Module « scipy.linalg »

Fonction clarkson_woodruff_transform - module scipy.linalg

Signature de la fonction clarkson_woodruff_transform

def clarkson_woodruff_transform(input_matrix, sketch_size, seed=None) 

Description

clarkson_woodruff_transform.__doc__

    Applies a Clarkson-Woodruff Transform/sketch to the input matrix.

    Given an input_matrix ``A`` of size ``(n, d)``, compute a matrix ``A'`` of
    size (sketch_size, d) so that

    .. math:: \|Ax\| \approx \|A'x\|

    with high probability via the Clarkson-Woodruff Transform, otherwise
    known as the CountSketch matrix.

    Parameters
    ----------
    input_matrix: array_like
        Input matrix, of shape ``(n, d)``.
    sketch_size: int
        Number of rows for the sketch.
    seed : {None, int, `numpy.random.Generator`,
            `numpy.random.RandomState`}, optional

        If `seed` is None (or `np.random`), the `numpy.random.RandomState`
        singleton is used.
        If `seed` is an int, a new ``RandomState`` instance is used,
        seeded with `seed`.
        If `seed` is already a ``Generator`` or ``RandomState`` instance then
        that instance is used.

    Returns
    -------
    A' : array_like
        Sketch of the input matrix ``A``, of size ``(sketch_size, d)``.

    Notes
    -----
    To make the statement

    .. math:: \|Ax\| \approx \|A'x\|

    precise, observe the following result which is adapted from the
    proof of Theorem 14 of [2]_ via Markov's Inequality. If we have
    a sketch size ``sketch_size=k`` which is at least

    .. math:: k \geq \frac{2}{\epsilon^2\delta}

    Then for any fixed vector ``x``,

    .. math:: \|Ax\| = (1\pm\epsilon)\|A'x\|

    with probability at least one minus delta.

    This implementation takes advantage of sparsity: computing
    a sketch takes time proportional to ``A.nnz``. Data ``A`` which
    is in ``scipy.sparse.csc_matrix`` format gives the quickest
    computation time for sparse input.

    >>> from scipy import linalg
    >>> from scipy import sparse
    >>> rng = np.random.default_rng()
    >>> n_rows, n_columns, density, sketch_n_rows = 15000, 100, 0.01, 200
    >>> A = sparse.rand(n_rows, n_columns, density=density, format='csc')
    >>> B = sparse.rand(n_rows, n_columns, density=density, format='csr')
    >>> C = sparse.rand(n_rows, n_columns, density=density, format='coo')
    >>> D = rng.standard_normal((n_rows, n_columns))
    >>> SA = linalg.clarkson_woodruff_transform(A, sketch_n_rows) # fastest
    >>> SB = linalg.clarkson_woodruff_transform(B, sketch_n_rows) # fast
    >>> SC = linalg.clarkson_woodruff_transform(C, sketch_n_rows) # slower
    >>> SD = linalg.clarkson_woodruff_transform(D, sketch_n_rows) # slowest

    That said, this method does perform well on dense inputs, just slower
    on a relative scale.

    Examples
    --------
    Given a big dense matrix ``A``:

    >>> from scipy import linalg
    >>> n_rows, n_columns, sketch_n_rows = 15000, 100, 200
    >>> rng = np.random.default_rng()
    >>> A = rng.standard_normal((n_rows, n_columns))
    >>> sketch = linalg.clarkson_woodruff_transform(A, sketch_n_rows)
    >>> sketch.shape
    (200, 100)
    >>> norm_A = np.linalg.norm(A)
    >>> norm_sketch = np.linalg.norm(sketch)

    Now with high probability, the true norm ``norm_A`` is close to
    the sketched norm ``norm_sketch`` in absolute value.

    Similarly, applying our sketch preserves the solution to a linear
    regression of :math:`\min \|Ax - b\|`.

    >>> from scipy import linalg
    >>> n_rows, n_columns, sketch_n_rows = 15000, 100, 200
    >>> rng = np.random.default_rng()
    >>> A = rng.standard_normal((n_rows, n_columns))
    >>> b = rng.standard_normal(n_rows)
    >>> x = np.linalg.lstsq(A, b, rcond=None)
    >>> Ab = np.hstack((A, b.reshape(-1,1)))
    >>> SAb = linalg.clarkson_woodruff_transform(Ab, sketch_n_rows)
    >>> SA, Sb = SAb[:,:-1], SAb[:,-1]
    >>> x_sketched = np.linalg.lstsq(SA, Sb, rcond=None)

    As with the matrix norm example, ``np.linalg.norm(A @ x - b)``
    is close to ``np.linalg.norm(A @ x_sketched - b)`` with high
    probability.

    References
    ----------
    .. [1] Kenneth L. Clarkson and David P. Woodruff. Low rank approximation and
           regression in input sparsity time. In STOC, 2013.

    .. [2] David P. Woodruff. Sketching as a tool for numerical linear algebra.
           In Foundations and Trends in Theoretical Computer Science, 2014.