755 lines
25 KiB
Python
755 lines
25 KiB
Python
"""Matrix factorization with Sparse PCA"""
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# Author: Vlad Niculae, Gael Varoquaux, Alexandre Gramfort
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# License: BSD
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import time
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import sys
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from math import sqrt, floor, ceil
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import itertools
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import numpy as np
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from numpy.lib.stride_tricks import as_strided
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from scipy import linalg
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from ..utils import check_random_state
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from ..utils import gen_even_slices
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from ..utils.extmath import fast_svd
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from ..linear_model import Lasso, lars_path, ridge_regression
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from ..externals.joblib import Parallel, delayed, cpu_count
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from ..base import BaseEstimator, TransformerMixin
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def _update_code(dictionary, Y, alpha, code=None, Gram=None, method='lars',
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tol=1e-8):
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"""Update the sparse code factor in the sparse_pca loop.
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Each column of the result is the solution to a Lasso problem.
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Parameters
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----------
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dictionary: array of shape (n_samples, n_components)
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Dictionary against which to optimize the sparse code.
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Y: array of shape (n_samples, n_features)
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Data matrix.
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alpha: float
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Regularization parameter for the Lasso problem.
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code: array of shape (n_components, n_features)
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Value of the sparse codes at the previous iteration.
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Gram: array of shape (n_features, n_features)
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Precomputed Gram matrix, (Y^T * Y).
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method: {'lars', 'cd'}
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lars: uses the least angle regression method (linear_model.lars_path)
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cd: uses the coordinate descent method to compute the
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Lasso solution (linear_model.Lasso). Lars will be faster if
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the estimated components are sparse.
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tol: float
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Numerical tolerance for coordinate descent Lasso convergence.
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Only used if `method='cd'`
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Returns
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-------
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new_code : array of shape (n_components, n_features)
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The sparse codes precomputed using this iteration's dictionary
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"""
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n_features = Y.shape[1]
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n_atoms = dictionary.shape[1]
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new_code = np.empty((n_atoms, n_features))
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if Gram is None:
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Gram = np.dot(dictionary.T, dictionary)
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if method == 'lars':
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XY = np.dot(dictionary.T, Y)
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try:
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err_mgt = np.seterr(all='ignore')
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for k in range(n_features):
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# A huge amount of time is spent in this loop. It needs to be
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# tight.
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_, _, coef_path_ = lars_path(dictionary, Y[:, k], Xy=XY[:, k],
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Gram=Gram, alpha_min=alpha,
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method='lasso')
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new_code[:, k] = coef_path_[:, -1]
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finally:
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np.seterr(**err_mgt)
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elif method == 'cd':
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clf = Lasso(alpha=alpha, fit_intercept=False, precompute=Gram,
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max_iter=1000, tol=tol)
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for k in range(n_features):
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# A huge amount of time is spent in this loop. It needs to be
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# tight.
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if code is not None:
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clf.coef_ = code[:, k] # Init with previous value of Vk
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clf.fit(dictionary, Y[:, k])
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new_code[:, k] = clf.coef_
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else:
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raise NotImplemented("Lasso method %s is not implemented." % method)
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return new_code
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def _update_code_parallel(dictionary, Y, alpha, code=None, Gram=None,
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method='lars', n_jobs=1, tol=1e-8):
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"""Update the sparse factor V in the sparse_pca loop in parallel.
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The computation is spread over all the available cores.
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Parameters
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----------
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dictionary: array of shape (n_samples, n_components)
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Dictionary against which to optimize the sparse code.
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Y: array of shape (n_samples, n_features)
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Data matrix.
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alpha: float
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Regularization parameter for the Lasso problem.
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code: array of shape (n_components, n_features)
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Previous iteration of the sparse code.
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Gram: array of shape (n_features, n_features)
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Precomputed Gram matrix, (Y^T * Y).
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method: 'lars' | 'cd'
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lars: uses the least angle regression method (linear_model.lars_path)
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cd: uses the coordinate descent method to compute the
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lasso solution (linear_model.Lasso). Lars will be faster if
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the components extracted are sparse.
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n_jobs: int
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Number of parallel jobs to run.
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tol: float
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Numerical tolerance for coordinate descent Lasso convergence.
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Only used if `method='cd`.
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"""
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n_samples, n_features = Y.shape
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n_atoms = dictionary.shape[1]
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if Gram is None:
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Gram = np.dot(dictionary.T, dictionary)
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if n_jobs == 1:
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return _update_code(dictionary, Y, alpha, code=code, Gram=Gram,
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method=method)
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if code is None:
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code = np.empty((n_atoms, n_features))
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slices = list(gen_even_slices(n_features, n_jobs))
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code_views = Parallel(n_jobs=n_jobs)(
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delayed(_update_code)(dictionary, Y[:, this_slice],
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code=code[:, this_slice], alpha=alpha,
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Gram=Gram, method=method, tol=tol)
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for this_slice in slices)
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for this_slice, this_view in zip(slices, code_views):
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code[:, this_slice] = this_view
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return code
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def _update_dict(dictionary, Y, code, verbose=False, return_r2=False,
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random_state=None):
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"""Update the dense dictionary factor in place.
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Parameters
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----------
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dictionary: array of shape (n_samples, n_components)
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Value of the dictionary at the previous iteration.
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Y: array of shape (n_samples, n_features)
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Data matrix.
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code: array of shape (n_components, n_features)
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Sparse coding of the data against which to optimize the dictionary.
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verbose:
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Degree of output the procedure will print.
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return_r2: bool
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Whether to compute and return the residual sum of squares corresponding
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to the computed solution.
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random_state: int or RandomState
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Pseudo number generator state used for random sampling.
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Returns
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-------
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dictionary: array of shape (n_samples, n_components)
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Updated dictionary.
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"""
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n_atoms = len(code)
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n_samples = Y.shape[0]
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random_state = check_random_state(random_state)
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# Residuals, computed 'in-place' for efficiency
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R = -np.dot(dictionary, code)
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R += Y
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R = np.asfortranarray(R)
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ger, = linalg.get_blas_funcs(('ger',), (dictionary, code))
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for k in xrange(n_atoms):
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# R <- 1.0 * U_k * V_k^T + R
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R = ger(1.0, dictionary[:, k], code[k, :], a=R, overwrite_a=True)
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dictionary[:, k] = np.dot(R, code[k, :].T)
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# Scale k'th atom
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atom_norm_square = np.dot(dictionary[:, k], dictionary[:, k])
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if atom_norm_square < 1e-20:
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if verbose == 1:
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sys.stdout.write("+")
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sys.stdout.flush()
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elif verbose:
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print "Adding new random atom"
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dictionary[:, k] = random_state.randn(n_samples)
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# Setting corresponding coefs to 0
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code[k, :] = 0.0
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dictionary[:, k] /= sqrt(np.dot(dictionary[:, k],
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dictionary[:, k]))
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else:
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dictionary[:, k] /= sqrt(atom_norm_square)
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# R <- -1.0 * U_k * V_k^T + R
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R = ger(-1.0, dictionary[:, k], code[k, :], a=R, overwrite_a=True)
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if return_r2:
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R **= 2
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# R is fortran-ordered. For numpy version < 1.6, sum does not
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# follow the quick striding first, and is thus inefficient on
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# fortran ordered data. We take a flat view of the data with no
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# striding
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R = as_strided(R, shape=(R.size, ), strides=(R.dtype.itemsize,))
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R = np.sum(R)
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return dictionary, R
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return dictionary
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def dict_learning(X, n_atoms, alpha, max_iter=100, tol=1e-8, method='lars',
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n_jobs=1, dict_init=None, code_init=None, callback=None,
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verbose=False, random_state=None):
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"""Solves a dictionary learning matrix factorization problem.
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Finds the best dictionary and the corresponding sparse code for
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approximating the data matrix X by solving::
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(U^*, V^*) = argmin 0.5 || X - U V ||_2^2 + alpha * || U ||_1
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(U,V)
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with || V_k ||_2 = 1 for all 0 <= k < n_atoms
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where V is the dictionary and U is the sparse code.
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Parameters
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----------
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X: array of shape (n_samples, n_features)
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Data matrix.
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n_atoms: int,
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Number of dictionary atoms to extract.
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alpha: int,
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Sparsity controlling parameter.
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max_iter: int,
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Maximum number of iterations to perform.
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tol: float,
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Tolerance for the stopping condition.
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method: {'lars', 'cd'}
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lars: uses the least angle regression method (linear_model.lars_path)
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cd: uses the coordinate descent method to compute the
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Lasso solution (linear_model.Lasso). Lars will be faster if
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the estimated components are sparse.
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n_jobs: int,
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Number of parallel jobs to run, or -1 to autodetect.
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dict_init: array of shape (n_atoms, n_features),
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Initial value for the dictionary for warm restart scenarios.
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code_init: array of shape (n_samples, n_atoms),
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Initial value for the sparse code for warm restart scenarios.
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callback:
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Callable that gets invoked every five iterations.
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verbose:
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Degree of output the procedure will print.
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random_state: int or RandomState
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Pseudo number generator state used for random sampling.
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Returns
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-------
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code: array of shape (n_samples, n_atoms)
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The sparse code factor in the matrix factorization.
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dictionary: array of shape (n_atoms, n_features),
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The dictionary factor in the matrix factorization.
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errors: array
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Vector of errors at each iteration.
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"""
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t0 = time.time()
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n_features = X.shape[1]
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# Avoid integer division problems
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alpha = float(alpha)
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random_state = check_random_state(random_state)
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if n_jobs == -1:
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n_jobs = cpu_count()
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# Init U and V with SVD of Y
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if code_init is not None and code_init is not None:
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code = np.array(code_init, order='F')
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# Don't copy V, it will happen below
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dictionary = dict_init
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else:
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code, S, dictionary = linalg.svd(X, full_matrices=False)
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dictionary = S[:, np.newaxis] * dictionary
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r = len(dictionary)
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if n_atoms <= r: # True even if n_atoms=None
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code = code[:, :n_atoms]
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dictionary = dictionary[:n_atoms, :]
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else:
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code = np.c_[code, np.zeros((len(code), n_atoms - r))]
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dictionary = np.r_[dictionary,
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np.zeros((n_atoms - r, dictionary.shape[1]))]
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# Fortran-order dict, as we are going to access its row vectors
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dictionary = np.array(dictionary, order='F')
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residuals = 0
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errors = []
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current_cost = np.nan
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if verbose == 1:
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print '[dict_learning]',
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for ii in xrange(max_iter):
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dt = (time.time() - t0)
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if verbose == 1:
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sys.stdout.write(".")
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sys.stdout.flush()
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elif verbose:
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print ("Iteration % 3i "
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"(elapsed time: % 3is, % 4.1fmn, current cost % 7.3f)" %
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(ii, dt, dt / 60, current_cost))
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# Update code
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code = _update_code_parallel(dictionary.T, X.T, alpha / n_features,
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code.T, method=method, n_jobs=n_jobs)
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code = code.T
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# Update dictionary
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dictionary, residuals = _update_dict(dictionary.T, X.T, code.T,
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verbose=verbose, return_r2=True,
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random_state=random_state)
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dictionary = dictionary.T
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# Cost function
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current_cost = 0.5 * residuals + alpha * np.sum(np.abs(code))
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errors.append(current_cost)
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if ii > 0:
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dE = errors[-2] - errors[-1]
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assert(dE >= -tol * errors[-1])
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if dE < tol * errors[-1]:
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if verbose == 1:
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# A line return
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print ""
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elif verbose:
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print "--- Convergence reached after %d iterations" % ii
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break
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if ii % 5 == 0 and callback is not None:
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callback(locals())
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return code, dictionary, errors
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def dict_learning_online(X, n_atoms, alpha, n_iter=100, return_code=True,
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dict_init=None, callback=None, chunk_size=3,
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verbose=False, shuffle=True, n_jobs=1,
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method='lars', iter_offset=0, random_state=None):
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"""Solves a dictionary learning matrix factorization problem online.
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Finds the best dictionary and the corresponding sparse code for
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approximating the data matrix X by solving:
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(U^*, V^*) = argmin 0.5 || X - U V ||_2^2 + alpha * || U ||_1
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(U,V)
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with || V_k ||_2 = 1 for all 0 <= k < n_atoms
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where V is the dictionary and U is the sparse code. This is
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accomplished by repeatedly iterating over mini-batches by slicing
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the input data.
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Parameters
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----------
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X: array of shape (n_samples, n_features)
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data matrix
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n_atoms: int,
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number of dictionary atoms to extract
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alpha: int,
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sparsity controlling parameter
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n_iter: int,
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number of iterations to perform
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return_code: boolean,
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whether to also return the code U or just the dictionary V
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dict_init: array of shape (n_atoms, n_features),
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initial value for the dictionary for warm restart scenarios
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callback:
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callable that gets invoked every five iterations
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chunk_size: int,
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the number of samples to take in each batch
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verbose:
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degree of output the procedure will print
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shuffle: boolean,
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whether to shuffle the data before splitting it in batches
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n_jobs: int,
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number of parallel jobs to run, or -1 to autodetect.
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method: {'lars', 'cd'}
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lars: uses the least angle regression method (linear_model.lars_path)
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cd: uses the coordinate descent method to compute the
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Lasso solution (linear_model.Lasso). Lars will be faster if
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the estimated components are sparse.
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iter_offset: int, default 0
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number of previous iterations completed on the dictionary used for
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initialization
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random_state: int or RandomState
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Pseudo number generator state used for random sampling.
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Returns
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-------
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dictionary: array of shape (n_atoms, n_features),
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the solutions to the dictionary learning problem
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code: array of shape (n_samples, n_atoms),
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the sparse code (only returned if `return_code=True`)
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"""
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t0 = time.time()
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n_samples, n_features = X.shape
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# Avoid integer division problems
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alpha = float(alpha)
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random_state = check_random_state(random_state)
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if n_jobs == -1:
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n_jobs = cpu_count()
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# Init V with SVD of X
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if dict_init is not None:
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dictionary = dict_init
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else:
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_, S, dictionary = fast_svd(X, n_atoms)
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dictionary = S[:, np.newaxis] * dictionary
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r = len(dictionary)
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if n_atoms <= r:
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dictionary = dictionary[:n_atoms, :]
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else:
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dictionary = np.r_[dictionary,
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np.zeros((n_atoms - r, dictionary.shape[1]))]
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dictionary = np.ascontiguousarray(dictionary.T)
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if verbose == 1:
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print '[dict_learning]',
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n_batches = floor(float(len(X)) / chunk_size)
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if shuffle:
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X_train = X.copy()
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random_state.shuffle(X_train)
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else:
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X_train = X
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batches = np.array_split(X_train, n_batches)
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batches = itertools.cycle(batches)
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# The covariance of the dictionary
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A = np.zeros((n_atoms, n_atoms))
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# The data approximation
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B = np.zeros((n_features, n_atoms))
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for ii, this_X in itertools.izip(xrange(iter_offset, iter_offset + n_iter),
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batches):
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dt = (time.time() - t0)
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if verbose == 1:
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sys.stdout.write(".")
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sys.stdout.flush()
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elif verbose:
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if verbose > 10 or ii % ceil(100. / verbose) == 0:
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print ("Iteration % 3i (elapsed time: % 3is, % 4.1fmn)" %
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(ii, dt, dt / 60))
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this_code = _update_code(dictionary, this_X.T, alpha, method=method)
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# Update the auxiliary variables
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if ii < chunk_size - 1:
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theta = float((ii + 1) * chunk_size)
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else:
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theta = float(chunk_size ** 2 + ii + 1 - chunk_size)
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beta = (theta + 1 - chunk_size) / (theta + 1)
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A *= beta
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A += np.dot(this_code, this_code.T)
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B *= beta
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B += np.dot(this_X.T, this_code.T)
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# Update dictionary
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dictionary = _update_dict(dictionary, B, A, verbose=verbose,
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random_state=random_state)
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# XXX: Can the residuals be of any use?
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# Maybe we need a stopping criteria based on the amount of
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# modification in the dictionary
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if callback is not None:
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callback(locals())
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if return_code:
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if verbose > 1:
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print 'Learning code...',
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elif verbose == 1:
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print '|',
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code = _update_code_parallel(dictionary, X.T, alpha, n_jobs=n_jobs,
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method=method)
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if verbose > 1:
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dt = (time.time() - t0)
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print 'done (total time: % 3is, % 4.1fmn)' % (dt, dt / 60)
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return code.T, dictionary.T
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return dictionary.T
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|
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class SparsePCA(BaseEstimator, TransformerMixin):
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|
"""Sparse Principal Components Analysis (SparsePCA)
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|
|
|
Finds the set of sparse components that can optimally reconstruct
|
|
the data. The amount of sparseness is controllable by the coefficient
|
|
of the L1 penalty, given by the parameter alpha.
|
|
|
|
Parameters
|
|
----------
|
|
n_components: int,
|
|
Number of sparse atoms to extract.
|
|
|
|
alpha: float,
|
|
Sparsity controlling parameter. Higher values lead to sparser
|
|
components.
|
|
|
|
ridge_alpha: float,
|
|
Amount of ridge shrinkage to apply in order to improve
|
|
conditioning when calling the transform method.
|
|
|
|
max_iter: int,
|
|
Maximum number of iterations to perform.
|
|
|
|
tol: float,
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|
Tolerance for the stopping condition.
|
|
|
|
method: {'lars', 'cd'}
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|
lars: uses the least angle regression method (linear_model.lars_path)
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|
cd: uses the coordinate descent method to compute the
|
|
Lasso solution (linear_model.Lasso). Lars will be faster if
|
|
the estimated components are sparse.
|
|
|
|
n_jobs: int,
|
|
Number of parallel jobs to run.
|
|
|
|
U_init: array of shape (n_samples, n_atoms),
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|
Initial values for the loadings for warm restart scenarios.
|
|
|
|
V_init: array of shape (n_atoms, n_features),
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|
Initial values for the components for warm restart scenarios.
|
|
|
|
verbose:
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|
Degree of verbosity of the printed output.
|
|
|
|
random_state: int or RandomState
|
|
Pseudo number generator state used for random sampling.
|
|
|
|
Attributes
|
|
----------
|
|
components_: array, [n_components, n_features]
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|
Sparse components extracted from the data.
|
|
|
|
error_: array
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|
Vector of errors at each iteration.
|
|
|
|
See also
|
|
--------
|
|
PCA
|
|
|
|
"""
|
|
def __init__(self, n_components, alpha=1, ridge_alpha=0.01, max_iter=1000,
|
|
tol=1e-8, method='lars', n_jobs=1, U_init=None, V_init=None,
|
|
verbose=False, random_state=None):
|
|
self.n_components = n_components
|
|
self.alpha = alpha
|
|
self.ridge_alpha = ridge_alpha
|
|
self.max_iter = max_iter
|
|
self.tol = tol
|
|
self.method = method
|
|
self.n_jobs = n_jobs
|
|
self.U_init = U_init
|
|
self.V_init = V_init
|
|
self.verbose = verbose
|
|
self.random_state = random_state
|
|
|
|
def fit(self, X, y=None):
|
|
"""Fit the model from data in X.
|
|
|
|
Parameters
|
|
----------
|
|
X: array-like, shape (n_samples, n_features)
|
|
Training vector, where n_samples in the number of samples
|
|
and n_features is the number of features.
|
|
|
|
Returns
|
|
-------
|
|
self : object
|
|
Returns the instance itself.
|
|
"""
|
|
self.random_state = check_random_state(self.random_state)
|
|
X = np.asanyarray(X)
|
|
code_init = self.V_init.T if self.V_init is not None else None
|
|
dict_init = self.U_init.T if self.U_init is not None else None
|
|
Vt, _, E = dict_learning(X.T, self.n_components, self.alpha,
|
|
tol=self.tol, max_iter=self.max_iter,
|
|
method=self.method, n_jobs=self.n_jobs,
|
|
verbose=self.verbose,
|
|
random_state=self.random_state,
|
|
code_init=code_init,
|
|
dict_init=dict_init)
|
|
self.components_ = Vt.T
|
|
self.error_ = E
|
|
return self
|
|
|
|
def transform(self, X, ridge_alpha=None):
|
|
"""Least Squares projection of the data onto the sparse components.
|
|
|
|
To avoid instability issues in case the system is under-determined,
|
|
regularization can be applied (Ridge regression) via the
|
|
`ridge_alpha` parameter.
|
|
|
|
Note that Sparse PCA components orthogonality is not enforced as in PCA
|
|
hence one cannot use a simple linear projection.
|
|
|
|
Parameters
|
|
----------
|
|
X: array of shape (n_samples, n_features)
|
|
Test data to be transformed, must have the same number of
|
|
features as the data used to train the model.
|
|
|
|
ridge_alpha: float, default: 0.01
|
|
Amount of ridge shrinkage to apply in order to improve
|
|
conditioning.
|
|
|
|
Returns
|
|
-------
|
|
X_new array, shape (n_samples, n_components)
|
|
Transformed data.
|
|
"""
|
|
ridge_alpha = self.ridge_alpha if ridge_alpha is None else ridge_alpha
|
|
U = ridge_regression(self.components_.T, X.T, ridge_alpha,
|
|
solver='dense_cholesky')
|
|
U /= np.sqrt((U ** 2).sum(axis=0))
|
|
return U
|
|
|
|
|
|
class MiniBatchSparsePCA(SparsePCA):
|
|
"""Mini-batch Sparse Principal Components Analysis
|
|
|
|
Finds the set of sparse components that can optimally reconstruct
|
|
the data. The amount of sparseness is controllable by the coefficient
|
|
of the L1 penalty, given by the parameter alpha.
|
|
|
|
Parameters
|
|
----------
|
|
n_components: int,
|
|
number of sparse atoms to extract
|
|
|
|
alpha: int,
|
|
Sparsity controlling parameter. Higher values lead to sparser
|
|
components.
|
|
|
|
ridge_alpha: float,
|
|
Amount of ridge shrinkage to apply in order to improve
|
|
conditioning when calling the transform method.
|
|
|
|
n_iter: int,
|
|
number of iterations to perform for each mini batch
|
|
|
|
callback: callable,
|
|
callable that gets invoked every five iterations
|
|
|
|
chunk_size: int,
|
|
the number of features to take in each mini batch
|
|
|
|
verbose:
|
|
degree of output the procedure will print
|
|
|
|
shuffle: boolean,
|
|
whether to shuffle the data before splitting it in batches
|
|
|
|
n_jobs: int,
|
|
number of parallel jobs to run, or -1 to autodetect.
|
|
|
|
method: {'lars', 'cd'}
|
|
lars: uses the least angle regression method (linear_model.lars_path)
|
|
cd: uses the coordinate descent method to compute the
|
|
Lasso solution (linear_model.Lasso). Lars will be faster if
|
|
the estimated components are sparse.
|
|
|
|
random_state: int or RandomState
|
|
Pseudo number generator state used for random sampling.
|
|
|
|
"""
|
|
def __init__(self, n_components, alpha=1, ridge_alpha=0.01, n_iter=100,
|
|
callback=None, chunk_size=3, verbose=False, shuffle=True,
|
|
n_jobs=1, method='lars', random_state=None):
|
|
self.n_components = n_components
|
|
self.alpha = alpha
|
|
self.ridge_alpha = ridge_alpha
|
|
self.n_iter = n_iter
|
|
self.callback = callback
|
|
self.chunk_size = chunk_size
|
|
self.verbose = verbose
|
|
self.shuffle = shuffle
|
|
self.n_jobs = n_jobs
|
|
self.method = method
|
|
self.random_state = random_state
|
|
|
|
def fit(self, X, y=None):
|
|
"""Fit the model from data in X.
|
|
|
|
Parameters
|
|
----------
|
|
X: array-like, shape (n_samples, n_features)
|
|
Training vector, where n_samples in the number of samples
|
|
and n_features is the number of features.
|
|
|
|
Returns
|
|
-------
|
|
self : object
|
|
Returns the instance itself.
|
|
"""
|
|
self.random_state = check_random_state(self.random_state)
|
|
X = np.asanyarray(X)
|
|
Vt, _ = dict_learning_online(X.T, self.n_components, alpha=self.alpha,
|
|
n_iter=self.n_iter, return_code=True,
|
|
dict_init=None, verbose=self.verbose,
|
|
callback=self.callback,
|
|
chunk_size=self.chunk_size,
|
|
shuffle=self.shuffle,
|
|
n_jobs=self.n_jobs, method=self.method,
|
|
random_state=self.random_state)
|
|
self.components_ = Vt.T
|
|
return self
|