# fbpca¶

Functions for principal component analysis (PCA) and accuracy checks

This module contains eight functions:

pca
principal component analysis (singular value decomposition)
eigens
eigenn
eigendecomposition of a nonnegative-definite self-adjoint matrix
diffsnorm
spectral-norm accuracy of a singular value decomposition
diffsnormc
spectral-norm accuracy of a centered singular value decomposition
diffsnorms
spectral-norm accuracy of a Schur decomposition
mult
default matrix multiplication
set_matrix_mult
re-definition of the matrix multiplication function “mult”

“Software” means the fbpca software distributed by Facebook Inc.

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fbpca.diffsnorm(A, U, s, Va, n_iter=20)

2-norm accuracy of an approx to a matrix.

Computes an estimate snorm of the spectral norm (the operator norm induced by the Euclidean vector norm) of A - U diag(s) Va, using n_iter iterations of the power method started with a random vector; n_iter must be a positive integer.

Increasing n_iter improves the accuracy of the estimate snorm of the spectral norm of A - U diag(s) Va.

Parameters: A : array_like first matrix in A - U diag(s) Va whose spectral norm is being estimated U : array_like second matrix in A - U diag(s) Va whose spectral norm is being estimated s : array_like vector in A - U diag(s) Va whose spectral norm is being estimated Va : array_like fourth matrix in A - U diag(s) Va whose spectral norm is being estimated n_iter : int, optional number of iterations of the power method to conduct; n_iter must be a positive integer, and defaults to 20 float an estimate of the spectral norm of A - U diag(s) Va (the estimate fails to be accurate with exponentially low prob. as n_iter increases; see references DM1, DM2, and DM3 below)

Notes

To obtain repeatable results, reset the seed for the pseudorandom number generator.

References

 [DM1] Jacek Kuczynski and Henryk Wozniakowski, Estimating the largest eigenvalues by the power and Lanczos methods with a random start, SIAM Journal on Matrix Analysis and Applications, 13 (4): 1094-1122, 1992.
 [DM2] Edo Liberty, Franco Woolfe, Per-Gunnar Martinsson, Vladimir Rokhlin, and Mark Tygert, Randomized algorithms for the low-rank approximation of matrices, Proceedings of the National Academy of Sciences (USA), 104 (51): 20167-20172, 2007. (See the appendix.)
 [DM3] Franco Woolfe, Edo Liberty, Vladimir Rokhlin, and Mark Tygert, A fast randomized algorithm for the approximation of matrices, Applied and Computational Harmonic Analysis, 25 (3): 335-366, 2008. (See Section 3.4.)

Examples

>>> from fbpca import diffsnorm, pca
>>> from numpy.random import uniform
>>> from scipy.linalg import svd
>>>
>>> A = uniform(low=-1.0, high=1.0, size=(100, 2))
>>> A = A.dot(uniform(low=-1.0, high=1.0, size=(2, 100)))
>>> (U, s, Va) = svd(A, full_matrices=False)
>>> A = A / s[0]
>>>
>>> (U, s, Va) = pca(A, 2, True)
>>> err = diffsnorm(A, U, s, Va)
>>> print(err)


This example produces a rank-2 approximation U diag(s) Va to A such that the columns of U are orthonormal, as are the rows of Va, and the entries of s are all nonnegative and are nonincreasing. diffsnorm(A, U, s, Va) outputs an estimate of the spectral norm of A - U diag(s) Va, which should be close to the machine precision.

fbpca.diffsnormc(A, U, s, Va, n_iter=20)

2-norm approx error to a matrix upon centering.

Computes an estimate snorm of the spectral norm (the operator norm induced by the Euclidean vector norm) of C(A) - U diag(s) Va, using n_iter iterations of the power method started with a random vector, where C(A) refers to A from the input, after centering its columns; n_iter must be a positive integer.

Increasing n_iter improves the accuracy of the estimate snorm of the spectral norm of C(A) - U diag(s) Va, where C(A) refers to A after centering its columns.

Parameters: A : array_like first matrix in the column-centered A - U diag(s) Va whose spectral norm is being estimated U : array_like second matrix in the column-centered A - U diag(s) Va whose spectral norm is being estimated s : array_like vector in the column-centered A - U diag(s) Va whose spectral norm is being estimated Va : array_like fourth matrix in the column-centered A - U diag(s) Va whose spectral norm is being estimated n_iter : int, optional number of iterations of the power method to conduct; n_iter must be a positive integer, and defaults to 20 float an estimate of the spectral norm of the column-centered A - U diag(s) Va (the estimate fails to be accurate with exponentially low probability as n_iter increases; see references DC1, DC2, and DC3 below)

Notes

To obtain repeatable results, reset the seed for the pseudorandom number generator.

References

 [DC1] Jacek Kuczynski and Henryk Wozniakowski, Estimating the largest eigenvalues by the power and Lanczos methods with a random start, SIAM Journal on Matrix Analysis and Applications, 13 (4): 1094-1122, 1992.
 [DC2] Edo Liberty, Franco Woolfe, Per-Gunnar Martinsson, Vladimir Rokhlin, and Mark Tygert, Randomized algorithms for the low-rank approximation of matrices, Proceedings of the National Academy of Sciences (USA), 104 (51): 20167-20172, 2007. (See the appendix.)
 [DC3] Franco Woolfe, Edo Liberty, Vladimir Rokhlin, and Mark Tygert, A fast randomized algorithm for the approximation of matrices, Applied and Computational Harmonic Analysis, 25 (3): 335-366, 2008. (See Section 3.4.)

Examples

>>> from fbpca import diffsnormc, pca
>>> from numpy.random import uniform
>>> from scipy.linalg import svd
>>>
>>> A = uniform(low=-1.0, high=1.0, size=(100, 2))
>>> A = A.dot(uniform(low=-1.0, high=1.0, size=(2, 100)))
>>> (U, s, Va) = svd(A, full_matrices=False)
>>> A = A / s[0]
>>>
>>> (U, s, Va) = pca(A, 2, False)
>>> err = diffsnormc(A, U, s, Va)
>>> print(err)


This example produces a rank-2 approximation U diag(s) Va to the column-centered A such that the columns of U are orthonormal, as are the rows of Va, and the entries of s are nonnegative and nonincreasing. diffsnormc(A, U, s, Va) outputs an estimate of the spectral norm of the column-centered A - U diag(s) Va, which should be close to the machine precision.

fbpca.diffsnorms(A, S, V, n_iter=20)

2-norm accuracy of a Schur decomp. of a matrix.

Computes an estimate snorm of the spectral norm (the operator norm induced by the Euclidean vector norm) of A-VSV’, using n_iter iterations of the power method started with a random vector; n_iter must be a positive integer.

Increasing n_iter improves the accuracy of the estimate snorm of the spectral norm of A-VSV’.

Parameters: A : array_like first matrix in A-VSV’ whose spectral norm is being estimated S : array_like third matrix in A-VSV’ whose spectral norm is being estimated V : array_like second matrix in A-VSV’ whose spectral norm is being estimated n_iter : int, optional number of iterations of the power method to conduct; n_iter must be a positive integer, and defaults to 20 float an estimate of the spectral norm of A-VSV’ (the estimate fails to be accurate with exponentially low probability as n_iter increases; see references DS1, DS2, and DS3 below)

Notes

To obtain repeatable results, reset the seed for the pseudorandom number generator.

References

 [DS1] Jacek Kuczynski and Henryk Wozniakowski, Estimating the largest eigenvalues by the power and Lanczos methods with a random start, SIAM Journal on Matrix Analysis and Applications, 13 (4): 1094-1122, 1992.
 [DS2] Edo Liberty, Franco Woolfe, Per-Gunnar Martinsson, Vladimir Rokhlin, and Mark Tygert, Randomized algorithms for the low-rank approximation of matrices, Proceedings of the National Academy of Sciences (USA), 104 (51): 20167-20172, 2007. (See the appendix.)
 [DS3] Franco Woolfe, Edo Liberty, Vladimir Rokhlin, and Mark Tygert, A fast randomized algorithm for the approximation of matrices, Applied and Computational Harmonic Analysis, 25 (3): 335-366, 2008. (See Section 3.4.)

Examples

>>> from fbpca import diffsnorms, eigenn
>>> from numpy import diag
>>> from numpy.random import uniform
>>> from scipy.linalg import svd
>>>
>>> A = uniform(low=-1.0, high=1.0, size=(2, 100))
>>> A = A.T.dot(A)
>>> (U, s, Va) = svd(A, full_matrices=False)
>>> A = A / s[0]
>>>
>>> (d, V) = eigenn(A, 2)
>>> err = diffsnorms(A, diag(d), V)
>>> print(err)


This example produces a rank-2 approximation V diag(d) V’ to A such that the columns of V are orthonormal and the entries of d are nonnegative and are nonincreasing. diffsnorms(A, diag(d), V) outputs an estimate of the spectral norm of A - V diag(d) V’, which should be close to the machine precision.

fbpca.eigenn(A, k=6, n_iter=4, l=None)

Eigendecomposition of a NONNEGATIVE-DEFINITE matrix.

Constructs a nearly optimal rank-k approximation V diag(d) V’ to a NONNEGATIVE-DEFINITE matrix A, using n_iter normalized power iterations, with block size l, started with an n x l random matrix, when A is n x n; the reference EGN below explains “nearly optimal.” k must be a positive integer <= the dimension n of A, n_iter must be a nonnegative integer, and l must be a positive integer >= k.

The rank-k approximation V diag(d) V’ comes in the form of an eigendecomposition – the columns of V are orthonormal and d is a real vector such that its entries are nonnegative and nonincreasing. V is n x k and len(d) = k, when A is n x n.

Increasing n_iter or l improves the accuracy of the approximation V diag(d) V’; the reference EGN below describes how the accuracy depends on n_iter and l. Please note that even n_iter=1 guarantees superb accuracy, whether or not there is any gap in the singular values of the matrix A being approximated, at least when measuring accuracy as the spectral norm || A - V diag(d) V’ || of the matrix A - V diag(d) V’ (relative to the spectral norm ||A|| of A).

Parameters: A : array_like, shape (n, n) matrix being approximated k : int, optional rank of the approximation being constructed; k must be a positive integer <= the dimension of A, and defaults to 6 n_iter : int, optional number of normalized power iterations to conduct; n_iter must be a nonnegative integer, and defaults to 4 l : int, optional block size of the normalized power iterations; l must be a positive integer >= k, and defaults to k+2 d : ndarray, shape (k,) vector of length k in the rank-k approximation V diag(d) V’ to A, such that its entries are nonnegative and nonincreasing V : ndarray, shape (n, k) n x k matrix in the rank-k approximation V diag(d) V’ to A, where A is n x n

Notes

THE MATRIX A MUST BE SELF-ADJOINT AND NONNEGATIVE DEFINITE.

To obtain repeatable results, reset the seed for the pseudorandom number generator.

The user may ascertain the accuracy of the approximation V diag(d) V’ to A by invoking diffsnorms(A, numpy.diag(d), V).

References

 [EGN] Nathan Halko, Per-Gunnar Martinsson, and Joel Tropp, Finding structure with randomness: probabilistic algorithms for constructing approximate matrix decompositions, arXiv:0909.4061 [math.NA; math.PR], 2009 (available at arXiv).

Examples

>>> from fbpca import diffsnorms, eigenn
>>> from numpy import diag
>>> from numpy.random import uniform
>>> from scipy.linalg import svd
>>>
>>> A = uniform(low=-1.0, high=1.0, size=(2, 100))
>>> A = A.T.dot(A)
>>> (U, s, Va) = svd(A, full_matrices=False)
>>> A = A / s[0]
>>>
>>> (d, V) = eigenn(A, 2)
>>> err = diffsnorms(A, diag(d), V)
>>> print(err)


This example produces a rank-2 approximation V diag(d) V’ to A such that the columns of V are orthonormal and the entries of d are nonnegative and nonincreasing. diffsnorms(A, diag(d), V) outputs an estimate of the spectral norm of A - V diag(d) V’, which should be close to the machine precision.

fbpca.eigens(A, k=6, n_iter=4, l=None)

Constructs a nearly optimal rank-k approximation V diag(d) V’ to a SELF-ADJOINT matrix A, using n_iter normalized power iterations, with block size l, started with an n x l random matrix, when A is n x n; the reference EGS below explains “nearly optimal.” k must be a positive integer <= the dimension n of A, n_iter must be a nonnegative integer, and l must be a positive integer >= k.

The rank-k approximation V diag(d) V’ comes in the form of an eigendecomposition – the columns of V are orthonormal and d is a vector whose entries are real-valued and their absolute values are nonincreasing. V is n x k and len(d) = k, when A is n x n.

Increasing n_iter or l improves the accuracy of the approximation V diag(d) V’; the reference EGS below describes how the accuracy depends on n_iter and l. Please note that even n_iter=1 guarantees superb accuracy, whether or not there is any gap in the singular values of the matrix A being approximated, at least when measuring accuracy as the spectral norm || A - V diag(d) V’ || of the matrix A - V diag(d) V’ (relative to the spectral norm ||A|| of A).

Parameters: A : array_like, shape (n, n) matrix being approximated k : int, optional rank of the approximation being constructed; k must be a positive integer <= the dimension of A, and defaults to 6 n_iter : int, optional number of normalized power iterations to conduct; n_iter must be a nonnegative integer, and defaults to 4 l : int, optional block size of the normalized power iterations; l must be a positive integer >= k, and defaults to k+2 d : ndarray, shape (k,) vector of length k in the rank-k approximation V diag(d) V’ to A, such that its entries are real-valued and their absolute values are nonincreasing V : ndarray, shape (n, k) n x k matrix in the rank-k approximation V diag(d) V’ to A, where A is n x n

Notes

THE MATRIX A MUST BE SELF-ADJOINT.

To obtain repeatable results, reset the seed for the pseudorandom number generator.

The user may ascertain the accuracy of the approximation V diag(d) V’ to A by invoking diffsnorms(A, numpy.diag(d), V).

References

 [EGS] Nathan Halko, Per-Gunnar Martinsson, and Joel Tropp, Finding structure with randomness: probabilistic algorithms for constructing approximate matrix decompositions, arXiv:0909.4061 [math.NA; math.PR], 2009 (available at arXiv).

Examples

>>> from fbpca import diffsnorms, eigens
>>> from numpy import diag
>>> from numpy.random import uniform
>>> from scipy.linalg import svd
>>>
>>> A = uniform(low=-1.0, high=1.0, size=(2, 100))
>>> A = A.T.dot(A)
>>> (U, s, Va) = svd(A, full_matrices=False)
>>> A = A / s[0]
>>>
>>> (d, V) = eigens(A, 2)
>>> err = diffsnorms(A, diag(d), V)
>>> print(err)


This example produces a rank-2 approximation V diag(d) V’ to A such that the columns of V are orthonormal, and the entries of d are real-valued and their absolute values are nonincreasing. diffsnorms(A, diag(d), V) outputs an estimate of the spectral norm of A - V diag(d) V’, which should be close to the machine precision.

fbpca.mult(A, B)

default matrix multiplication.

Multiplies A and B together via the “dot” method.

Parameters: A : array_like first matrix in the product A*B being calculated B : array_like second matrix in the product A*B being calculated array_like product of the inputs A and B

Examples

>>> from fbpca import mult
>>> from numpy import array
>>> from numpy.linalg import norm
>>>
>>> A = array([[1., 2.], [3., 4.]])
>>> B = array([[5., 6.], [7., 8.]])
>>> norm(mult(A, B) - A.dot(B))


This example multiplies two matrices two ways – once with mult, and once with the usual “dot” method – and then calculates the (Frobenius) norm of the difference (which should be near 0).

fbpca.pca(A, k=6, raw=False, n_iter=2, l=None)

Principal component analysis.

Constructs a nearly optimal rank-k approximation U diag(s) Va to A, centering the columns of A first when raw is False, using n_iter normalized power iterations, with block size l, started with a min(m,n) x l random matrix, when A is m x n; the reference PCA below explains “nearly optimal.” k must be a positive integer <= the smaller dimension of A, n_iter must be a nonnegative integer, and l must be a positive integer >= k.

The rank-k approximation U diag(s) Va comes in the form of a singular value decomposition (SVD) – the columns of U are orthonormal, as are the rows of Va, and the entries of s are all nonnegative and nonincreasing. U is m x k, Va is k x n, and len(s)=k, when A is m x n.

Increasing n_iter or l improves the accuracy of the approximation U diag(s) Va; the reference PCA below describes how the accuracy depends on n_iter and l. Please note that even n_iter=1 guarantees superb accuracy, whether or not there is any gap in the singular values of the matrix A being approximated, at least when measuring accuracy as the spectral norm || A - U diag(s) Va || of the matrix A - U diag(s) Va (relative to the spectral norm ||A|| of A, and accounting for centering when raw is False).

Parameters: A : array_like, shape (m, n) matrix being approximated k : int, optional rank of the approximation being constructed; k must be a positive integer <= the smaller dimension of A, and defaults to 6 raw : bool, optional centers A when raw is False but does not when raw is True; raw must be a Boolean and defaults to False n_iter : int, optional number of normalized power iterations to conduct; n_iter must be a nonnegative integer, and defaults to 2 l : int, optional block size of the normalized power iterations; l must be a positive integer >= k, and defaults to k+2 U : ndarray, shape (m, k) m x k matrix in the rank-k approximation U diag(s) Va to A or C(A), where A is m x n, and C(A) refers to A after centering its columns; the columns of U are orthonormal s : ndarray, shape (k,) vector of length k in the rank-k approximation U diag(s) Va to A or C(A), where A is m x n, and C(A) refers to A after centering its columns; the entries of s are all nonnegative and nonincreasing Va : ndarray, shape (k, n) k x n matrix in the rank-k approximation U diag(s) Va to A or C(A), where A is m x n, and C(A) refers to A after centering its columns; the rows of Va are orthonormal

Notes

To obtain repeatable results, reset the seed for the pseudorandom number generator.

The user may ascertain the accuracy of the approximation U diag(s) Va to A by invoking diffsnorm(A, U, s, Va), when raw is True. The user may ascertain the accuracy of the approximation U diag(s) Va to C(A), where C(A) refers to A after centering its columns, by invoking diffsnormc(A, U, s, Va), when raw is False.

References

 [PCA] Nathan Halko, Per-Gunnar Martinsson, and Joel Tropp, Finding structure with randomness: probabilistic algorithms for constructing approximate matrix decompositions, arXiv:0909.4061 [math.NA; math.PR], 2009 (available at arXiv).

Examples

>>> from fbpca import diffsnorm, pca
>>> from numpy.random import uniform
>>> from scipy.linalg import svd
>>>
>>> A = uniform(low=-1.0, high=1.0, size=(100, 2))
>>> A = A.dot(uniform(low=-1.0, high=1.0, size=(2, 100)))
>>> (U, s, Va) = svd(A, full_matrices=False)
>>> A = A / s[0]
>>>
>>> (U, s, Va) = pca(A, 2, True)
>>> err = diffsnorm(A, U, s, Va)
>>> print(err)


This example produces a rank-2 approximation U diag(s) Va to A such that the columns of U are orthonormal, as are the rows of Va, and the entries of s are all nonnegative and are nonincreasing. diffsnorm(A, U, s, Va) outputs an estimate of the spectral norm of A - U diag(s) Va, which should be close to the machine precision.

fbpca.set_matrix_mult(newmult)

re-definition of the matrix multiplication function “mult”.

Sets the matrix multiplication function “mult” used in fbpca to be the input “newmult” – which must return the product A*B of its two inputs A and B, i.e., newmult(A, B) must be the product of A and B.

Parameters: newmult : callable matrix multiplication replacing mult in fbpca; newmult must return the product of its two array_like inputs None

Examples

>>> from fbpca import set_matrix_mult
>>>
>>> def newmult(A, B):
...     return A*B
...
>>> set_matrix_mult(newmult)


This example redefines the matrix multiplication used in fbpca to be the entrywise product.