a 7Sic* @sdZddgZddlmZddlZddlmZddlmZm Z dd l m Z m Z dee e e e eed d d Zdee e e e e ee eeefd ddZdee e e e e ee eeefd ddZdee e ee e eeefdddZdS)zCImplement various linear algebra algorithms for low rank matrices. svd_lowrank pca_lowrank)TensorN) _linalg_utils)has_torch_functionhandle_torch_function)OptionalTuple)AqniterMreturnc Cs|dur dn|}|jdd\}}t|}tj}tj||||jd}t|} |durtj |||j } t |D],} tj || | j } tj ||| j } qtnrt|} tj ||||||j } t |D]@} tj || | || | j } tj ||| ||| j } q| S)a8Return 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`. .. 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:`(*, 1, 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 `_). Nr dtypedevice) shape_utilsget_floating_dtypematmultorchrandnrZ transjugatelinalgqrQrange) r r rrmnrrRZA_HriZM_Hr#J/var/www/html/django/DPS/env/lib/python3.9/site-packages/torch/_lowrank.pyget_approximate_basis s"/      r%cCs\tjsL||f}ttt|tjtdfsLt|rLt t |||||dSt ||||dS)a_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 diag(S) V^T`. 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:: To obtain repeatable results, reset the seed for the pseudorandom number generator .. note:: The input is assumed to be a low-rank matrix. .. note:: In general, use the full-rank SVD implementation :func:`torch.linalg.svd` for dense matrices due to its 10-fold 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:`(*, 1, 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 `_). N)r rr) rjit is_scriptingsetmaptypeissubsetrrrr _svd_lowrank)r r rrZ tensor_opsr#r#r$rUs ) &cCs|dur dn|}|jdd\}}tj}|dur6d}n t|}t|}||ks\||krt||||d} t| } |dur||| } n||| ||| } | jd|ksJ| j|f| jd|ksJ| j|f| jd| jdksJ| jtjj| dd\} } }|j }| |}nt||||d} t| } |durL||| }n||| ||| }t|} | jd|ksJ| j|f| jd|ksJ| j|f| jd| jdksJ| jtjj| dd\} } }|j }| | } | | |fS)Nr&rrrF) full_matrices) rrr transposer% conjugaterrsvdmH)r r rrrr rZM_tZA_trZQ_cZB_tUSVhVBr#r#r$r-s@           r-T)r r centerrrcCstjs8t|tjur8t|fr8tt|f||||dS|jdd\}}|dur`t d||}n,|dkrv|t ||kst d |t |||dkst d |t |}|st|||ddSt |rt|jd krt d tjj|d d |}|d}tjd t||j|jd } || d<tj| ||df||jd } tj|jddd|f||jd } t tj| | } t|||| dS|jd dd} t|| ||ddSdS)aPerforms 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 = U diag(S) V^T`. .. 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 `_). )r r:rrNr&rzEq(={}) must be non-negative integer and not greater than min(m, n)={}z'niter(={}) must be non-negative integerr.r z8pca_lowrank input is expected to be 2-dimensional tensor)r)dimrrT)r;keepdim)rr'r(r+rrrrrmin ValueErrorformatrrr- is_sparselensparsesumindiceszerosrrsparse_coo_tensorvaluesonesr1mmmean)r r r:rrr rcZcolumn_indicesrDZC_tZ ones_m1_trCr#r#r$rsF>      $)r N)r&r N)r&r N)NTr )__doc____all__rrrr overridesrrtypingr r intr%rr-boolrr#r#r#r$s:   H 0 /