tfp.experimental.psd_kernels.AdditiveKernel

Additive Kernel.

Inherits From: AutoCompositeTensorPsdKernel, PositiveSemidefiniteKernel, AutoCompositeTensor

tfp.experimental.psd_kernels.AdditiveKernel(
    kernel, amplitudes, validate_args=False, name='AdditiveKernel'
)

This kernel has the following form

k(x, y) = sum k_add_i(x, y)
k_add_n(x, y) = a_n**2 sum_{1<=i1<i2<...in} prod k_i(x[i], y[i])

Where \(k_i\) is the one-dimensional base kernel for the ith dimension.

In other words, this computes sums of elementary symmetric polynomials over k_i(x[i], y[i]).

This kernel is very related to the ANOVA kernel defined as: k_{ANOVA}(x, y) = prod (1 + k_i(x[i], x[i]). k_{ANOVA} is equivalent to a special case of this kernel where the amplitudes are all one, along with a constant shift by 1.

References

[1] D. Duvenaud, H. Nickish, C. E. Rasmussen, Additive Gaussian Process. https://hannes.nickisch.org/papers/conferences/duvenaud11gpadditive.pdf

[2] M. Stitson, A. Gammerman, V. Vapnik, V. Vovk, et al. Support Vector Regression with ANOVA Decomposition Kernels http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.34.7818&rep=rep1&type=pdf

Args
`kernel`	An instance of `PositiveSemidefiniteKernel`s that are defined within this class (specifically they allow for reinterpreting batch dimensions as feature dimensions) that act on inputs of the form `[B1, ...., Bk, D, 1]`; that is, `kernel` is a batch of D-kernels, each acting on 1-dimensional inputs. We assume that the kernel has a batch dimension broadcastable with `[D]`. `kernel` must inherit from `tf.__internal__.CompositeTensor`.
`amplitudes`	`Tensor` of shape `[B1, ...., Bk, M]`, where `M` is the order of the additive kernel. `M` must be statically identifiable.
`validate_args`	Python `bool`, default `False`. When `True` kernel parameters are checked for validity despite possibly degrading runtime performance. When `False` invalid inputs may silently render incorrect outputs.
`name`	Python `str` name prefixed to Ops created by this class. Default: subclass name.

Raises
`TypeError`	if `kernel` is not an instance of `tf.__internal__.CompositeTensor`.

Attributes
`amplitudes`	Amplitude parameter for each additive kernel.
`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.
`kernel`	Inner kernel used for scalar kernel computations.
`name`	Name prepended to all ops created by this class.
`name_scope`	Returns a `tf.name_scope` instance for this class.
`non_trainable_variables`	Sequence of non-trainable variables owned by this module and its submodules. Note: this method uses reflection to find variables on the current instance and submodules. For performance reasons you may wish to cache the result of calling this method if you don't expect the return value to change.
`parameters`	Dictionary of parameters used to instantiate this `PSDKernel`.
`submodules`	Sequence of all sub-modules. Submodules are modules which are properties of this module, or found as properties of modules which are properties of this module (and so on). `a = tf.Module()` `b = tf.Module()` `c = tf.Module()` `a.b = b` `b.c = c` `list(a.submodules) == [b, c]` `True` `list(b.submodules) == [c]` `True` `list(c.submodules) == []` `True`
`trainable_variables`	Sequence of trainable variables owned by this module and its submodules. Note: this method uses reflection to find variables on the current instance and submodules. For performance reasons you may wish to cache the result of calling this method if you don't expect the return value to change.
`validate_args`	Python `bool` indicating possibly expensive checks are enabled.
`variables`	Sequence of variables owned by this module and its submodules. Note: this method uses reflection to find variables on the current instance and submodules. For performance reasons you may wish to cache the result of calling this method if you don't expect the return value to change.

Methods

`apply`

View source

apply(
    x1, x2, example_ndims=0, name='apply'
)

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

# 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

batch_shape_tensor(
    name='batch_shape_tensor'
)

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

copy(
    **override_parameters_kwargs
)

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

matrix(
    x1, x2, name='matrix'
)

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

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

@classmethod
parameter_properties(
    dtype=tf.float32
)

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.

Returns
`parameter_properties`	A `str ->`tfp.python.internal.parameter_properties.ParameterProperties`dict mapping constructor argument names to`ParameterProperties` instances.

`tensor`

View source

tensor(
    x1, x2, x1_example_ndims, x2_example_ndims, name='tensor'
)

Construct (batched) tensors 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` 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 (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` 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 (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.
`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

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.

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

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.

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]

`with_name_scope`

@classmethod
with_name_scope(
    method
)

Decorator to automatically enter the module name scope.

class MyModule(tf.Module):
  @tf.Module.with_name_scope
  def __call__(self, x):
    if not hasattr(self, 'w'):
      self.w = tf.Variable(tf.random.normal([x.shape[1], 3]))
    return tf.matmul(x, self.w)

Using the above module would produce tf.Variables and tf.Tensors whose names included the module name:

mod = MyModule()
mod(tf.ones([1, 2]))
<tf.Tensor: shape=(1, 3), dtype=float32, numpy=..., dtype=float32)>
mod.w
<tf.Variable 'my_module/Variable:0' shape=(2, 3) dtype=float32,
numpy=..., dtype=float32)>

Args
`method`	The method to wrap.

Returns
The original method wrapped such that it enters the module's name scope.

`add`

View source

__add__(
    k
)

`getitem`

View source

__getitem__(
    slices
)

Slices the batch axes of this kernel, returning a new instance.

k = tfpk.ExponentiatedQuadratic(
  amplitude=tf.ones([3, 5, 7, 9]),
  length_scale=tf.ones([3, 5, 7, 9]))
k.batch_shape  # => [3, 5, 7, 9]
k2 = k[:, tf.newaxis, ..., -2:, 1::2]
k2.batch_shape  # => [3, 1, 5, 2, 4]

Args
`slices`	slices from the [] operator

Returns
`dist`	A new `PositiveSemidefiniteKernel` instance with sliced parameters.

`iter`

View source

__iter__()

`mul`

View source

__mul__(
    k
)

`radd`

View source

__radd__(
    lhs
)

tfp.experimental.psd_kernels.AdditiveKernel

References

Args

Raises

Attributes

Methods

apply

Examples

Fix #1

Fix #2

batch_shape_tensor

copy

matrix

Examples

parameter_properties

tensor

Examples

with_name_scope

__add__

__getitem__

__iter__

__mul__

__radd__

`apply`

`batch_shape_tensor`

`copy`

`matrix`

`parameter_properties`

`tensor`

`with_name_scope`

`add`

`getitem`

`iter`

`mul`

`radd`