View source on GitHub |
Input transformed kernel.
Inherits From: AutoCompositeTensorPsdKernel
, PositiveSemidefiniteKernel
, AutoCompositeTensor
tfp.math.psd_kernels.FeatureTransformed(
kernel,
transformation_fn,
validate_args=False,
parameters=None,
name='FeatureTransformed'
)
Given a kernel k
and function f
, compute k_{new}(x, y) = k(f(x), f(y))
.
Examples
import tensorflow as tf
import tensorflow_probability as tfp
from tensorflow_probability.positive_semidefinite_kernel.internal import util
tfpk = tfp.math.psd_kernels
base_kernel = tfpk.ExponentiatedQuadratic(amplitude=2., length_scale=1.)
- Identity function.
# This is the same as base_kernel
same_kernel = tfpk.FeatureTransformed(
base_kernel,
transformation_fn=lambda x, _, _: x)
- Exponential transformation.
exp_kernel = tfpk.FeatureTransformed(
base_kernel,
transformation_fn=lambda x, _, _: tf.exp(x))
- Transformation with broadcasting parameters.
# Exponentiate inputs
p = np.random.uniform(low=2., high=3., size=[10, 2])
def inputs_to_power(x, feature_ndims, param_expansion_ndims):
# Make sure we account for extra feature dimensions for
# broadcasting purposes.
power = util.pad_shape_with_ones(
p,
ndims=feature_ndims + param_expansion_ndims,
start=-(feature_ndims + 1))
return x ** power
power_kernel = tfpk.FeatureTransformed(
base_kernel, transformation_fn=inputs_to_power)
<!-- Tabular view -->
<table class="responsive fixed orange">
<colgroup><col width="214px"><col></colgroup>
<tr><th colspan="2"><h2 class="add-link">Args</h2></th></tr>
<tr>
<td>
`kernel`<a id="kernel"></a>
</td>
<td>
`PositiveSemidefiniteKernel` instance. Inputs are transformed and
passed in to this kernel. Parameters to `kernel` must be broadcastable
with parameters of `transformation_fn`.
</td>
</tr><tr>
<td>
`transformation_fn`<a id="transformation_fn"></a>
</td>
<td>
Callable. `transformation_fn` takes in an input
`Tensor`, a Python integer representing the number of feature
dimensions, and a Python integer representing the
`param_expansion_ndims` arg of `_apply`. Computations in
`transformation_fn` must be broadcastable with parameters of `kernel`.
</td>
</tr><tr>
<td>
`validate_args`<a id="validate_args"></a>
</td>
<td>
If `True`, parameters are checked for validity despite
possibly degrading runtime performance
</td>
</tr><tr>
<td>
`parameters`<a id="parameters"></a>
</td>
<td>
When subclassing, a dict of constructor arguments.
</td>
</tr><tr>
<td>
`name`<a id="name"></a>
</td>
<td>
Python `str` name prefixed to Ops created by this class.
</td>
</tr>
</table>
<!-- Tabular view -->
<table class="responsive fixed orange">
<colgroup><col width="214px"><col></colgroup>
<tr><th colspan="2"><h2 class="add-link">Attributes</h2></th></tr>
<tr>
<td>
`batch_shape`<a id="batch_shape"></a>
</td>
<td>
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`.
</td>
</tr><tr>
<td>
`dtype`<a id="dtype"></a>
</td>
<td>
(Nested) dype over which the kernel operates.
</td>
</tr><tr>
<td>
`feature_ndims`<a id="feature_ndims"></a>
</td>
<td>
The number of feature dimensions.
Kernel functions generally act on pairs of inputs from some space like
```none
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
name
name_scope
tf.name_scope
instance for this class.
non_trainable_variables
parameters
PSDKernel
.
submodules
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
transformation_fn
kernel
.
validate_args
bool
indicating possibly expensive checks are enabled.
variables
Methods
apply
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
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
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
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
@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.ParameterPropertiesdict mapping constructor argument names to ParameterProperties`
instances.
|
tensor
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 | |
---|---|
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.
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.Variable
s and tf.Tensor
s 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__
__add__(
k
)
__getitem__
__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__
__iter__()
__mul__
__mul__(
k
)
__radd__
__radd__(
lhs
)