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

Linear Kernel.

Inherits From: `Polynomial`, `PositiveSemidefiniteKernel`

Is based on the dot product covariance function and can be obtained from linear regression. This kernel, when parameterizing a Gaussian Process, results in random linear functions. The Linear kernel is based on the Polynomial kernel without the exponent.

``````k(x, y) = bias_variance**2 + slope_variance**2 *
((x - shift) dot (y - shift))
``````

`bias_variance` Positive floating point `Tensor` that controls the variance from the origin. If bias = 0, there is no variance and the fitted function goes through the origin (also known as the homogeneous linear kernel). Must be broadcastable with `slope_variance`, `shift` and inputs to `apply` and `matrix` methods. A value of `None` is treated like 0. Default Value: `None`
`slope_variance` Positive floating point `Tensor` that controls the variance of the regression line slope. Must be broadcastable with `bias_variance`, `shift`, and inputs to `apply` and `matrix` methods. A value of `None` is treated like 1. Default Value: `None`
`shift` Floating point `Tensor` that controls the intercept with the x-axis of the linear interpolation. Must be broadcastable with `bias_variance`, `slope_variance`, and inputs to `apply` and `matrix` methods. A value of `None` is treated like 0, which results in having the intercept at the origin.
`feature_ndims` Python `int` number of rightmost dims to include in kernel computation. Default Value: 1
`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: `'Linear'`

`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.

`bias_variance` Variance on bias parameter.
`dtype` DType over which the kernel operates.
`exponent` Exponent of the polynomial term.
`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.

`name` Name prepended to all ops created by this class.
`shift` Shift of linear function that is exponentiated.
`slope_variance` Variance on slope parameter.
`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="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]
``````

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