scipy.sparse.linalg.

svds

scipy.sparse.linalg.svds(A, k=6, ncv=None, tol=0, which='LM', v0=None, maxiter=None, return_singular_vectors=True, solver='arpack', rng=None, options=None)

Partial singular value decomposition of a sparse matrix.

Compute the largest or smallest k singular values and corresponding singular vectors of a sparse matrix A. The order in which the singular values are returned is not guaranteed.

In the descriptions below, let M, N = A.shape.

Parameters:

Andarray, sparse matrix, or LinearOperator

Matrix to decompose of a floating point numeric dtype.

kint, default: 6

Number of singular values and singular vectors to compute. Must satisfy 1 <= k <= kmax, where kmax=min(M, N) for solver='propack' and kmax=min(M, N) - 1 otherwise.

ncvint, optional

When solver='arpack', this is the number of Lanczos vectors generated. See ‘arpack’ for details. When solver='lobpcg' or solver='propack', this parameter is ignored.

tolfloat, optional

Tolerance for singular values. Zero (default) means machine precision.

which{‘LM’, ‘SM’}

Which k singular values to find: either the largest magnitude (‘LM’) or smallest magnitude (‘SM’) singular values.

v0ndarray, optional

The starting vector for iteration; see method-specific documentation (‘arpack’, ‘lobpcg’), or ‘propack’ for details.

maxiterint, optional

Maximum number of iterations; see method-specific documentation (‘arpack’, ‘lobpcg’), or ‘propack’ for details.

return_singular_vectors{True, False, “u”, “vh”}

Singular values are always computed and returned; this parameter controls the computation and return of singular vectors.

• True: return singular vectors.

• False: do not return singular vectors.

• "u": if M <= N, compute only the left singular vectors and return None for the right singular vectors. Otherwise, compute all singular vectors.

• "vh": if M > N, compute only the right singular vectors and return None for the left singular vectors. Otherwise, compute all singular vectors.

If solver='propack', the option is respected regardless of the matrix shape.

solver{‘arpack’, ‘propack’, ‘lobpcg’}, optional

The solver used. ‘arpack’, ‘lobpcg’, and ‘propack’ are supported. Default: ‘arpack’.

rng{None, int, numpy.random.Generator}, optional

If rng is passed by keyword, types other than numpy.random.Generator are passed to numpy.random.default_rng to instantiate a Generator. If rng is already a Generator instance, then the provided instance is used. Specify rng for repeatable function behavior.

If this argument is passed by position or random_state is passed by keyword, legacy behavior for the argument random_state applies:

• If random_state is None (or numpy.random), the numpy.random.RandomState singleton is used.

• If random_state is an int, a new RandomState instance is used, seeded with random_state.

• If random_state is already a Generator or RandomState instance then that instance is used.

Changed in version 1.15.0: As part of the SPEC-007 transition from use of numpy.random.RandomState to numpy.random.Generator, this keyword was changed from random_state to rng. For an interim period, both keywords will continue to work, although only one may be specified at a time. After the interim period, function calls using the random_state keyword will emit warnings. The behavior of both random_state and rng are outlined above, but only the rng keyword should be used in new code.

optionsdict, optional

A dictionary of solver-specific options. No solver-specific options are currently supported; this parameter is reserved for future use.

Returns:

undarray, shape=(M, k)

Unitary matrix having left singular vectors as columns.

sndarray, shape=(k,)

The singular values.

vhndarray, shape=(k, N)

Unitary matrix having right singular vectors as rows.

Notes

This is a naive implementation using ARPACK or LOBPCG as an eigensolver on the matrix A.conj().T @ A or A @ A.conj().T, depending on which one is smaller size, followed by the Rayleigh-Ritz method as postprocessing; see Using the normal matrix, in Rayleigh-Ritz method, (2022, Nov. 19), Wikipedia, https://w.wiki/4zms.

Alternatively, the PROPACK solver can be called.

Choices of the input matrix A numeric dtype may be limited. Only solver="lobpcg" supports all floating point dtypes real: ‘np.float32’, ‘np.float64’, ‘np.longdouble’ and complex: ‘np.complex64’, ‘np.complex128’, ‘np.clongdouble’. The solver="arpack" supports only ‘np.float32’, ‘np.float64’, and ‘np.complex128’.

Examples

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Construct a matrix A from singular values and vectors.

>>> import numpy as np
>>> from scipy import sparse, linalg, stats
>>> from scipy.sparse.linalg import svds, aslinearoperator, LinearOperator

Construct a dense matrix A from singular values and vectors.

>>> rng = np.random.default_rng()
>>> orthogonal = stats.ortho_group.rvs(10, random_state=rng)
>>> s = [1e-3, 1, 2, 3, 4]  # non-zero singular values
>>> u = orthogonal[:, :5]         # left singular vectors
>>> vT = orthogonal[:, 5:].T      # right singular vectors
>>> A = u @ np.diag(s) @ vT

With only four singular values/vectors, the SVD approximates the original matrix.

>>> u4, s4, vT4 = svds(A, k=4)
>>> A4 = u4 @ np.diag(s4) @ vT4
>>> np.allclose(A4, A, atol=1e-3)
True

With all five non-zero singular values/vectors, we can reproduce the original matrix more accurately.

>>> u5, s5, vT5 = svds(A, k=5)
>>> A5 = u5 @ np.diag(s5) @ vT5
>>> np.allclose(A5, A)
True

The singular values match the expected singular values.

>>> np.allclose(s5, s)
True

Since the singular values are not close to each other in this example, every singular vector matches as expected up to a difference in sign.

>>> (np.allclose(np.abs(u5), np.abs(u)) and
...  np.allclose(np.abs(vT5), np.abs(vT)))
True

The singular vectors are also orthogonal.

>>> (np.allclose(u5.T @ u5, np.eye(5)) and
...  np.allclose(vT5 @ vT5.T, np.eye(5)))
True

If there are (nearly) multiple singular values, the corresponding individual singular vectors may be unstable, but the whole invariant subspace containing all such singular vectors is computed accurately as can be measured by angles between subspaces via ‘subspace_angles’.

>>> rng = np.random.default_rng()
>>> s = [1, 1 + 1e-6]  # non-zero singular values
>>> u, _ = np.linalg.qr(rng.standard_normal((99, 2)))
>>> v, _ = np.linalg.qr(rng.standard_normal((99, 2)))
>>> vT = v.T
>>> A = u @ np.diag(s) @ vT
>>> A = A.astype(np.float32)
>>> u2, s2, vT2 = svds(A, k=2, rng=rng)
>>> np.allclose(s2, s)
True

The angles between the individual exact and computed singular vectors may not be so small. To check use:

>>> (linalg.subspace_angles(u2[:, :1], u[:, :1]) +
...  linalg.subspace_angles(u2[:, 1:], u[:, 1:]))
array([0.06562513])  # may vary
>>> (linalg.subspace_angles(vT2[:1, :].T, vT[:1, :].T) +
...  linalg.subspace_angles(vT2[1:, :].T, vT[1:, :].T))
array([0.06562507])  # may vary

As opposed to the angles between the 2-dimensional invariant subspaces that these vectors span, which are small for rights singular vectors

>>> linalg.subspace_angles(u2, u).sum() < 1e-6
True

as well as for left singular vectors.

>>> linalg.subspace_angles(vT2.T, vT.T).sum() < 1e-6
True

The next example follows that of ‘sklearn.decomposition.TruncatedSVD’.

>>> rng = np.random.default_rng()
>>> X_dense = rng.random(size=(100, 100))
>>> X_dense[:, 2 * np.arange(50)] = 0
>>> X = sparse.csr_array(X_dense)
>>> _, singular_values, _ = svds(X, k=5, rng=rng)
>>> print(singular_values)
[ 4.3221...  4.4043...  4.4907...  4.5858... 35.4549...]

The function can be called without the transpose of the input matrix ever explicitly constructed.

>>> rng = np.random.default_rng()
>>> G = sparse.random_array((8, 9), density=0.5, rng=rng)
>>> Glo = aslinearoperator(G)
>>> _, singular_values_svds, _ = svds(Glo, k=5, rng=rng)
>>> _, singular_values_svd, _ = linalg.svd(G.toarray())
>>> np.allclose(singular_values_svds, singular_values_svd[-4::-1])
True

The most memory efficient scenario is where neither the original matrix, nor its transpose, is explicitly constructed. Our example computes the smallest singular values and vectors of ‘LinearOperator’ constructed from the numpy function ‘np.diff’ used column-wise to be consistent with ‘LinearOperator’ operating on columns.

>>> diff0 = lambda a: np.diff(a, axis=0)

Let us create the matrix from ‘diff0’ to be used for validation only.

>>> n = 5  # The dimension of the space.
>>> M_from_diff0 = diff0(np.eye(n))
>>> print(M_from_diff0.astype(int))
[[-1  1  0  0  0]
 [ 0 -1  1  0  0]
 [ 0  0 -1  1  0]
 [ 0  0  0 -1  1]]

The matrix ‘M_from_diff0’ is bi-diagonal and could be alternatively created directly by

>>> M = - np.eye(n - 1, n, dtype=int)
>>> np.fill_diagonal(M[:,1:], 1)
>>> np.allclose(M, M_from_diff0)
True

Its transpose

>>> print(M.T)
[[-1  0  0  0]
 [ 1 -1  0  0]
 [ 0  1 -1  0]
 [ 0  0  1 -1]
 [ 0  0  0  1]]

can be viewed as the incidence matrix; see Incidence matrix, (2022, Nov. 19), Wikipedia, https://w.wiki/5YXU, of a linear graph with 5 vertices and 4 edges. The 5x5 normal matrix M.T @ M thus is

>>> print(M.T @ M)
[[ 1 -1  0  0  0]
 [-1  2 -1  0  0]
 [ 0 -1  2 -1  0]
 [ 0  0 -1  2 -1]
 [ 0  0  0 -1  1]]

the graph Laplacian, while the actually used in ‘svds’ smaller size 4x4 normal matrix M @ M.T

>>> print(M @ M.T)
[[ 2 -1  0  0]
 [-1  2 -1  0]
 [ 0 -1  2 -1]
 [ 0  0 -1  2]]

is the so-called edge-based Laplacian; see Symmetric Laplacian via the incidence matrix, in Laplacian matrix, (2022, Nov. 19), Wikipedia, https://w.wiki/5YXW.

The ‘LinearOperator’ setup needs the options ‘rmatvec’ and ‘rmatmat’ of multiplication by the matrix transpose M.T, but we want to be matrix-free to save memory, so knowing how M.T looks like, we manually construct the following function to be used in rmatmat=diff0t.

>>> def diff0t(a):
...     if a.ndim == 1:
...         a = a[:,np.newaxis]  # Turn 1D into 2D array
...     d = np.zeros((a.shape[0] + 1, a.shape[1]), dtype=a.dtype)
...     d[0, :] = - a[0, :]
...     d[1:-1, :] = a[0:-1, :] - a[1:, :]
...     d[-1, :] = a[-1, :]
...     return d

We check that our function ‘diff0t’ for the matrix transpose is valid.

>>> np.allclose(M.T, diff0t(np.eye(n-1)))
True

Now we setup our matrix-free ‘LinearOperator’ called ‘diff0_func_aslo’ and for validation the matrix-based ‘diff0_matrix_aslo’.

>>> def diff0_func_aslo_def(n):
...     return LinearOperator(matvec=diff0,
...                           matmat=diff0,
...                           rmatvec=diff0t,
...                           rmatmat=diff0t,
...                           shape=(n - 1, n))
>>> diff0_func_aslo = diff0_func_aslo_def(n)
>>> diff0_matrix_aslo = aslinearoperator(M_from_diff0)

And validate both the matrix and its transpose in ‘LinearOperator’.

>>> np.allclose(diff0_func_aslo(np.eye(n)),
...             diff0_matrix_aslo(np.eye(n)))
True
>>> np.allclose(diff0_func_aslo.T(np.eye(n-1)),
...             diff0_matrix_aslo.T(np.eye(n-1)))
True

Having the ‘LinearOperator’ setup validated, we run the solver.

>>> n = 100
>>> diff0_func_aslo = diff0_func_aslo_def(n)
>>> u, s, vT = svds(diff0_func_aslo, k=3, which='SM')

The singular values squared and the singular vectors are known explicitly; see Pure Dirichlet boundary conditions, in Eigenvalues and eigenvectors of the second derivative, (2022, Nov. 19), Wikipedia, https://w.wiki/5YX6, since ‘diff’ corresponds to first derivative, and its smaller size n-1 x n-1 normal matrix M @ M.T represent the discrete second derivative with the Dirichlet boundary conditions. We use these analytic expressions for validation.

>>> se = 2. * np.sin(np.pi * np.arange(1, 4) / (2. * n))
>>> ue = np.sqrt(2 / n) * np.sin(np.pi * np.outer(np.arange(1, n),
...                              np.arange(1, 4)) / n)
>>> np.allclose(s, se, atol=1e-3)
True
>>> np.allclose(np.abs(u), np.abs(ue), atol=1e-6)
True

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