Class MaternThreeHalves
Inherits From: PositiveSemidefiniteKernel
Matern Kernel with parameter 3/2.
This kernel is part of the Matern family of kernels, with parameter 3/2.
z = sqrt(3) * ||x - y|| / length_scale
k(x, y) = (1 + z) * exp(-z)
where the double-bars represent vector length (i.e. Euclidean, or L2 Norm).
This kernel, acts over the space S = R^(D1 x D2 .. x Dd)
.
__init__
__init__(
amplitude=None,
length_scale=None,
feature_ndims=1,
validate_args=False,
name='MaternThreeHalves'
)
Construct a MaternThreeHalves kernel instance.
Args:
amplitude
: Positive floating pointTensor
that controls the maximum value of the kernel. Must be broadcastable withlength_scale
and inputs toapply
andmatrix
methods. A value ofNone
is treated like 1.length_scale
: Positive floating pointTensor
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 withamplitude
and inputs toapply
andmatrix
methods. A value ofNone
is treated like 1.feature_ndims
: Pythonint
number of rightmost dims to include in the squared difference norm in the exponential.validate_args
: IfTrue
, parameters are checked for validity despite possibly degrading runtime performancename
: Pythonstr
name prefixed to Ops created by this class.
Properties
amplitude
Amplitude parameter.
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.
Returns:
TensorShape
instance describing the fully broadcast shape of all
kernel parameters.
dtype
DType 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.
Returns:
The number of feature dimensions (feature rank) of this kernel.
length_scale
Length scale parameter.
name
Name prepended to all ops created by this class.
Methods
__add__
__add__(k)
__iadd__
__iadd__(k)
__imul__
__imul__(k)
__mul__
__mul__(k)
apply
apply(
x1,
x2
)
Apply the kernel function to a pair of (batches of) inputs.
Args:
x1
:Tensor
input to the first positional parameter of the kernel, of shape[b1, ..., bB, f1, ..., fF]
, whereB
may be zero (ie, no batching) andF
(number of feature dimensions) must equal the kernel'sfeature_ndims
property. Batch shape must broadcast with the batch shape ofx2
and with the kernel's parameters.x2
:Tensor
input to the second positional parameter of the kernel, shape[c1, ..., cC, f1, ..., fF]
, whereC
may be zero (ie, no batching) andF
(number of feature dimensions) must equal the kernel'sfeature_ndims
property. Batch shape must broadcast with the batch shape ofx1
and with the kernel's parameters.
Returns:
Tensor
containing the (batch of) results of applying the kernel function
to inputs x1
and x2
. If the kernel parameters' batch shape is
[k1, ..., kK]
then the shape of the Tensor
resulting from this method
call is broadcast([b1, ..., bB], [c1, ..., cC], [k1, ..., kK])
.
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. Given the same kernel and, say, batched inputs of shape
[b1, ..., bB, f1, ..., fF]
, it will yield a batch of scalars of shape
[b1, ..., bB]
.
Examples
import tensorflow_probability as tfp
# Suppose `SomeKernel` acts on vectors (rank-1 tensors)
scalar_kernel = tfp.positive_semidefinite_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.positive_semidefinite_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 by giving the parameter a shape of
[2, 1]
which will correctly broadcast with [5]
to yield [2, 5]
:
batch_kernel = tfp.positive_semidefinite_kernels.SomeKernel(
param=[[.2], [.5]])
batch_kernel.batch_shape
# ==> [2, 1]
batch_kernel.apply(x, y).shape
# ==> [2, 5]
batch_shape_tensor
batch_shape_tensor()
The batch_shape property of a PositiveSemidefiniteKernel as a Tensor
.
Returns:
Tensor
which evaluates to a vector of integers which are the
fully-broadcast shapes of the kernel parameters.
matrix
matrix(
x1,
x2
)
Construct (batched) matrices from (batches of) collections of inputs.
Args:
x1
:Tensor
input to the first positional parameter of the kernel, of shape[b1, ..., bB, e1, f1, ..., fF]
, whereB
may be zero (ie, no batching), e1 is an integer greater than zero, andF
(number of feature dimensions) must equal the kernel'sfeature_ndims
property. Batch shape must broadcast with the batch shape ofx2
and with the kernel's parameters after parameter expansion (seeparam_expansion_ndims
argument).x2
:Tensor
input to the second positional parameter of the kernel, shape[c1, ..., cC, e2, f1, ..., fF]
, whereC
may be zero (ie, no batching), e2 is an integer greater than zero, andF
(number of feature dimensions) must equal the kernel'sfeature_ndims
property. Batch shape must broadcast with the batch shape ofx1
and with the kernel's parameters after parameter expansion (seeparam_expansion_ndims
argument).
Returns:
Tensor containing (batch of) matrices of kernel applications to pairs
from inputs
x1and
x2. If the kernel parameters' batch shape is
[k1, ..., kK], then the shape of the resulting
Tensoris
broadcast([b1, ..., bB], [c1, ..., cC], [k1, ..., kK]) + [e1, e2]`.
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).
N.B., this method can only be used to compute the pairwise application of
the kernel function on rank-1 collections. E.g., it does support inputs of
shape [e1, f]
and [e2, f]
, yielding a matrix of shape [e1, e2]
. It
does not support inputs of shape [e1, e2, f]
and [e3, e4, f]
, yielding
a Tensor
of shape [e1, e2, e3, e4]
. To do this, one should instead
reshape the inputs and pass them to apply
, e.g.:
k = psd_kernels.SomeKernel()
t1 = tf.placeholder([4, 4, 3], tf.float32)
t2 = tf.placeholder([5, 5, 3], tf.float32)
k.apply(
tf.reshape(t1, [4, 4, 1, 1, 3]),
tf.reshape(t2, [1, 1, 5, 5, 3])).shape
# ==> [4, 4, 5, 5, 3]
matrix
is a special case of the above, where there is only one example
dimension; indeed, its implementation looks almost exactly like the above
(reshaped inputs passed to the private version of _apply
).
Examples
First, consider a kernel with a single scalar parameter.
import tensorflow_probability as tfp
scalar_kernel = tfp.positive_semidefinite_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.positive_semidefinite_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.positive_semidefinite_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.positive_semidefinite_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.positive_semidefinite_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.