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bundles / scipy latest / scipy / sparse / linalg / _eigen / _svds / svds

function

scipy.sparse.linalg._eigen._svds:svds

source: /scipy/sparse/linalg/_eigen/_svds.py :108

Signature

def   svds ( A ,  k = 6 ,  ncv = None ,  tol = 0 ,  which = LM ,  v0 = None ,  maxiter = None ,  return_singular_vectors = True ,  solver = arpack ,  rng = None ,  options = None ,  * ,  random_state = None )

Summary

Partial singular value decomposition of a sparse matrix.

Extended Summary

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

A : ndarray, sparse matrix, or LinearOperator

Matrix to decompose of a floating point numeric dtype.

k : int, 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.

ncv : int, optional

When solver='arpack', this is the number of Lanczos vectors generated. See 'arpack' <sparse.linalg.svds-arpack> for details. When solver='lobpcg' or solver='propack', this parameter is ignored.

tol : float, 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.

v0 : ndarray, optional

The starting vector for iteration; see method-specific documentation ('arpack' <sparse.linalg.svds-arpack>, 'lobpcg' <sparse.linalg.svds-lobpcg>), or 'propack' <sparse.linalg.svds-propack> for details.

maxiter : int, optional

Maximum number of iterations; see method-specific documentation ('arpack' <sparse.linalg.svds-arpack>, 'lobpcg' <sparse.linalg.svds-lobpcg>), or 'propack' <sparse.linalg.svds-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' <sparse.linalg.svds-arpack>, 'lobpcg' <sparse.linalg.svds-lobpcg>, and 'propack' <sparse.linalg.svds-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.

options : dict, optional

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

Returns

u : ndarray, shape=(M, k)

Unitary matrix having left singular vectors as columns.

s : ndarray, shape=(k,)

The singular values.

vh : ndarray, shape=(k, N)

Unitary matrix having right singular vectors as rows.

Notes

When ARPACK or LOBPCG is selected as the method, the singular values and singular vectors are computed as the eigenvalues and eigenvectors of the corresponding Gram matrix, either A.conj().T @ A or A @ A.conj().T, depending on which one is computationally cheaper. It is then 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.

Dtype of A is mapped, if possible, to float(32,64) and complex(64,128) for internal computations.

Examples

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(258265244568965474821194062361901728911)
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)
✓
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)
✓
The singular values match the expected singular values. 
np.allclose(s5, s)
✓
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)))
✓
The singular vectors are also orthogonal. 
(np.allclose(u5.T @ u5, np.eye(5)) and
 np.allclose(vT5 @ vT5.T, np.eye(5)))
✓
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(178686584221410808734965903901790843963)
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)
✓
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:]))
(linalg.subspace_angles(vT2[:1, :].T, vT[:1, :].T) +
 linalg.subspace_angles(vT2[1:, :].T, vT[1:, :].T))
✗
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
✗
as well as for left singular vectors. 
linalg.subspace_angles(vT2.T, vT.T).sum() < 1e-6
✗
The next example follows that of 'sklearn.decomposition.TruncatedSVD'. 
rng = np.random.default_rng(0)
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)
✓
The function can be called without the transpose of the input matrix ever explicitly constructed. 
rng = np.random.default_rng(102524723947864966825913730119128190974)
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])
✓
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))
✓
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)
✓
Its transpose 
print(M.T)
✓
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)
✓
the graph Laplacian, while the actually used in 'svds' smaller size 4x4 normal matrix ``M @ M.T`` 
print(M @ M.T)
✓
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)))
✓
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)))
np.allclose(diff0_func_aslo.T(np.eye(n-1)),
            diff0_matrix_aslo.T(np.eye(n-1)))
✓
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)
np.allclose(np.abs(u), np.abs(ue), atol=1e-6)
✓

Aliases

  • scipy.sparse.linalg.svds

Referenced by

This package