103 lines
2.9 KiB
Python
103 lines
2.9 KiB
Python
# Sebastian Raschka 2014-2020
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# mlxtend Machine Learning Library Extensions
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#
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# Functions for different linear algebra operations.
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# Author: Sebastian Raschka <sebastianraschka.com>
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#
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# License: BSD 3 clause
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import numpy as np
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def vectorspace_orthonormalization(ary, # method='gram-schmidt',
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eps=1e-13):
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"""Transforms a set of column vectors to a orthonormal basis.
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Given a set of orthogonal vectors, this functions converts such
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column vectors, arranged in a matrix, into orthonormal basis
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vectors.
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Parameters
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----------
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ary : array-like, shape=[num_vectors, num_vectors]
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An orthogonal set of vectors (arranged as columns in a matrix)
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eps : float (default: 1e-13)
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A small tolerance value to determine whether
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the vector norm is zero or not.
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Returns
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----------
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arr : array-like, shape=[num_vectors, num_vectors]
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An orthonormal set of vectors (arranged as columns)
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Examples
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-----------
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For usage examples, please see
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http://rasbt.github.io/mlxtend/user_guide/math/vectorspace_orthonormalization/
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"""
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# Gram-Schmidt Process
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# 1) col i for i = 0: keep & normalize:
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# 2) col i for i > 1:
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# 2a) project column i+1 onto i
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# 2b) subtract column i from projection vector
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# 2c) Normalize if linearly independent,
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# and set to zero otherwise
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arr = ary.astype(np.float).copy()
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for i in range(arr.shape[1]):
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for j in range(i):
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# 2a) & 2b)
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arr[:, i] -= np.dot(arr[:, i], arr[:, j]) * arr[:, j]
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# 2c
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tmp = np.linalg.norm(arr[:, i])
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is_linearly_indepedent = tmp > eps
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if is_linearly_indepedent:
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arr[:, i] /= tmp
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else:
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arr[:, i] = np.zeros(arr[:, i].shape)
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# elif method == 'qr-factorization':
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# Q, R = np.linalg.qr(ary)
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# arr = Q
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# QR factorization is not used here because of non-useful
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# results in cases of linear dependence
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# else:
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# raise ValueError("Method must be 'gram-schmidt'"
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# "or 'qr-factorization'")
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return arr
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def vectorspace_dimensionality(ary):
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"""Computes the hyper-volume spanned by a vector set
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Parameters
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----------
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ary : array-like, shape=[num_vectors, num_vectors]
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An orthogonal set of vectors (arranged as columns in a matrix)
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Returns
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----------
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dimensions : int
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An integer indicating the "dimensionality" hyper-volume spanned by
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the vector set
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Examples
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-----------
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For usage examples, please see
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http://rasbt.github.io/mlxtend/user_guide/math/vectorspace_dimensionality/
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"""
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# Note that since the vectors of
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# an orthonormal vectoset have unit length or are zero,
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# the sum of the individual
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# norms equals the dimensionality of that vector space
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return int(np.sum(np.linalg.norm(
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vectorspace_orthonormalization(ary), axis=0)))
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