tfp.experimental.linalg.LinearOperatorInterpolatedPSDKernel

Structured interpolation to approximate a large matrix.

This implements a component of the Structured Kernel Interpolation [1] algorithm. We approximate the pairwise kernel values for two inputs x1, x2 with the following product:

k(x1, x2) = r(x1) @ k(u, u) @ r(x2)^T

where u is a set of points regularly spaced on grid and r is an interpolation matrix. In short, instead of evaluating kernel on x1 and x2, we evaluate it on u, which is chosen to contain fewer points than x1 and x2 and the interpolate using the interpolation matrix.

This construction lets us compute matrix products efficiently. If x1/x2 are of shape O(n) and u is of shape O(m), this reduces the compute cost and memory to O(n + m^2) from O(n^2).

In practice, the interpolation matrix is implicitly defined using interp_fn, of which tfp.math.batch_interp_regular_nd_grid linear interpolation is a prototypical example. When x1 == x2 we can relatively cheaply compute the diagonal of that matrix exactly, to preserve positive-semi-definiteness.

Since we rely on a dense grid u, this method works best when the the kernel inputs are low dimensional (2 or 3).

References

[1]: Wilson, A. G., & Nickisch, H. Kernel Interpolation for Scalable Structured Gaussian Processes (KISS-GP). 2005. arXiv. http://arxiv.org/abs/1503.01057

kernel Instance of tfp.math.psd_kernels.PositiveSemidefiniteKernel'. Must havefeature_ndims == 1. </td> </tr><tr> <td>bounds_min<a id="bounds_min"></a> </td> <td> Floating pointTensor. Minimum bounds of the interpolation grid. Shape: [D] </td> </tr><tr> <td>bounds_max<a id="bounds_max"></a> </td> <td> Floating pointTensor. Maximum bounds of the interpolation grid. Shape: [D] </td> </tr><tr> <td>num_interpolation_points<a id="num_interpolation_points"></a> </td> <td> Python integer. Number of inducing points per dimension. </td> </tr><tr> <td>x1<a id="x1"></a> </td> <td> Floating pointTensor. First input to the kernel. Shape: [B, R, D] </td> </tr><tr> <td>x2<a id="x2"></a> </td> <td> Optional floating pointTensor. Second input to the kernel. Shape: [B, C, D] Omit this argument to statically indicate that this operator is square, self-adjoint, positive definite and non-singular. </td> </tr><tr> <td>interp_fn<a id="interp_fn"></a> </td> <td> Interpolation function with an API same as <a href="../../../tfp/math/batch_interp_regular_nd_grid"><code>tfp.math.batch_interp_regular_nd_grid</code></a>. This must implicitly define a matrix as in the class docstring, i.e. it must be linear in the value of they_refargument. </td> </tr><tr> <td>diag_shift<a id="diag_shift"></a> </td> <td> Optional floating pointTensor. A diagonal offset to add to the resultant matrix. Must beNoneif the operator is not square and self-adjoint. Shape: [B] </td> </tr><tr> <td>is_non_singular<a id="is_non_singular"></a> </td> <td> Expect that this operator is non-singular. </td> </tr><tr> <td>is_self_adjoint<a id="is_self_adjoint"></a> </td> <td> Expect that this operator is equal to its hermitian transpose. Ifdtypeis real, this is equivalent to being symmetric. </td> </tr><tr> <td>is_positive_definite<a id="is_positive_definite"></a> </td> <td> Expect that this operator is positive definite, meaning the quadratic formx^H A xhas positive real part for all nonzerox. </td> </tr><tr> <td>is_square<a id="is_square"></a> </td> <td> Expect that this operator acts like square [B] matrices. </td> </tr><tr> <td>name` Name of the operator. Default: LinearOperatorInterpolatedPSDKernel

H Returns the adjoint of the current LinearOperator.

Given A representing this LinearOperator, return A*. Note that calling self.adjoint() and self.H are equivalent.

batch_shape TensorShape of batch dimensions of this LinearOperator.

If this operator acts like the batch matrix A with A.shape = [B1,...,Bb, M, N], then this returns TensorShape([B1,...,Bb]), equivalent to A.shape[:-2]

bounds_max

bounds_min

diag_shift

domain_dimension Dimension (in the sense of vector spaces) of the domain of this operator.

If this operator acts like the batch matrix A with A.shape = [B1,...,Bb, M, N], then this returns N.

dtype The DType of Tensors handled by this LinearOperator.
graph_parents List of graph dependencies of this LinearOperator. (deprecated)

is_non_singular

is_positive_definite

is_self_adjoint

is_square Return True/False depending on if this operator is square.
kernel

name Name prepended to all ops created by this LinearOperator.
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_interpolation_points

parameters Dictionary of parameters used to instantiate this LinearOperator.
range_dimension Dimension (in the sense of vector spaces) of the range of this operator.

If this operator acts like the batch matrix A with A.shape = [B1,...,Bb, M, N], then this returns M.

shape TensorShape of this LinearOperator.

If this operator acts like the batch matrix A with A.shape = [B1,...,Bb, M, N], then this returns TensorShape([B1,...,Bb, M, N]), equivalent to A.shape.

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

tensor_rank Rank (in the sense of tensors) of matrix corresponding to this operator.

If this operator acts like the batch matrix A with A.shape = [B1,...,Bb, M, N], then this returns b + 2.

trainable_variables Sequence of trainable variables owned by this module and its submodules.

variables Sequence of variables owned by this module and its submodules.
x1

x2

Methods

add_to_tensor

Add matrix represented by this operator to x. Equivalent to A + x.

Args
x Tensor with same dtype and shape broadcastable to self.shape.
name A name to give this Op.

Returns
A Tensor with broadcast shape and same dtype as self.

adjoint

Returns the adjoint of the current LinearOperator.

Given A representing this LinearOperator, return A*. Note that calling self.adjoint() and self.H are equivalent.

Args
name A name for this Op.

Returns
LinearOperator which represents the adjoint of this LinearOperator.

assert_non_singular

Returns an Op that asserts this operator is non singular.

This operator is considered non-singular if

ConditionNumber < max{100, range_dimension, domain_dimension} * eps,
eps := np.finfo(self.dtype.as_numpy_dtype).eps

Args
name A string name to prepend to created ops.

Returns
An Assert Op, that, when run, will raise an InvalidArgumentError if the operator is singular.

assert_positive_definite

Returns an Op that asserts this operator is positive definite.

Here, positive definite means that the quadratic form x^H A x has positive real part for all nonzero x. Note that we do not require the operator to be self-adjoint to be positive definite.

Args
name A name to give this Op.

Returns
An Assert Op, that, when run, will raise an InvalidArgumentError if the operator is not positive definite.

assert_self_adjoint

Returns an Op that asserts this operator is self-adjoint.

Here we check that this operator is exactly equal to its hermitian transpose.

Args
name A string name to prepend to created ops.

Returns
An Assert Op, that, when run, will raise an InvalidArgumentError if the operator is not self-adjoint.

batch_shape_tensor

Shape of batch dimensions of this operator, determined at runtime.

If this operator acts like the batch matrix A with A.shape = [B1,...,Bb, M, N], then this returns a Tensor holding [B1,...,Bb].

Args
name A name for this Op.

Returns
int32 Tensor

cholesky

Returns a Cholesky factor as a LinearOperator.

Given A representing this LinearOperator, if A is positive definite self-adjoint, return L, where A = L L^T, i.e. the cholesky decomposition.

Args
name A name for this Op.

Returns
LinearOperator which represents the lower triangular matrix in the Cholesky decomposition.

Raises
ValueError When the LinearOperator is not hinted to be positive definite and self adjoint.

cond

Returns the condition number of this linear operator.

Args
name A name for this Op.

Returns
Shape [B1,...,Bb] Tensor of same dtype as self.

determinant

Determinant for every batch member.

Args
name A name for this Op.

Returns
Tensor with shape self.batch_shape and same dtype as self.

Raises
NotImplementedError If self.is_square is False.

diag_part

Efficiently get the [batch] diagonal part of this operator.

If this operator has shape [B1,...,Bb, M, N], this returns a Tensor diagonal, of shape [B1,...,Bb, min(M, N)], where diagonal[b1,...,bb, i] = self.to_dense()[b1,...,bb, i, i].

my_operator = LinearOperatorDiag([1., 2.])

# Efficiently get the diagonal
my_operator.diag_part()
==> [1., 2.]

# Equivalent, but inefficient method
tf.linalg.diag_part(my_operator.to_dense())
==> [1., 2.]

Args
name A name for this Op.

Returns
diag_part A Tensor of same dtype as self.

domain_dimension_tensor

Dimension (in the sense of vector spaces) of the domain of this operator.

Determined at runtime.

If this operator acts like the batch matrix A with A.shape = [B1,...,Bb, M, N], then this returns N.

Args
name A name for this Op.

Returns
int32 Tensor

eigvals

Returns the eigenvalues of this linear operator.

If the operator is marked as self-adjoint (via is_self_adjoint) this computation can be more efficient.

Args
name A name for this Op.

Returns
Shape [B1,...,Bb, N] Tensor of same dtype as self.

inverse

Returns the Inverse of this LinearOperator.

Given A representing this LinearOperator, return a LinearOperator representing A^-1.

Args
name A name scope to use for ops added by this method.

Returns
LinearOperator representing inverse of this matrix.

Raises
ValueError When the LinearOperator is not hinted to be non_singular.

log_abs_determinant

Log absolute value of determinant for every batch member.

Args
name A name for this Op.

Returns
Tensor with shape self.batch_shape and same dtype as self.

Raises
NotImplementedError If self.is_square is False.

matmul

Transform [batch] matrix x with left multiplication: x --> Ax.

# Make an operator acting like batch matrix A.  Assume A.shape = [..., M, N]
operator = LinearOperator(...)
operator.shape = [..., M, N]

X = ... # shape [..., N, R], batch matrix, R > 0.

Y = operator.matmul(X)
Y.shape
==> [..., M, R]

Y[..., :, r] = sum_j A[..., :, j] X[j, r]

Args
x LinearOperator or Tensor with compatible shape and same dtype as self. See class docstring for definition of compatibility.
adjoint Python bool. If True, left multiply by the adjoint: A^H x.
adjoint_arg Python bool. If True, compute A x^H where x^H is the hermitian transpose (transposition and complex conjugation).
name A name for this Op.

Returns
A LinearOperator or Tensor with shape [..., M, R] and same dtype as self.

matvec

Transform [batch] vector x with left multiplication: x --> Ax.

# Make an operator acting like batch matrix A.  Assume A.shape = [..., M, N]
operator = LinearOperator(...)

X = ... # shape [..., N], batch vector

Y = operator.matvec(X)
Y.shape
==> [..., M]

Y[..., :] = sum_j A[..., :, j] X[..., j]

Args
x Tensor with compatible shape and same dtype as self. x is treated as a [batch] vector meaning for every set of leading dimensions, the last dimension defines a vector. See class docstring for definition of compatibility.
adjoint Python bool. If True, left multiply by the adjoint: A^H x.
name A name for this Op.

Returns
A Tensor with shape [..., M] and same dtype as self.

range_dimension_tensor

Dimension (in the sense of vector spaces) of the range of this operator.

Determined at runtime.

If this operator acts like the batch matrix A with A.shape = [B1,...,Bb, M, N], then this returns M.

Args
name A name for this Op.

Returns
int32 Tensor

row

View source

Gets a row from the dense operator.

Args
index The index (indices) of the row[s] to get, may be scalar or up to batch shape.

Returns
rows Row[s] of the matrix, with shape (...batch_shape..., num_cols). Effectively the same as operator.to_dense()[..., index, :] for a scalar index, analogous to gather for non-scalar.

shape_tensor

Shape of this LinearOperator, determined at runtime.

If this operator acts like the batch matrix A with A.shape = [B1,...,Bb, M, N], then this returns a Tensor holding [B1,...,Bb, M, N], equivalent to tf.shape(A).

Args
name A name for this Op.

Returns
int32 Tensor

solve

Solve (exact or approx) R (batch) systems of equations: A X = rhs.

The returned Tensor will be close to an exact solution if A is well conditioned. Otherwise closeness will vary. See class docstring for details.

Examples:

# Make an operator acting like batch matrix A.  Assume A.shape = [..., M, N]
operator = LinearOperator(...)
operator.shape = [..., M, N]

# Solve R > 0 linear systems for every member of the batch.
RHS = ... # shape [..., M, R]

X = operator.solve(RHS)
# X[..., :, r] is the solution to the r'th linear system
# sum_j A[..., :, j] X[..., j, r] = RHS[..., :, r]

operator.matmul(X)
==> RHS

Args
rhs Tensor with same dtype as this operator and compatible shape. rhs is treated like a [batch] matrix meaning for every set of leading dimensions, the last two dimensions defines a matrix. See class docstring for definition of compatibility.
adjoint Python bool. If True, solve the system involving the adjoint of this LinearOperator: A^H X = rhs.
adjoint_arg Python bool. If True, solve A X = rhs^H where rhs^H is the hermitian transpose (transposition and complex conjugation).
name A name scope to use for ops added by this method.

Returns
Tensor with shape [...,N, R] and same dtype as rhs.

Raises
NotImplementedError If self.is_non_singular or is_square is False.

solvevec

Solve single equation with best effort: A X = rhs.

The returned Tensor will be close to an exact solution if A is well conditioned. Otherwise closeness will vary. See class docstring for details.

Examples:

# Make an operator acting like batch matrix A.  Assume A.shape = [..., M, N]
operator = LinearOperator(...)
operator.shape = [..., M, N]

# Solve one linear system for every member of the batch.
RHS = ... # shape [..., M]

X = operator.solvevec(RHS)
# X is the solution to the linear system
# sum_j A[..., :, j] X[..., j] = RHS[..., :]

operator.matvec(X)
==> RHS

Args
rhs Tensor with same dtype as this operator. rhs is treated like a [batch] vector meaning for every set of leading dimensions, the last dimension defines a vector. See class docstring for definition of compatibility regarding batch dimensions.
adjoint Python bool. If True, solve the system involving the adjoint of this LinearOperator: A^H X = rhs.
name A name scope to use for ops added by this method.

Returns
Tensor with shape [...,N] and same dtype as rhs.

Raises
NotImplementedError If self.is_non_singular or is_square is False.

tensor_rank_tensor

Rank (in the sense of tensors) of matrix corresponding to this operator.

If this operator acts like the batch matrix A with A.shape = [B1,...,Bb, M, N], then this returns b + 2.

Args
name A name for this Op.

Returns
int32 Tensor, determined at runtime.

to_dense

Return a dense (batch) matrix representing this operator.

trace

Trace of the linear operator, equal to sum of self.diag_part().

If the operator is square, this is also the sum of the eigenvalues.

Args
name A name for this Op.

Returns
Shape [B1,...,Bb] Tensor of same dtype as self.

with_name_scope

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.

__getitem__

__matmul__