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This kernel function has the form:

``````k(x, y) = amplitude**2 * (1. + ||x - y|| ** 2 / (
2 * scale_mixture_rate * length_scale**2)) ** -scale_mixture_rate
``````

where the double-bars represent vector length (i.e. Euclidean, or L2 Norm). This kernel acts over the space `S = R^(D1 x D2 .. Dd)`. When `scale_mixture_rate` tends towards infinity, this kernel acts like an ExponentiatedQuadratic kernel.

The name `scale_mixture_rate` comes from the interpretation that that this kernel is an Inverse-Gamma weighted mixture of ExponentiatedQuadratic Kernels with different length scales.

More formally, if `r = ||x - y|||`, then:

``````integral from 0 to infinity (k_EQ(r | sqrt(t)) p(t | a, a * l ** 2)) dt =
(1 + r**2 / (2 * a * l ** 2)) ** -a = k(r)
``````

where:

• `a = scale_mixture_rate`
• `l = length_scale`
• `p(t | a, b)` is the Inverse-Gamma density. https://en.wikipedia.org/wiki/Inverse-gamma_distribution
• `k_EQ(r | l) = exp(-r ** 2 / (2 * l ** 2)` is the ExponentiatedQuadratic positive semidefinite kernel.

#### References

[1]: Filip Tronarp, Toni Karvonen, and Simo Saarka. Mixture representation of the Matern class with applications in state space approximations and Bayesian quadrature. In 28th IEEE International Workshop on Machine Learning for Signal Processing, 2018. https://acris.aalto.fi/ws/portalfiles/portal/30548539/ELEC_Tronarp_etal_Mixture_Presentation_of_the_Matern_MLSP2018.pdf

`amplitude` Positive floating point `Tensor` that controls the maximum value of the kernel. Must be broadcastable with `length_scale` and `scale_mixture_rate` and inputs to `apply` and `matrix` methods. A value of `None` is treated like 1. Default value: None
`length_scale` Positive floating point `Tensor` that controls how sharp or wide the kernel shape is. This provides a characteristic "unit" of length against which `||x - y||` can be compared for scale. Must be broadcastable with `amplitude`, `scale_mixture_rate` and inputs to `apply` and `matrix` methods. A value of `None` is treated like 1. Default value: None
`inverse_length_scale` Non-negative floating point `Tensor` that is treated as `1 / length_scale`. Only one of `length_scale` or `inverse_length_scale` should be provided. Default value: None
`scale_mixture_rate` Positive floating point `Tensor` that controls how the ExponentiatedQuadratic kernels are mixed. Must be broadcastable with `amplitude`, `length_scale` and inputs to `apply` and `matrix` methods. A value of `None` is treated like 1. Default value: None
`feature_ndims` Python `int` number of rightmost dims to include in the squared difference norm in the exponential.
`validate_args` If `True`, parameters are checked for validity despite possibly degrading runtime performance
`name` Python `str` name prefixed to Ops created by this class.

`amplitude` Amplitude parameter.
`batch_shape` Shape of a single sample from a single event index as a `TensorShape`.

May be partially defined or unknown.

The batch dimensions are indexes into independent, non-identical parameterizations of this `PositiveSemidefiniteKernel`.

`dtype` (Nested) dype 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. Inputs may also be nested structures, in which case `feature_ndims` is a parallel nested structure containing the feature rank of each component.

`inverse_length_scale` Inverse length scale parameter.
`length_scale` Length scale parameter.
`name` Name prepended to all ops created by this class.
`parameters` Dictionary of parameters used to instantiate this `PSDKernel`.
`scale_mixture_rate` scale_mixture_rate 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` (Nested) `Tensor` input to the kernel, of shape `B1 + E1 + F`, where `B1` and `E1` may be empty (ie, no batch/example dims, resp.). If nested, `B1` and `E1` must broadcast across elements of the structure. `F` (the feature shape) must have rank equal to the kernel's `feature_ndims` property, or to the corresponding element of the `feature_ndims` nested structure. 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` (Nested) `Tensor` input to the kernel, of shape `B2 + E2 + F`, where `B2` and `E2` may be empty (ie, no batch/example dims, resp.). If nested, `B1` and `E1` must broadcast across elements of the structure. `F` (the feature shape) must have rank equal to the kernel's `feature_ndims` property, or to the corresponding element of the `feature_ndims` nested structure. 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).
`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.substrates.jax

# 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:

##### Fix #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]
``````
##### Fix #2

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]
``````

### `batch_shape_tensor`

View source

Shape of a single sample from a single event index as a 1-D `Tensor`.

The batch dimensions are indexes into independent, non-identical parameterizations of this `PositiveSemidefiniteKernel`.

Args
`name` name to give to the op

Returns
`batch_shape` `Tensor`.

### `copy`

View source

Creates a copy of the kernel.

Args
`**override_parameters_kwargs` String/value dictionary of initialization arguments to override with new values.

Returns
`copied_kernel` A new instance of `type(self)` initialized from the union of self.parameters and override_parameters_kwargs, i.e., `dict(self.parameters, **override_parameters_kwargs)`.

### `matrix`

View source

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

Args
`x1` (Nested) `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 (or to the corresponding element of `feature_ndims`, if nested). Batch shape must broadcast with the batch shape of `x2` and with the kernel's batch shape.
`x2` (Nested) `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 (or to the corresponding element of `feature_ndims`, if nested). Batch shape must broadcast with the batch shape of `x1` and with the kernel's batch shape.
`name` name to give to the op

Returns
`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 are broadcast together).

Given inputs `x1` and `x2` of shapes

``````[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.substrates.jax

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.

### `parameter_properties`

View source

Returns a dict mapping constructor arg names to property annotations.

This dict should include an entry for each of the kernel's `Tensor`-valued constructor arguments.

Args
`dtype` Optional float `dtype` to assume for continuous-valued parameters. Some constraining bijectors require advance knowledge of the dtype because certain constants (e.g., `tfb.Softplus.low`) must be instantiated with the same dtype as the values to be transformed.