468 lines
15 KiB
Python
468 lines
15 KiB
Python
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"""Sparse block 1-norm estimator.
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"""
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import numpy as np
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from scipy.sparse.linalg import aslinearoperator
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__all__ = ['onenormest']
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def onenormest(A, t=2, itmax=5, compute_v=False, compute_w=False):
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"""
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Compute a lower bound of the 1-norm of a sparse matrix.
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Parameters
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----------
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A : ndarray or other linear operator
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A linear operator that can be transposed and that can
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produce matrix products.
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t : int, optional
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A positive parameter controlling the tradeoff between
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accuracy versus time and memory usage.
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Larger values take longer and use more memory
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but give more accurate output.
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itmax : int, optional
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Use at most this many iterations.
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compute_v : bool, optional
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Request a norm-maximizing linear operator input vector if True.
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compute_w : bool, optional
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Request a norm-maximizing linear operator output vector if True.
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Returns
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-------
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est : float
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An underestimate of the 1-norm of the sparse matrix.
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v : ndarray, optional
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The vector such that ||Av||_1 == est*||v||_1.
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It can be thought of as an input to the linear operator
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that gives an output with particularly large norm.
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w : ndarray, optional
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The vector Av which has relatively large 1-norm.
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It can be thought of as an output of the linear operator
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that is relatively large in norm compared to the input.
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Notes
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-----
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This is algorithm 2.4 of [1].
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In [2] it is described as follows.
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"This algorithm typically requires the evaluation of
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about 4t matrix-vector products and almost invariably
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produces a norm estimate (which is, in fact, a lower
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bound on the norm) correct to within a factor 3."
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.. versionadded:: 0.13.0
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References
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----------
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.. [1] Nicholas J. Higham and Francoise Tisseur (2000),
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"A Block Algorithm for Matrix 1-Norm Estimation,
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with an Application to 1-Norm Pseudospectra."
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SIAM J. Matrix Anal. Appl. Vol. 21, No. 4, pp. 1185-1201.
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.. [2] Awad H. Al-Mohy and Nicholas J. Higham (2009),
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"A new scaling and squaring algorithm for the matrix exponential."
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SIAM J. Matrix Anal. Appl. Vol. 31, No. 3, pp. 970-989.
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Examples
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--------
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>>> import numpy as np
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>>> from scipy.sparse import csc_matrix
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>>> from scipy.sparse.linalg import onenormest
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>>> A = csc_matrix([[1., 0., 0.], [5., 8., 2.], [0., -1., 0.]], dtype=float)
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>>> A.toarray()
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array([[ 1., 0., 0.],
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[ 5., 8., 2.],
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[ 0., -1., 0.]])
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>>> onenormest(A)
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9.0
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>>> np.linalg.norm(A.toarray(), ord=1)
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9.0
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"""
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# Check the input.
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A = aslinearoperator(A)
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if A.shape[0] != A.shape[1]:
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raise ValueError('expected the operator to act like a square matrix')
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# If the operator size is small compared to t,
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# then it is easier to compute the exact norm.
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# Otherwise estimate the norm.
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n = A.shape[1]
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if t >= n:
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A_explicit = np.asarray(aslinearoperator(A).matmat(np.identity(n)))
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if A_explicit.shape != (n, n):
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raise Exception('internal error: ',
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'unexpected shape ' + str(A_explicit.shape))
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col_abs_sums = abs(A_explicit).sum(axis=0)
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if col_abs_sums.shape != (n, ):
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raise Exception('internal error: ',
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'unexpected shape ' + str(col_abs_sums.shape))
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argmax_j = np.argmax(col_abs_sums)
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v = elementary_vector(n, argmax_j)
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w = A_explicit[:, argmax_j]
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est = col_abs_sums[argmax_j]
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else:
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est, v, w, nmults, nresamples = _onenormest_core(A, A.H, t, itmax)
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# Report the norm estimate along with some certificates of the estimate.
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if compute_v or compute_w:
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result = (est,)
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if compute_v:
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result += (v,)
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if compute_w:
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result += (w,)
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return result
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else:
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return est
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def _blocked_elementwise(func):
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"""
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Decorator for an elementwise function, to apply it blockwise along
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first dimension, to avoid excessive memory usage in temporaries.
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"""
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block_size = 2**20
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def wrapper(x):
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if x.shape[0] < block_size:
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return func(x)
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else:
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y0 = func(x[:block_size])
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y = np.zeros((x.shape[0],) + y0.shape[1:], dtype=y0.dtype)
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y[:block_size] = y0
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del y0
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for j in range(block_size, x.shape[0], block_size):
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y[j:j+block_size] = func(x[j:j+block_size])
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return y
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return wrapper
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@_blocked_elementwise
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def sign_round_up(X):
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"""
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This should do the right thing for both real and complex matrices.
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From Higham and Tisseur:
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"Everything in this section remains valid for complex matrices
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provided that sign(A) is redefined as the matrix (aij / |aij|)
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(and sign(0) = 1) transposes are replaced by conjugate transposes."
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"""
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Y = X.copy()
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Y[Y == 0] = 1
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Y /= np.abs(Y)
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return Y
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@_blocked_elementwise
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def _max_abs_axis1(X):
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return np.max(np.abs(X), axis=1)
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def _sum_abs_axis0(X):
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block_size = 2**20
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r = None
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for j in range(0, X.shape[0], block_size):
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y = np.sum(np.abs(X[j:j+block_size]), axis=0)
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if r is None:
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r = y
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else:
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r += y
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return r
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def elementary_vector(n, i):
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v = np.zeros(n, dtype=float)
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v[i] = 1
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return v
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def vectors_are_parallel(v, w):
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# Columns are considered parallel when they are equal or negative.
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# Entries are required to be in {-1, 1},
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# which guarantees that the magnitudes of the vectors are identical.
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if v.ndim != 1 or v.shape != w.shape:
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raise ValueError('expected conformant vectors with entries in {-1,1}')
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n = v.shape[0]
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return np.dot(v, w) == n
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def every_col_of_X_is_parallel_to_a_col_of_Y(X, Y):
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for v in X.T:
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if not any(vectors_are_parallel(v, w) for w in Y.T):
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return False
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return True
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def column_needs_resampling(i, X, Y=None):
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# column i of X needs resampling if either
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# it is parallel to a previous column of X or
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# it is parallel to a column of Y
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n, t = X.shape
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v = X[:, i]
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if any(vectors_are_parallel(v, X[:, j]) for j in range(i)):
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return True
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if Y is not None:
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if any(vectors_are_parallel(v, w) for w in Y.T):
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return True
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return False
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def resample_column(i, X):
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X[:, i] = np.random.randint(0, 2, size=X.shape[0])*2 - 1
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def less_than_or_close(a, b):
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return np.allclose(a, b) or (a < b)
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def _algorithm_2_2(A, AT, t):
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"""
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This is Algorithm 2.2.
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Parameters
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----------
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A : ndarray or other linear operator
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A linear operator that can produce matrix products.
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AT : ndarray or other linear operator
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The transpose of A.
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t : int, optional
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A positive parameter controlling the tradeoff between
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accuracy versus time and memory usage.
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Returns
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-------
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g : sequence
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A non-negative decreasing vector
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such that g[j] is a lower bound for the 1-norm
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of the column of A of jth largest 1-norm.
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The first entry of this vector is therefore a lower bound
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on the 1-norm of the linear operator A.
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This sequence has length t.
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ind : sequence
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The ith entry of ind is the index of the column A whose 1-norm
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is given by g[i].
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This sequence of indices has length t, and its entries are
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chosen from range(n), possibly with repetition,
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where n is the order of the operator A.
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Notes
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-----
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This algorithm is mainly for testing.
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It uses the 'ind' array in a way that is similar to
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its usage in algorithm 2.4. This algorithm 2.2 may be easier to test,
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so it gives a chance of uncovering bugs related to indexing
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which could have propagated less noticeably to algorithm 2.4.
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"""
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A_linear_operator = aslinearoperator(A)
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AT_linear_operator = aslinearoperator(AT)
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n = A_linear_operator.shape[0]
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# Initialize the X block with columns of unit 1-norm.
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X = np.ones((n, t))
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if t > 1:
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X[:, 1:] = np.random.randint(0, 2, size=(n, t-1))*2 - 1
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X /= float(n)
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# Iteratively improve the lower bounds.
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# Track extra things, to assert invariants for debugging.
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g_prev = None
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h_prev = None
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k = 1
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ind = range(t)
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while True:
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Y = np.asarray(A_linear_operator.matmat(X))
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g = _sum_abs_axis0(Y)
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best_j = np.argmax(g)
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g.sort()
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g = g[::-1]
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S = sign_round_up(Y)
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Z = np.asarray(AT_linear_operator.matmat(S))
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h = _max_abs_axis1(Z)
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# If this algorithm runs for fewer than two iterations,
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# then its return values do not have the properties indicated
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# in the description of the algorithm.
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# In particular, the entries of g are not 1-norms of any
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# column of A until the second iteration.
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# Therefore we will require the algorithm to run for at least
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# two iterations, even though this requirement is not stated
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# in the description of the algorithm.
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if k >= 2:
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if less_than_or_close(max(h), np.dot(Z[:, best_j], X[:, best_j])):
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break
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ind = np.argsort(h)[::-1][:t]
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h = h[ind]
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for j in range(t):
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X[:, j] = elementary_vector(n, ind[j])
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# Check invariant (2.2).
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if k >= 2:
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if not less_than_or_close(g_prev[0], h_prev[0]):
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raise Exception('invariant (2.2) is violated')
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if not less_than_or_close(h_prev[0], g[0]):
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raise Exception('invariant (2.2) is violated')
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# Check invariant (2.3).
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if k >= 3:
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for j in range(t):
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if not less_than_or_close(g[j], g_prev[j]):
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raise Exception('invariant (2.3) is violated')
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# Update for the next iteration.
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g_prev = g
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h_prev = h
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k += 1
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# Return the lower bounds and the corresponding column indices.
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return g, ind
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def _onenormest_core(A, AT, t, itmax):
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"""
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Compute a lower bound of the 1-norm of a sparse matrix.
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Parameters
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----------
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A : ndarray or other linear operator
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A linear operator that can produce matrix products.
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AT : ndarray or other linear operator
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The transpose of A.
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t : int, optional
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A positive parameter controlling the tradeoff between
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accuracy versus time and memory usage.
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itmax : int, optional
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Use at most this many iterations.
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Returns
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-------
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est : float
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An underestimate of the 1-norm of the sparse matrix.
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v : ndarray, optional
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The vector such that ||Av||_1 == est*||v||_1.
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It can be thought of as an input to the linear operator
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that gives an output with particularly large norm.
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w : ndarray, optional
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The vector Av which has relatively large 1-norm.
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It can be thought of as an output of the linear operator
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that is relatively large in norm compared to the input.
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nmults : int, optional
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The number of matrix products that were computed.
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nresamples : int, optional
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The number of times a parallel column was observed,
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necessitating a re-randomization of the column.
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Notes
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-----
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This is algorithm 2.4.
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"""
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# This function is a more or less direct translation
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# of Algorithm 2.4 from the Higham and Tisseur (2000) paper.
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A_linear_operator = aslinearoperator(A)
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AT_linear_operator = aslinearoperator(AT)
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if itmax < 2:
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raise ValueError('at least two iterations are required')
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if t < 1:
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raise ValueError('at least one column is required')
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n = A.shape[0]
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if t >= n:
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raise ValueError('t should be smaller than the order of A')
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# Track the number of big*small matrix multiplications
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# and the number of resamplings.
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nmults = 0
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nresamples = 0
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# "We now explain our choice of starting matrix. We take the first
|
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|
# column of X to be the vector of 1s [...] This has the advantage that
|
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|
|
# for a matrix with nonnegative elements the algorithm converges
|
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|
# with an exact estimate on the second iteration, and such matrices
|
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|
|
# arise in applications [...]"
|
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|
X = np.ones((n, t), dtype=float)
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|
|
# "The remaining columns are chosen as rand{-1,1},
|
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|
|
# with a check for and correction of parallel columns,
|
||
|
|
# exactly as for S in the body of the algorithm."
|
||
|
|
if t > 1:
|
||
|
|
for i in range(1, t):
|
||
|
|
# These are technically initial samples, not resamples,
|
||
|
|
# so the resampling count is not incremented.
|
||
|
|
resample_column(i, X)
|
||
|
|
for i in range(t):
|
||
|
|
while column_needs_resampling(i, X):
|
||
|
|
resample_column(i, X)
|
||
|
|
nresamples += 1
|
||
|
|
# "Choose starting matrix X with columns of unit 1-norm."
|
||
|
|
X /= float(n)
|
||
|
|
# "indices of used unit vectors e_j"
|
||
|
|
ind_hist = np.zeros(0, dtype=np.intp)
|
||
|
|
est_old = 0
|
||
|
|
S = np.zeros((n, t), dtype=float)
|
||
|
|
k = 1
|
||
|
|
ind = None
|
||
|
|
while True:
|
||
|
|
Y = np.asarray(A_linear_operator.matmat(X))
|
||
|
|
nmults += 1
|
||
|
|
mags = _sum_abs_axis0(Y)
|
||
|
|
est = np.max(mags)
|
||
|
|
best_j = np.argmax(mags)
|
||
|
|
if est > est_old or k == 2:
|
||
|
|
if k >= 2:
|
||
|
|
ind_best = ind[best_j]
|
||
|
|
w = Y[:, best_j]
|
||
|
|
# (1)
|
||
|
|
if k >= 2 and est <= est_old:
|
||
|
|
est = est_old
|
||
|
|
break
|
||
|
|
est_old = est
|
||
|
|
S_old = S
|
||
|
|
if k > itmax:
|
||
|
|
break
|
||
|
|
S = sign_round_up(Y)
|
||
|
|
del Y
|
||
|
|
# (2)
|
||
|
|
if every_col_of_X_is_parallel_to_a_col_of_Y(S, S_old):
|
||
|
|
break
|
||
|
|
if t > 1:
|
||
|
|
# "Ensure that no column of S is parallel to another column of S
|
||
|
|
# or to a column of S_old by replacing columns of S by rand{-1,1}."
|
||
|
|
for i in range(t):
|
||
|
|
while column_needs_resampling(i, S, S_old):
|
||
|
|
resample_column(i, S)
|
||
|
|
nresamples += 1
|
||
|
|
del S_old
|
||
|
|
# (3)
|
||
|
|
Z = np.asarray(AT_linear_operator.matmat(S))
|
||
|
|
nmults += 1
|
||
|
|
h = _max_abs_axis1(Z)
|
||
|
|
del Z
|
||
|
|
# (4)
|
||
|
|
if k >= 2 and max(h) == h[ind_best]:
|
||
|
|
break
|
||
|
|
# "Sort h so that h_first >= ... >= h_last
|
||
|
|
# and re-order ind correspondingly."
|
||
|
|
#
|
||
|
|
# Later on, we will need at most t+len(ind_hist) largest
|
||
|
|
# entries, so drop the rest
|
||
|
|
ind = np.argsort(h)[::-1][:t+len(ind_hist)].copy()
|
||
|
|
del h
|
||
|
|
if t > 1:
|
||
|
|
# (5)
|
||
|
|
# Break if the most promising t vectors have been visited already.
|
||
|
|
if np.in1d(ind[:t], ind_hist).all():
|
||
|
|
break
|
||
|
|
# Put the most promising unvisited vectors at the front of the list
|
||
|
|
# and put the visited vectors at the end of the list.
|
||
|
|
# Preserve the order of the indices induced by the ordering of h.
|
||
|
|
seen = np.in1d(ind, ind_hist)
|
||
|
|
ind = np.concatenate((ind[~seen], ind[seen]))
|
||
|
|
for j in range(t):
|
||
|
|
X[:, j] = elementary_vector(n, ind[j])
|
||
|
|
|
||
|
|
new_ind = ind[:t][~np.in1d(ind[:t], ind_hist)]
|
||
|
|
ind_hist = np.concatenate((ind_hist, new_ind))
|
||
|
|
k += 1
|
||
|
|
v = elementary_vector(n, ind_best)
|
||
|
|
return est, v, w, nmults, nresamples
|