tfp.experimental.psd_kernels.Independent

Represents a multi-task kernel whose computations are independent of task.

Inherits From: MultiTaskKernel, AutoCompositeTensorPsdKernel, PositiveSemidefiniteKernel, AutoCompositeTensor

tfp.experimental.psd_kernels.Independent(
    num_tasks, base_kernel, validate_args=False, name='Independent'
)

num_tasks Python integer indicating the number of tasks.
dtype DType on which this kernel operates.
feature_ndims Python integer indicating the number of dims (the rank) of the feature space this kernel acts on.
name Python str name prefixed to Ops created by this class. Default: subclass name.
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.
parameters Python dict of constructor arguments.

ValueError if num_tasks is not an integer greater than 0.

base_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.

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.
num_tasks Number of tasks for this kernel.
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.
validate_args Python bool indicating possibly expensive checks are enabled.
variables Sequence of variables owned by this module and its submodules.

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.

matrix_over_all_tasks

View source

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

Returns matrix computation assuming all tasks are observed.

Given x1 and x2 of shape B1 + [E1, M] and B2 + [E2, M], returns a LinearOperator S of shape broadcast(B1, B2) + [E1 * N, E2 * N], where N is num_tasks. Each N x N block of this LinearOperator represents the action of this kernel at a fixed pair of inputs, over all tasks.

Args
x1 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.
x2 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.
name name to give to the op

Returns
Tensor containing the matrix (possibly batched) of kernel applications to pairs from inputs x1 and x2, over all tasks. If the kernel parameters' batch shape is Bk then the shape of the LinearOperator resulting from this method call is broadcast(Bk, B1, B2) + [N * e1, N * e2].

Examples

import tensorflow_probability as tfp

base_kernel = tfp.math.psd_kernels.ExponentiatedQuadratic(1., 1.)
kernel = tfs.psd_kernels.SomeVectorKernel(num_tasks=2, base_kernel)
kernel.batch_shape
# ==> []

# Our inputs are two lists of 3-D vectors
x = np.ones([3, 3], np.float32)
y = np.ones([3, 3], np.float32)
kernel.matrix_over_all_tasks(x, y).shape
# ==> [6, 6]

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

<a target="_blank" class="external" href="https://github.com/tensorflow/probability/blob/v0.23.0/tensorflow_probability/python/math/psd_kernels/positive_semidefinite_kernel.py#L281-L300">View source</a>

<pre class="devsite-click-to-copy prettyprint lang-py tfo-signature-link">
<code>@classmethod</code>
<code>parameter_properties(
    dtype=tf.float32
)
</code></pre>

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.

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

<tr>
<td>
`dtype`
</td>
<td>
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.
</td>
</tr>
</table>



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

<tr>
<td>
`parameter_properties`
</td>
<td>
A
`str -> `tfp.python.internal.parameter_properties.ParameterProperties`
dict mapping constructor argument names to `ParameterProperties`
instances.
</td>
</tr>
</table>



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

<a target="_blank" class="external" href="https://github.com/tensorflow/probability/blob/v0.23.0/tensorflow_probability/python/math/psd_kernels/positive_semidefinite_kernel.py#L902-L1065">View source</a>

<pre class="devsite-click-to-copy prettyprint lang-py tfo-signature-link">
<code>tensor(
    x1, x2, x1_example_ndims, x2_example_ndims, name=&#x27;tensor&#x27;
)
</code></pre>

Construct (batched) tensors 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>
(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.
</td>
</tr><tr>
<td>
`x2`
</td>
<td>
(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.
</td>
</tr><tr>
<td>
`x1_example_ndims`
</td>
<td>
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.
</td>
</tr><tr>
<td>
`x2_example_ndims`
</td>
<td>
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.
</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 (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.
</td>
</tr>

</table>


#### Examples

First, consider a kernel with a single scalar parameter.

```python
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.

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]

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
)