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tfp.positive_semidefinite_kernels.SchurComplement

Class SchurComplement

The SchurComplement kernel.

Inherits From: PositiveSemidefiniteKernel

Defined in python/positive_semidefinite_kernels/schur_complement.py.

Given a block matrix M = [[A, B], [C, D]], the Schur complement of D in M is written M / D = A - B @ Inverse(D) @ C.

This class represents a PositiveSemidefiniteKernel whose behavior is as follows. We compute a matrix, analogous to D in the above definition, by calling base_kernel.matrix(fixed_inputs, fixed_inputs). Then given new input locations x and y, we can construct the remaining pieces of M above, and compute the Schur complement of D in M (see Mathematical Details, below).

Notably, this kernel uses a bijector (Invert(CholeskyOuterProduct)), as an intermediary for the requisite matrix solve, which means we get a caching benefit after the first use.

Mathematical Details

Suppose we have a kernel k and some fixed collection of inputs Z = [z0, z1, ..., zN]. Given new inputs x and y, we can form a block matrix

   M = [
     [k(x, y), k(x, z0), ..., k(x, zN)],
     [k(z0, y), k(z0, z0), ..., k(z0, zN)],
     ...,
     [k(zN, y), k(z0, zN), ..., k(zN, zN)],
   ]

We might write this, so as to emphasize the block structure,

   M = [
     [xy, xZ],
     [yZ^T, ZZ],
   ],

   xy = [k(x, y)]
   xZ = [k(x, z0), ..., k(x, zN)]
   yZ = [k(y, z0), ..., k(y, zN)]
   ZZ = "the matrix of k(zi, zj)'s"

Then we have the definition of this kernel's apply method:

schur_comp.apply(x, y) = xy - xZ @ ZZ^{-1} @ yZ^T

and similarly, if x and y are collections of inputs.

As with other PSDKernels, the apply method acts as a (possibly vectorized) scalar function of 2 inputs. Given a single x and y, apply will yield a scalar output. Given two (equal size!) collections X and Y, it will yield another (equal size!) collection of scalar outputs.

Examples

Here's a simple example usage, with no particular motivation.

from tensorflow_probability import positive_semidefinite_kernels as psd_kernel

base_kernel = psd_kernel.ExponentiatedQuadratic(amplitude=np.float64(1.))
# 3 points in 1-dimensional space (shape [3, 1]).
z = [[0.], [3.], [4.]]

schur_kernel = psd_kernel.SchurComplement(
    base_kernel=base_kernel,
    fixed_inputs=z)

# Two individual 1-d points
x = [1.]
y = [2.]
print(schur_kernel.apply(x, y))
# ==> k(x, y) - k(x, z) @ Inverse(k(z, z)) @ k(z, y)

A more motivating application of this kernel is in constructing a Gaussian process that is conditioned on some observed data.

from tensorflow_probability import distributions as tfd
from tensorflow_probability import positive_semidefinite_kernels as psd_kernel

base_kernel = psd_kernel.ExponentiatedQuadratic(amplitude=np.float64(1.))
observation_index_points = np.random.uniform(-1., 1., [50, 1])
observations = np.sin(2 * np.pi * observation_index_points[..., 0])

posterior_kernel = psd_kernel.SchurComplement(
    base_kernel=base_kernel,
    fixed_inputs=observation_index_points)

# Assume we use a zero prior mean, and compute the posterior mean.
def posterior_mean_fn(x):
  k_x_obs_linop = tf.linalg.LinearOperatorFullMatrix(
      base_kernel.matrix(x, observation_index_points))
  chol_linop = tf.linalg.LinearOperatorLowerTriangular(
      posterior_kernel.divisor_matrix_cholesky)

  return k_x_obs_linop.matvec(
      chol_linop.solvevec(
          chol_linop.solvevec(observations),
          adjoint=True))

# Construct the GP posterior distribution at some new points.
gp_posterior = tfp.distributions.GaussianProcess(
    index_points=np.linspace(-1., 1., 100)[..., np.newaxis],
    kernel=posterior_kernel,
    mean_fn=posterior_mean_fn)

# Draw 5 samples on the above 100-point grid
samples = gp_posterior.sample(5)

__init__

__init__(
    base_kernel,
    fixed_inputs,
    diag_shift=None,
    validate_args=False,
    name='SchurComplement'
)

Construct a SchurComplement kernel instance.

Args:

  • base_kernel: A PositiveSemidefiniteKernel instance, the kernel used to build the block matrices of which this kernel computes the Schur complement.
  • fixed_inputs: A Tensor, representing a collection of inputs. The Schur complement that this kernel computes comes from a block matrix, whose bottom-right corner is derived from base_kernel.matrix(fixed_inputs, fixed_inputs), and whose top-right and bottom-left pieces are constructed by computing the base_kernel at pairs of input locations together with these fixed_inputs. fixed_inputs is allowed to be an empty collection (either None or having a zero shape entry), in which case the kernel falls back to the trivial application of base_kernel to inputs. See class-level docstring for more details on the exact computation this does; fixed_inputs correspond to the Z structure discussed there. fixed_inputs is assumed to have shape [b1, ..., bB, N, f1, ..., fF] where the b's are batch shape entries, the f's are feature_shape entries, and N is the number of fixed inputs. Use of this kernel entails a 1-time O(N^3) cost of computing the Cholesky decomposition of the k(Z, Z) matrix. The batch shape elements of fixed_inputs must be broadcast compatible with base_kernel.batch_shape.
  • diag_shift: A floating point scalar to be added to the diagonal of the divisor_matrix before computing its Cholesky.
  • validate_args: If True, parameters are checked for validity despite possibly degrading runtime performance. Default value: False
  • name: Python str name prefixed to Ops created by this class. Default value: "SchurComplement"

Properties

base_kernel

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.

cholesky_bijector

divisor_matrix

divisor_matrix_cholesky

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.

fixed_inputs

name

Name prepended to all ops created by this class.

Methods

__add__

__add__(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], where B may be zero (ie, no batching) and F (number of feature dimensions) must equal the kernel's feature_ndims property. Batch shape must broadcast with the batch shape of x2 and with the kernel's parameters.
  • x2: Tensor input to the second positional parameter of the kernel, shape [c1, ..., cC, f1, ..., fF], where C may be zero (ie, no batching) and F (number of feature dimensions) must equal the kernel's feature_ndims property. Batch shape must broadcast with the batch shape of x1 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], where B may be zero (ie, no batching), e1 is an integer greater than zero, and F (number of feature dimensions) must equal the kernel's feature_ndims property. Batch shape must broadcast with the batch shape of x2 and with the kernel's parameters after parameter expansion (see param_expansion_ndims argument).
  • x2: Tensor input to the second positional parameter of the kernel, shape [c1, ..., cC, e2, f1, ..., fF], where C may be zero (ie, no batching), e2 is an integer greater than zero, and F (number of feature dimensions) must equal the kernel's feature_ndims property. Batch shape must broadcast with the batch shape of x1 and with the kernel's parameters after parameter expansion (see param_expansion_ndims argument).

Returns:

Tensor containing (batch of) matrices of kernel applications to pairs from inputsx1andx2. If the kernel parameters' batch shape is[k1, ..., kK], then the shape of the resultingTensorisbroadcast([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 = tfpk.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.