122 lines
3.8 KiB
Python
122 lines
3.8 KiB
Python
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""" Sketching-based Matrix Computations """
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# Author: Jordi Montes <jomsdev@gmail.com>
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# August 28, 2017
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from __future__ import division, print_function, absolute_import
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import numpy as np
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from scipy._lib._util import check_random_state
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__all__ = ['clarkson_woodruff_transform']
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def cwt_matrix(n_rows, n_columns, seed=None):
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r""""
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Generate a matrix S for the Clarkson-Woodruff sketch.
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Given the desired size of matrix, the method returns a matrix S of size
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(n_rows, n_columns) where each column has all the entries set to 0 less one
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position which has been randomly set to +1 or -1 with equal probability.
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Parameters
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----------
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n_rows: int
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Number of rows of S
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n_columns: int
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Number of columns of S
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seed : None or int or `numpy.random.RandomState` instance, optional
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This parameter defines the ``RandomState`` object to use for drawing
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random variates.
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If None (or ``np.random``), the global ``np.random`` state is used.
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If integer, it is used to seed the local ``RandomState`` instance.
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Default is None.
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Returns
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-------
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S : (n_rows, n_columns) array_like
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Notes
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-----
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Given a matrix A, with probability at least 9/10,
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.. math:: ||SA|| == (1 \pm \epsilon)||A||
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Where epsilon is related to the size of S
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"""
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S = np.zeros((n_rows, n_columns))
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nz_positions = np.random.randint(0, n_rows, n_columns)
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rng = check_random_state(seed)
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values = rng.choice([1, -1], n_columns)
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for i in range(n_columns):
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S[nz_positions[i]][i] = values[i]
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return S
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def clarkson_woodruff_transform(input_matrix, sketch_size, seed=None):
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r""""
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Find low-rank matrix approximation via the Clarkson-Woodruff Transform.
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Given an input_matrix ``A`` of size ``(n, d)``, compute a matrix ``A'`` of
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size (sketch_size, d) which holds:
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.. math:: ||Ax|| = (1 \pm \epsilon)||A'x||
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with high probability.
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The error is related to the number of rows of the sketch and it is bounded
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.. math:: poly(r(\epsilon^{-1}))
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Parameters
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----------
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input_matrix: array_like
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Input matrix, of shape ``(n, d)``.
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sketch_size: int
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Number of rows for the sketch.
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seed : None or int or `numpy.random.RandomState` instance, optional
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This parameter defines the ``RandomState`` object to use for drawing
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random variates.
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If None (or ``np.random``), the global ``np.random`` state is used.
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If integer, it is used to seed the local ``RandomState`` instance.
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Default is None.
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Returns
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-------
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A' : array_like
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Sketch of the input matrix ``A``, of size ``(sketch_size, d)``.
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Notes
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-----
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This is an implementation of the Clarkson-Woodruff Transform (CountSketch).
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``A'`` can be computed in principle in ``O(nnz(A))`` (with ``nnz`` meaning
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the number of nonzero entries), however we don't take advantage of sparse
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matrices in this implementation.
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Examples
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--------
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Given a big dense matrix ``A``:
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>>> from scipy import linalg
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>>> n_rows, n_columns, sketch_n_rows = (2000, 100, 100)
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>>> threshold = 0.1
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>>> tmp = np.random.normal(0, 0.1, n_rows*n_columns)
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>>> A = np.reshape(tmp, (n_rows, n_columns))
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>>> sketch = linalg.clarkson_woodruff_transform(A, sketch_n_rows)
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>>> sketch.shape
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(100, 100)
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>>> normA = linalg.norm(A)
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>>> norm_sketch = linalg.norm(sketch)
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Now with high probability, the condition ``abs(normA-normSketch) <
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threshold`` holds.
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References
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----------
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.. [1] Kenneth L. Clarkson and David P. Woodruff. Low rank approximation and
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regression in input sparsity time. In STOC, 2013.
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"""
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S = cwt_matrix(sketch_size, input_matrix.shape[0], seed)
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return np.dot(S, input_matrix)
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