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# Source code for torch._lowrank

"""Implement various linear algebra algorithms for low rank matrices."""

__all__ = ["svd_lowrank", "pca_lowrank"]

from typing import Optional, Tuple

import torch
from torch import _linalg_utils as _utils, Tensor
from torch.overrides import handle_torch_function, has_torch_function

def get_approximate_basis(
A: Tensor,
q: int,
niter: Optional[int] = 2,
M: Optional[Tensor] = None,
) -> Tensor:
"""Return tensor :math:Q with :math:q orthonormal columns such
that :math:Q Q^H A approximates :math:A. If :math:M is
specified, then :math:Q is such that :math:Q Q^H (A - M)
approximates :math:A - M. without instantiating any tensors
of the size of :math:A or :math:M.

.. note:: The implementation is based on the Algorithm 4.4 from
Halko et al., 2009.

.. note:: For an adequate approximation of a k-rank matrix
:math:A, where k is not known in advance but could be
estimated, the number of :math:Q columns, q, can be
choosen according to the following criteria: in general,
:math:k <= q <= min(2*k, m, n). For large low-rank
matrices, take :math:q = k + 5..10.  If k is
relatively small compared to :math:min(m, n), choosing
:math:q = k + 0..2 may be sufficient.

.. note:: To obtain repeatable results, reset the seed for the
pseudorandom number generator

Args::
A (Tensor): the input tensor of size :math:(*, m, n)

q (int): the dimension of subspace spanned by :math:Q
columns.

niter (int, optional): the number of subspace iterations to
conduct; niter must be a
nonnegative integer. In most cases, the
default value 2 is more than enough.

M (Tensor, optional): the input tensor's mean of size
:math:(*, m, n).

References::
- Nathan Halko, Per-Gunnar Martinsson, and Joel Tropp, Finding
structure with randomness: probabilistic algorithms for
constructing approximate matrix decompositions,
arXiv:0909.4061 [math.NA; math.PR], 2009 (available at
arXiv <http://arxiv.org/abs/0909.4061>_).
"""

niter = 2 if niter is None else niter
dtype = _utils.get_floating_dtype(A) if not A.is_complex() else A.dtype
matmul = _utils.matmul

R = torch.randn(A.shape[-1], q, dtype=dtype, device=A.device)

# The following code could be made faster using torch.geqrf + torch.ormqr
# but geqrf is not differentiable

X = matmul(A, R)
if M is not None:
X = X - matmul(M, R)
Q = torch.linalg.qr(X).Q
for i in range(niter):
X = matmul(A.mH, Q)
if M is not None:
X = X - matmul(M.mH, Q)
Q = torch.linalg.qr(X).Q
X = matmul(A, Q)
if M is not None:
X = X - matmul(M, Q)
Q = torch.linalg.qr(X).Q
return Q

[docs]def svd_lowrank(
A: Tensor,
q: Optional[int] = 6,
niter: Optional[int] = 2,
M: Optional[Tensor] = None,
) -> Tuple[Tensor, Tensor, Tensor]:
r"""Return the singular value decomposition (U, S, V) of a matrix,
batches of matrices, or a sparse matrix :math:A such that
:math:A \approx U \operatorname{diag}(S) V^{\text{H}}. In case :math:M is given, then
SVD is computed for the matrix :math:A - M.

.. note:: The implementation is based on the Algorithm 5.1 from
Halko et al., 2009.

.. note:: For an adequate approximation of a k-rank matrix
:math:A, where k is not known in advance but could be
estimated, the number of :math:Q columns, q, can be
choosen according to the following criteria: in general,
:math:k <= q <= min(2*k, m, n). For large low-rank
matrices, take :math:q = k + 5..10.  If k is
relatively small compared to :math:min(m, n), choosing
:math:q = k + 0..2 may be sufficient.

.. note:: This is a randomized method. To obtain repeatable results,
set the seed for the pseudorandom number generator

.. note:: In general, use the full-rank SVD implementation
:func:torch.linalg.svd for dense matrices due to its 10x
higher performance characteristics. The low-rank SVD
will be useful for huge sparse matrices that
:func:torch.linalg.svd cannot handle.

Args::
A (Tensor): the input tensor of size :math:(*, m, n)

q (int, optional): a slightly overestimated rank of A.

niter (int, optional): the number of subspace iterations to
conduct; niter must be a nonnegative
integer, and defaults to 2

M (Tensor, optional): the input tensor's mean of size
:math:(*, m, n), which will be broadcasted
to the size of A in this function.

References::
- Nathan Halko, Per-Gunnar Martinsson, and Joel Tropp, Finding
structure with randomness: probabilistic algorithms for
constructing approximate matrix decompositions,
arXiv:0909.4061 [math.NA; math.PR], 2009 (available at
arXiv <https://arxiv.org/abs/0909.4061>_).

"""
if not torch.jit.is_scripting():
tensor_ops = (A, M)
if not set(map(type, tensor_ops)).issubset(
(torch.Tensor, type(None))
) and has_torch_function(tensor_ops):
return handle_torch_function(
svd_lowrank, tensor_ops, A, q=q, niter=niter, M=M
)
return _svd_lowrank(A, q=q, niter=niter, M=M)

def _svd_lowrank(
A: Tensor,
q: Optional[int] = 6,
niter: Optional[int] = 2,
M: Optional[Tensor] = None,
) -> Tuple[Tensor, Tensor, Tensor]:
# Algorithm 5.1 in Halko et al., 2009

q = 6 if q is None else q
m, n = A.shape[-2:]
matmul = _utils.matmul
if M is not None:

# Assume that A is tall
if m < n:
A = A.mH
if M is not None:
M = M.mH

Q = get_approximate_basis(A, q, niter=niter, M=M)
B = matmul(Q.mH, A)
if M is not None:
B = B - matmul(Q.mH, M)
U, S, Vh = torch.linalg.svd(B, full_matrices=False)
V = Vh.mH
U = Q.matmul(U)

if m < n:
U, V = V, U

return U, S, V

[docs]def pca_lowrank(
A: Tensor,
q: Optional[int] = None,
center: bool = True,
niter: int = 2,
) -> Tuple[Tensor, Tensor, Tensor]:
r"""Performs linear Principal Component Analysis (PCA) on a low-rank
matrix, batches of such matrices, or sparse matrix.

This function returns a namedtuple (U, S, V) which is the
nearly optimal approximation of a singular value decomposition of
a centered matrix :math:A such that :math:A \approx U \operatorname{diag}(S) V^{\text{H}}

.. note:: The relation of (U, S, V) to PCA is as follows:

- :math:A is a data matrix with m samples and
n features

- the :math:V columns represent the principal directions

- :math:S ** 2 / (m - 1) contains the eigenvalues of
:math:A^T A / (m - 1) which is the covariance of
A when center=True is provided.

- matmul(A, V[:, :k]) projects data to the first k
principal components

.. note:: Different from the standard SVD, the size of returned
matrices depend on the specified rank and q
values as follows:

- :math:U is m x q matrix

- :math:S is q-vector

- :math:V is n x q matrix

.. note:: To obtain repeatable results, reset the seed for the
pseudorandom number generator

Args:

A (Tensor): the input tensor of size :math:(*, m, n)

q (int, optional): a slightly overestimated rank of
:math:A. By default, q = min(6, m,
n).

center (bool, optional): if True, center the input tensor,
otherwise, assume that the input is
centered.

niter (int, optional): the number of subspace iterations to
conduct; niter must be a nonnegative
integer, and defaults to 2.

References::

- Nathan Halko, Per-Gunnar Martinsson, and Joel Tropp, Finding
structure with randomness: probabilistic algorithms for
constructing approximate matrix decompositions,
arXiv:0909.4061 [math.NA; math.PR], 2009 (available at
arXiv <http://arxiv.org/abs/0909.4061>_).

"""

if not torch.jit.is_scripting():
if type(A) is not torch.Tensor and has_torch_function((A,)):
return handle_torch_function(
pca_lowrank, (A,), A, q=q, center=center, niter=niter
)

(m, n) = A.shape[-2:]

if q is None:
q = min(6, m, n)
elif not (q >= 0 and q <= min(m, n)):
raise ValueError(
f"q(={q}) must be non-negative integer and not greater than min(m, n)={min(m, n)}"
)
if not (niter >= 0):
raise ValueError(f"niter(={niter}) must be non-negative integer")

dtype = _utils.get_floating_dtype(A)

if not center:
return _svd_lowrank(A, q, niter=niter, M=None)

if _utils.is_sparse(A):
if len(A.shape) != 2:
raise ValueError("pca_lowrank input is expected to be 2-dimensional tensor")
c = torch.sparse.sum(A, dim=(-2,)) / m
# reshape c
column_indices = c.indices()[0]
indices = torch.zeros(
2,
len(column_indices),
dtype=column_indices.dtype,
device=column_indices.device,
)
indices[0] = column_indices
C_t = torch.sparse_coo_tensor(
indices, c.values(), (n, 1), dtype=dtype, device=A.device
)

ones_m1_t = torch.ones(A.shape[:-2] + (1, m), dtype=dtype, device=A.device)
M = torch.sparse.mm(C_t, ones_m1_t).mT
return _svd_lowrank(A, q, niter=niter, M=M)
else:
C = A.mean(dim=(-2,), keepdim=True)
return _svd_lowrank(A - C, q, niter=niter, M=None)


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