D vectors
x = np.ones([5, 6, 3], np.float32)
y = np.ones([7, 8, 3], np.float32)
scalar_kernel.tensor(x, y, x1_example_ndims=2, x2_example_ndims=2).shape
# ==> [2, 5, 6, 7, 8]
``````

We also support batching of the inputs. First, let's look at that with the scalar kernel again.

``````# Batch of 10 lists of 5x6 collections of dimension 3
x = np.ones([10, 5, 6, 3], np.float32)

# Batch of 10 lists of 7x8 collections of dimension 3
y = np.ones([10, 7, 8, 3], np.float32)

scalar_kernel.tensor(x, y, x1_example_ndims=2, x2_example_ndims=2).shape
# ==> [10, 5, 6, 7, 8]
``````

The result is a batch of 10 tensors built from the batch of 10 rank-2 collections of input vectors. The batch shapes have to be broadcastable. The following will not work:

``````x = np.ones([10, 5, 3], np.float32)
y = np.ones([20, 4, 3], np.float32)
scalar_kernel.tensor(x, y, x1_example_ndims=1, x2_example_ndims=1).shape
# ==> Error! [10] and [20] can't broadcast.
``````

Now let's consider batches of inputs in conjunction with batches of kernel parameters. We require that the input batch shapes be broadcastable with the kernel parameter batch shapes, otherwise we get an error:

``````x = np.ones([10, 5, 6, 3], np.float32)
y = np.ones([10, 7, 8, 3], np.float32)

batch_kernel = tfp.math.psd_kernels.SomeKernel(params=[1., .5])
batch_kernel.batch_shape
# ==> [2]
batch_kernel.tensor(x, y, x1_example_ndims=2, x2_example_ndims=2).shape
# ==> Error! [2] and [10] can't broadcast.
``````

The fix is to make the kernel parameter shape broadcastable with `[10]` (or reshape the inputs to be broadcastable!):

``````x = np.ones([10, 5, 6, 3], np.float32)
y = np.ones([10, 7, 8, 3], np.float32)

batch_kernel = tfp.math.psd_kernels.SomeKernel(
params=[[1.], [.5]])
batch_kernel.batch_shape
# ==> [2, 1]
batch_kernel.tensor(x, y, x1_example_ndims=2, x2_example_ndims=2).shape
# ==> [2, 10, 5, 6, 7, 8]

# Or, make the inputs broadcastable:
x = np.ones([10, 1, 5, 6, 3], np.float32)
y = np.ones([10, 1, 7, 8, 3], np.float32)

batch_kernel = tfp.math.psd_kernels.SomeKernel(
params=[1., .5])
batch_kernel.batch_shape
# ==> [2]
batch_kernel.tensor(x, y, x1_example_ndims=2, x2_example_ndims=2).shape
# ==> [10, 2, 5, 6, 7, 8]
``````

View source

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