graphdot.linalg.low_rank module

Low-rank approximation of square matrices.

class graphdot.linalg.low_rank.LATR(lhs, rhs)

Bases: graphdot.linalg.low_rank.LowRankBase

Represents an N-by-N square matrix A as :py:math:`L \cdot R`, where L and R are N-by-k and k-by-N (:py:math:`k << N`) rectangular matrices.

T

diagonal()

lhs

quadratic(a, b)

Computes a @ X @ b.

quadratic_diag(a, b)

Computes diag(a @ X @ b).

rhs

todense()

trace()

class graphdot.linalg.low_rank.LLT(X, rcond=0, mode='truncate')

Bases: graphdot.linalg.low_rank.LATR

A special case of factor approximation where the matrix is symmetric and positive-semidefinite. In this case, the matrix can be represented as :py:math:`L \cdot L^\mathsf{T}` from a spectral decomposition.

cond()

diagonal()

lhs

logdet()

pinv()

rhs

class graphdot.linalg.low_rank.LowRankBase

Bases: object

__add__(other)

class graphdot.linalg.low_rank.Sum(factors)

Bases: graphdot.linalg.low_rank.LowRankBase

Represents summations of factor approximations. Due to the bilinear nature of matrix inner product, it is best to store the summation as-is so as to preserve the low-rank structure of the matrices.

T

diagonal()

quadratic(a, b)

Computes a @ X @ b.

todense()

trace()

graphdot.linalg.low_rank.add(A, B)

graphdot.linalg.low_rank.dot(X, Y=None, method='auto', rcond=0, mode='truncate')

A utility method that creates low-rank matrices :py:math:`A \doteq X \cdot Y`.

Parameters:

• X (ndarray) – Left hand side of the product.
• Y (ndarray) – Right hand side of the product. If None, Y will be assumed to be the transposition of X.
• method ('auto' or 'direct' or 'spectral') – If ‘direct’, store the matrix as the product of X and Y. If ‘spectral’ and Y is None, store the matrix as the product of the singular vectors and singular values of X. ‘auto’ is equivalent to ‘spectral’ when Y is None and ‘direct’ otherwise.
• rcond (float) – Threshold for small singular values when method is ‘spectral’.
• mode ('truncate' or 'clamp') – Determines how small singular values of the original matrix are handled. For ‘truncate’, small values are discarded; for ‘clamp’, they are fixed to be the product of the largest singular value and rcond.

graphdot.linalg.low_rank.matmul(A, B)

graphdot.linalg.low_rank.pinvh(A: graphdot.linalg.low_rank.LATR, d, k='auto', rcond=1e-10, mode='truncate')

Calculate the low-rank approximated pseudoinverse of a low-rank symmetric matrix with optional diagonal regularization.

Parameters:

• A (LATR.) – A low-rank symmetric positive semidefinite matrix.
• d (array) – An optional regularization vector that will be added elementwise to the diagonal of A.
• k (int or 'auto') – Number of eigenvalues to resolve. If ‘auto’, k will be set to be the sum of the rank of A plus the number of nonzeros in d.
• rcond (float) – Cutoff for small eigenvalues. Eigenvalues less than or equal to rcond * largest_eigenvalue and associated eigenvators are discarded in forming the pseudoinverse.
• mode (str) – Determines how small eigenvalues of the original matrix are handled. For ‘truncate’, small eigenvalues are discarded; for ‘clamp’, they are fixed to be the product of the largest eigenvalue and rcond.

Returns:

Ainv – A low-rank representation of the pseudoinverse of A.

Return type:

LLT.

graphdot.linalg.low_rank.sub(A, B)