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# 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', random_state=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:
    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’ 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’, ‘lobpcg’), or ‘propack’ for details.

    maxiter int, 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’.

    random_state{None, int, `numpy.random.Generator`,

      > `numpy.random.RandomState`}, optional

      Pseudorandom number generator state used to generate resamples.

      If random_state is `None` (or np.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

  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

  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, random_state=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.RandomState(0)
      >>> X_dense = rng.random(size=(100, 100))
      >>> X_dense[:, 2 * np.arange(50)] = 0
      >>> X = sparse.csr_matrix(X_dense)
      >>> _, singular_values, _ = svds(X, k=5, random_state=rng)
      >>> print(singular_values)
      [ 4.3293...  4.4491...  4.5420...  4.5987... 35.2410...]

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

      >>> rng = np.random.default_rng()
      >>> G = sparse.rand(8, 9, density=0.5, random_state=rng)
      >>> Glo = aslinearoperator(G)
      >>> _, singular_values_svds, _ = svds(Glo, k=5, random_state=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
      >>> print(np.allclose(np.abs(u), np.abs(ue), atol=1e-6))
      True