scipy.linalg.clarkson_woodruff_transform

scipy.linalg.clarkson_woodruff_transform(input_matrix, sketch_size, seed=None)

” Find low-rank matrix approximation via the Clarkson-Woodruff Transform.

Given an input_matrix A of size (n, d), compute a matrix A' of size (sketch_size, d) which holds:

\[||Ax|| = (1 \pm \epsilon)||A'x||\]

with high probability.

The error is related to the number of rows of the sketch and it is bounded

\[poly(r(\epsilon^{-1}))\]

Parameters:

input_matrix: array_like

Input matrix, of shape (n, d).

sketch_size: int

Number of rows for the sketch.

seed : None or int or numpy.random.RandomState instance, optional

This parameter defines the RandomState object to use for drawing random variates. If None (or np.random), the global np.random state is used. If integer, it is used to seed the local RandomState instance. Default is None.

Returns:

A’ : array_like

Sketch of the input matrix A, of size (sketch_size, d).

Notes

This is an implementation of the Clarkson-Woodruff Transform (CountSketch). A' can be computed in principle in O(nnz(A)) (with nnz meaning the number of nonzero entries), however we don’t take advantage of sparse matrices in this implementation.

References

[R122]

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

Examples

Given a big dense matrix A:

>>> from scipy import linalg
>>> n_rows, n_columns, sketch_n_rows = (2000, 100, 100)
>>> threshold = 0.1
>>> tmp = np.random.normal(0, 0.1, n_rows*n_columns)
>>> A = np.reshape(tmp, (n_rows, n_columns))
>>> sketch = linalg.clarkson_woodruff_transform(A, sketch_n_rows)
>>> sketch.shape
(100, 100)
>>> normA = linalg.norm(A)
>>> norm_sketch = linalg.norm(sketch)

Now with high probability, the condition abs(normA-normSketch) < threshold holds.