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decompositions.py
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decompositions.py
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# -*- coding: utf-8 -*-
from __future__ import absolute_import, division, print_function
import math
from multiprocessing import Array, Value
from numbers import Number
from typing import Optional
from typing_extensions import Literal
import numpy as np
from scipy import linalg
from six import string_types
from sklearn.decomposition import PCA, IncrementalPCA
from sklearn.utils import (as_float_array, check_array, check_random_state,
gen_batches)
from sklearn.utils.extmath import (_incremental_mean_and_var, randomized_svd,
svd_flip)
from sklearn.utils.validation import check_is_fitted
from odin.ml.base import BaseEstimator, TransformerMixin
from odin.utils import Progbar, minibatch, ctext, flatten_list
from odin.utils.mpi import MPI
__all__ = [
"fast_pca",
"MiniBatchPCA",
"PPCA",
"SupervisedPPCA",
]
def fast_pca(
*x,
n_components: Optional[int] = None,
algo: Literal['pca', 'ipca', 'ppca', 'sppca', 'plda', 'rpca'] = 'pca',
y=None,
batch_size: int = 1024,
return_model: bool = False,
random_state: int = 1,
):
r""" A shortcut for many different PCA algorithms
Arguments:
x : {list, tuple}
list of matrices for transformation, the first matrix will
be used for training
n_components : {None, int}
number of PCA components
algo : {'pca', 'ipca', 'ppca', 'sppca', 'plda', 'rpca'}
different PCA algorithm:
'ipca' - IncrementalPCA,
'ppca' - Probabilistic PCA,
'sppca' - Supervised Probabilistic PCA,
'plda' - Probabilistic LDA,
'rpca' - randomized PCA using randomized SVD
'pca' - Normal PCA
y : {numpy.ndarray, None}
required for labels in case of `sppca`
batch_size : int (default: 1024)
batch size, only used for IncrementalPCA
return_model : bool (default: False)
if True, return the trained PCA model as the FIRST return
"""
try:
from cuml.decomposition import PCA as cuPCA
except ImportError:
cuPCA = None
batch_size = int(batch_size)
algo = str(algo).lower()
if algo not in ('pca', 'ipca', 'ppca', 'sppca', 'plda', 'rpca'):
raise ValueError("`algo` must be one of the following: 'pca', "
"'ppca', 'plda', 'sppca', or 'rpca'; but given: '%s'" %
algo)
if algo in ('sppca', 'plda') and y is None:
raise RuntimeError("`y` must be not None if `algo='sppca'`")
x = flatten_list(x, level=None)
# ====== check input ====== #
x_train = x[0]
x_test = x[1:]
input_shape = None
if x_train.ndim > 2: # only 2D for PCA
input_shape = (-1,) + x_train.shape[1:]
new_shape = (-1, np.prod(input_shape[1:]))
x_train = np.reshape(x_train, new_shape)
x_test = [np.reshape(x, new_shape) for x in x_test]
if n_components is not None: # no need to reshape back
input_shape = None
# ====== train PCA ====== #
if algo == 'sppca':
pca = SupervisedPPCA(n_components=n_components, random_state=random_state)
pca.fit(x_train, y)
elif algo == 'plda':
from odin.ml import PLDA
pca = PLDA(n_phi=n_components, random_state=random_state)
pca.fit(x_train, y)
elif algo == 'pca':
if x_train.shape[1] > 1000 and x_train.shape[0] > 1e5 and cuPCA is not None:
pca = cuPCA(n_components=n_components, random_state=random_state)
else:
pca = PCA(n_components=n_components, random_state=random_state)
pca.fit(x_train)
elif algo == 'rpca':
# we copy the implementation of RandomizedPCA because
# it is significantly faster than PCA(svd_solver='randomize')
pca = RandomizedPCA(n_components=n_components,
iterated_power=2,
random_state=random_state)
pca.fit(x_train)
elif algo == 'ipca':
pca = IncrementalPCA(n_components=n_components, batch_size=batch_size)
prog = Progbar(target=x_train.shape[0],
print_report=False,
print_summary=False,
name="Fitting PCA")
for start, end in minibatch(batch_size=batch_size,
n=x_train.shape[0],
seed=1234):
pca.partial_fit(x_train[start:end], check_input=False)
prog.add(end - start)
elif algo == 'ppca':
pca = PPCA(n_components=n_components, random_state=random_state)
pca.fit(x_train)
# ====== transform ====== #
x_train = pca.transform(x_train)
x_test = [pca.transform(x) for x in x_test]
# reshape back to original shape if necessary
if input_shape is not None:
x_train = np.reshape(x_train, input_shape)
x_test = [np.reshape(x, input_shape) for x in x_test]
# return the results
if len(x_test) == 0:
return x_train if not return_model else (pca, x_train)
if return_model:
return tuple([pca, x_train] + x_test)
del pca
return tuple([x_train] + x_test)
# ===========================================================================
# PPCA
# ===========================================================================
class PPCA(BaseEstimator, TransformerMixin):
""" Probabilistic Principal Components Analysis
(C) Copyright University of Eastern Finland (UEF).
Ville Vestman, [email protected],
Tomi Kinnunen, [email protected].
Parameters
----------
n_components : {int, None}
if None, keep the same dimensions as input features
bias : {vector, 'auto'} [feat_dim,]
if 'auto' take mean of training data
n_iter : {integer, 'auto'}
if 'auto', keep iterating until no more improvement (i.e. reduction in `sigma` value)
compared to the `improve_threshold`
improve_threshold : scalar
Only used in case `n_iter='auto'`
solver : {'traditional', 'simple'}
verbose: {0, 1}
showing logging information during fitting
random_state : {None, integer, numpy.random.RandomState}
Attributes
----------
V_ : [feat_dim, n_components]
total variability matrix
bias_ : [feat_dim]
bias vector
sigma_ : scalar
variance of error term
References
----------
[1] Ville Vestman and Tomi Kinnunen, "Supervector Compression
Strategies to Speed up i-vector System Development",
submitted to Speaker Odyssey 2018.
"""
def __init__(self,
n_components=None,
bias='auto',
n_iter='auto',
improve_threshold=1e-3,
solver='traditional',
verbose=0,
random_state=None):
super(PPCA, self).__init__()
if isinstance(n_components, Number):
assert n_components > 0, \
"`n_components` must be greater than 0, but given: %d" % n_components
n_components = int(n_components)
elif n_components is not None:
raise ValueError("`n_components` can be None or integer")
self.n_components_ = n_components
# ====== checking bias ====== #
if isinstance(bias, string_types):
bias = bias.strip().lower()
assert bias == 'auto', 'Invalid value for `bias`: %s' % bias
elif not isinstance(bias, (np.ndarray, Number)):
raise ValueError("`bias` can be 'auto', numpy.ndarray or a number")
self.bias_ = bias
# ====== checking solver ====== #
if solver not in ('traditional', 'simple'):
raise ValueError("`solver` must be: 'traditional', or 'simple'")
self.solver_ = solver
# ====== checking n_iter ====== #
if isinstance(n_iter, string_types):
n_iter = n_iter.lower()
assert n_iter == 'auto', 'Invalid `n_iter` value: %s' % n_iter
elif isinstance(n_iter, Number):
assert n_iter > 0, "`n_iter` must greater than 0, but given: %d" % n_iter
self.n_iter_ = n_iter
# ====== checking random_state ====== #
if random_state is None:
rand = np.random.RandomState(seed=None)
elif isinstance(random_state, Number):
rand = np.random.RandomState(seed=None)
elif isinstance(random_state, np.random.RandomState):
rand = random_state
else:
raise ValueError("No suppport for `random_state` value: %s" %
str(random_state))
self.random_state_ = rand
# ====== other dimension ====== #
self.improve_threshold_ = float(improve_threshold)
self.feat_dim_ = None
self.verbose_ = int(verbose)
def fit(self, X, y=None):
# ====== initialize ====== #
num_samples, feat_dim = X.shape
n_components = feat_dim if self.n_components_ is None else self.n_components_
if self.bias_ == 'auto':
bias = np.mean(X, 0)
elif isinstance(self.bias_, Number):
bias = np.full(shape=(feat_dim,), fill_value=self.bias_)
else:
bias = self.bias_
assert bias.shape == (feat_dim,), \
"Invialid `bias` given shape: %s, require shape: %s" % (str(bias.shape), str((feat_dim,)))
# ====== initialize parameters ====== #
V = self.random_state_.rand(feat_dim, n_components)
last_sigma = None
sigma = 1
centeredM = X - bias[np.newaxis, :]
varianceM = np.sum(centeredM**2) / (num_samples * feat_dim)
# ====== training ====== #
if self.verbose_:
print(
'[PPCA]n_components: %d n_sample: %d feat_dim: %d n_iter: %d threshold: %f solver: %s'
% (n_components, num_samples, feat_dim, -1 if self.n_iter_ == 'auto'
else self.n_iter_, self.improve_threshold_, self.solver_))
curr_n_iter = 0
while True:
B = (V * 1 / sigma).T # [feat_dim, n_components]
Sigma = np.linalg.inv(np.eye(n_components) +
np.dot(B, V)) # [n_components, n_components]
my = np.dot(np.dot(Sigma, B), centeredM.T) # [n_components, num_samples]
if self.solver_ == 'traditional':
sumEmm = num_samples * Sigma + np.dot(my, my.T)
elif self.solver_ == 'simple':
sumEmm = np.dot(my, my.T)
sumEmmInv = np.linalg.inv(sumEmm) # [n_components, n_components]
# updating V and sigma for next iteration
V = np.dot(np.dot(centeredM.T, my.T),
sumEmmInv) # [feat_dim, n_components]
last_sigma = sigma
sigma = varianceM - np.sum(
sumEmm * np.dot(V.T, V)) / (feat_dim * num_samples)
improvement = last_sigma - sigma
# log
if self.verbose_ > 0:
print("Iteration: %d sigma: %.3f improvement: %.3f" %
(curr_n_iter, sigma, improvement))
# check iteration escape
curr_n_iter += 1
if isinstance(self.n_iter_, Number):
if curr_n_iter >= self.n_iter_:
break
elif curr_n_iter > 1 and improvement < self.improve_threshold_:
break
# ====== save the model ====== #
# record new dimensions
self.feat_dim_ = feat_dim
self.n_components_ = n_components
# trained vectors and matrices
self.V_ = V
self.bias_ = bias
self.sigma_ = sigma
# pre-calculate matrix for transform
B = (V * 1 / sigma).T
Sigma = np.linalg.inv(np.eye(n_components) + np.dot(B, V))
self.extractorMatrix_ = np.dot(Sigma, B) # [n_components, feat_dim]
def transform(self, X):
"""
Parameters
----------
X : matrix [num_samples, feat_dim]
"""
assert hasattr(self, 'extractorMatrix_'), "The model hasn't `fit` on data"
assert X.shape[1] == self.feat_dim_, \
"Expect input matrix with shape: [?, %d], but give: %s" % (self.feat_dim_, str(X.shape))
ivec = np.dot(self.extractorMatrix_, (X - self.bias_[np.newaxis, :]).T)
return ivec.T
class SupervisedPPCA(PPCA):
""" Supervised Probabilistic Principal Components Analysis
(C) Copyright University of Eastern Finland (UEF).
Ville Vestman, [email protected],
Tomi Kinnunen, [email protected].
Parameters
----------
n_components : {int, None}
if None, keep the same dimensions as input features
bias : {vector, 'auto'} [feat_dim,]
if 'auto' take mean of training data
beta : scalar (default: 1)
a weight parameter (use beta = 1 as default)
n_iter : {integer, 'auto'}
if 'auto', keep iterating until no more improvement (i.e. reduction in `sigma` value)
compared to the `improve_threshold`
improve_threshold : scalar
Only used in case `n_iter='auto'`
solver : {'traditional', 'simple'}
extractor : {'supervised', 'unsupervised'}
'supervised' is the probabilistic partial least squares extractor using
both unsupervised and supervised information
verbose: {0, 1}
showing logging information during fitting
random_state : {None, integer, numpy.random.RandomState}
Attributes
----------
V_ : [feat_dim, n_components]
total variability matrix
Q_ : [feat_dim, n_components]
matrix for mapping speaker-dependent supervectors to i-vectors
sigma_ : scalar
variance of error term
rho_ : scalar
variance of error term in speaker-dependent supervector model
bias_ : [feat_dim,]
bias vector
classBias_ : [feat_dim,]
mean of speaker-dependent supervectors
"""
def __init__(self,
n_components=None,
bias='auto',
beta=1,
n_iter='auto',
improve_threshold=1e-3,
solver='traditional',
extractor='supervised',
verbose=0,
random_state=None):
super(SupervisedPPCA, self).__init__(n_components=n_components,
bias=bias,
n_iter=n_iter,
solver=solver,
improve_threshold=improve_threshold,
verbose=verbose,
random_state=random_state)
self.beta_ = float(beta)
# ====== check extractor ====== #
extractor = str(extractor).lower()
if extractor not in ('supervised', 'unsupervised'):
raise ValueError(
"`extractor` can only be: 'unsupervised' or 'supervised'")
self.extractor_ = extractor
def fit(self, X, y, z=None):
"""
Parameters
----------
X : matrix [num_samples, feat_dim]
y : vector (int) [num_samples,]
z : matrix [num_classes, feat_dim]
class-dependent feature vectors for each class from 0 to `num_classes - 1`
(in this order).
"""
# ====== initialize ====== #
num_samples, feat_dim = X.shape
num_classes = z.shape[0] if z is not None else len(np.unique(y))
n_components = feat_dim if self.n_components_ is None else self.n_components_
if self.bias_ == 'auto':
bias = np.mean(X, 0)
elif isinstance(self.bias_, Number):
bias = np.full(shape=(feat_dim,), fill_value=self.bias_)
else:
bias = self.bias_
assert bias.shape == (feat_dim,), \
"Invialid `bias` given shape: %s, require shape: %s" % (str(bias.shape), str((feat_dim,)))
# checking `y`
y = y.ravel().astype('int32')
assert y.shape[0] == num_samples, \
"Number of samples incosistent in `X`(%s) and `y`(%s)" % (str(X.shape), str(y.shape))
# checking `z`
if z is None:
z = np.empty(shape=(max(np.max(y) + 1, num_classes), feat_dim),
dtype=X.dtype)
for i in np.unique(y):
z[i, :] = np.mean(X[y == i], axis=0, keepdims=True)
else:
assert z.shape[0] == num_classes
assert z.shape[1] == feat_dim
# ====== initialize parameters ====== #
V = self.random_state_.rand(feat_dim, n_components)
Q = self.random_state_.rand(feat_dim, n_components)
last_sigma = None
sigma = 1
last_rho = None
rho = 1
centeredM = X - bias[np.newaxis, :]
varianceM = np.sum(centeredM**2) / (num_samples * feat_dim)
centeredY = z[y]
classBias = np.mean(centeredY, 0)
centeredY = centeredY - classBias[np.newaxis, :]
varianceY = np.sum(centeredY**2) / (num_samples * feat_dim)
# ====== training ====== #
if self.verbose_:
print(
'[S-PPCA]n_components: %d n_sample: %d feat_dim: %d n_iter: %d threshold: %f solver: %s'
% (n_components, num_samples, feat_dim, -1 if self.n_iter_ == 'auto'
else self.n_iter_, self.improve_threshold_, self.solver_))
curr_n_iter = 0
while True:
B = (V * 1 / sigma).T # [feat_dim, n_components]
C = (Q * self.beta_ * 1 / rho).T # [feat_dim, n_components]
Sigma = np.linalg.inv(np.eye(n_components) + np.dot(B, V) +
np.dot(C, Q)) # [n_components, n_components]
# [n_components, num_samples]
my = np.dot(Sigma, np.dot(B, centeredM.T) + np.dot(C, centeredY.T))
if self.solver_ == 'traditional':
sumEmm = num_samples * Sigma + np.dot(my, my.T)
elif self.solver_ == 'simple':
sumEmm = np.dot(my, my.T)
sumEmmInv = np.linalg.inv(sumEmm) # [n_components, n_components]
# updating V and sigma for next iteration
V = np.dot(np.dot(centeredM.T, my.T),
sumEmmInv) # [feat_dim, n_components]
Q = np.dot(np.dot(centeredY.T, my.T),
sumEmmInv) # [feat_dim, n_components]
last_sigma = sigma
sigma = varianceM - np.sum(
sumEmm * np.dot(V.T, V)) / (feat_dim * num_samples)
improvement_sigma = last_sigma - sigma
last_rho = rho
rho = varianceY - np.sum(
sumEmm * np.dot(Q.T, Q)) / (feat_dim * num_samples)
improvement_rho = last_rho - rho
# log
if self.verbose_ > 0:
print(
"Iteration: %d sigma: %.3f rho: %.3f improvement: %.3f:%.3f"
% (curr_n_iter, sigma, rho, improvement_sigma, improvement_rho))
# check iteration escape
curr_n_iter += 1
if isinstance(self.n_iter_, Number):
if curr_n_iter >= self.n_iter_:
break
elif curr_n_iter > 1 and \
improvement_sigma < self.improve_threshold_ and \
improvement_rho < self.improve_threshold_:
break
# ====== save the model ====== #
# record new dimensions
self.feat_dim_ = feat_dim
self.n_components_ = n_components
self.num_classes_ = num_classes
# trained vectors and matrices
self.V_ = V
self.Q_ = Q
self.bias_ = bias
self.classBias_ = classBias
self.sigma_ = sigma
self.rho_ = rho
# pre-calculate matrix for PPCA transform
B = (V * 1 / sigma).T
Sigma = np.linalg.inv(np.eye(n_components) + np.dot(B, V))
self.extractorMatrix_ = np.dot(Sigma, B) # [n_components, feat_dim]
# pre-calculate matrix for PPLS transform
A = np.concatenate([V, Q], axis=0) # [2 * feat_dim, n_components]
B = np.concatenate([(V * 1 / sigma).T, (Q * 1 / rho).T],
axis=-1) # [n_components, 2 * feat_dim]
sigmaW = np.linalg.inv(np.eye(n_components) +
np.dot(B, A)) # [n_components, n_components]
self.extractorMatrixPPLS_ = np.dot(sigmaW,
B) # [n_components, 2 * feat_dim]
C = np.dot(V.T,
V) + sigma * np.eye(n_components) # [n_components, n_components]
self.labelMatrix_ = np.dot(Q, np.linalg.solve(C,
V.T)) # [feat_dim, feat_dim]
def transform(self, X):
if self.extractor_ == 'unsupervised':
return super(SupervisedPPCA, self).transform(X)
else:
centeredM = X - self.bias_[np.newaxis, :]
labels = np.dot(self.labelMatrix_,
centeredM.T) + self.classBias_[:, np.newaxis]
ivec = np.dot(
self.extractorMatrixPPLS_,
np.concatenate([X.T, labels], axis=0) -
np.concatenate([self.bias_, self.classBias_])[:, np.newaxis])
return ivec.T
# ===========================================================================
# PCA
# ===========================================================================
class RandomizedPCA(BaseEstimator, TransformerMixin):
"""Principal component analysis (PCA) using randomized SVD
Linear dimensionality reduction using approximated Singular Value
Decomposition of the data and keeping only the most significant
singular vectors to project the data to a lower dimensional space.
Parameters
----------
n_components : int, optional
Maximum number of components to keep. When not given or None, this
is set to n_features (the second dimension of the training data).
copy : bool
If False, data passed to fit are overwritten and running
fit(X).transform(X) will not yield the expected results,
use fit_transform(X) instead.
iterated_power : int, default=2
Number of iterations for the power method.
whiten : bool, optional
When True (False by default) the `components_` vectors are multiplied
by the square root of (n_samples) and divided by the singular values to
ensure uncorrelated outputs with unit component-wise variances.
Whitening will remove some information from the transformed signal
(the relative variance scales of the components) but can sometime
improve the predictive accuracy of the downstream estimators by
making their data respect some hard-wired assumptions.
random_state : int, RandomState instance or None, optional, default=None
If int, random_state is the seed used by the random number generator;
If RandomState instance, random_state is the random number generator;
If None, the random number generator is the RandomState instance used
by `np.random`.
Attributes
----------
components_ : array, shape (n_components, n_features)
Components with maximum variance.
explained_variance_ratio_ : array, shape (n_components,)
Percentage of variance explained by each of the selected components.
If k is not set then all components are stored and the sum of explained
variances is equal to 1.0.
singular_values_ : array, shape (n_components,)
The singular values corresponding to each of the selected components.
The singular values are equal to the 2-norms of the ``n_components``
variables in the lower-dimensional space.
mean_ : array, shape (n_features,)
Per-feature empirical mean, estimated from the training set.
Examples
--------
>>> import numpy as np
>>> from sklearn.decomposition import RandomizedPCA
>>> X = np.array([[-1, -1], [-2, -1], [-3, -2], [1, 1], [2, 1], [3, 2]])
>>> pca = RandomizedPCA(n_components=2)
>>> pca.fit(X) # doctest: +ELLIPSIS +NORMALIZE_WHITESPACE
RandomizedPCA(copy=True, iterated_power=2, n_components=2,
random_state=None, whiten=False)
>>> print(pca.explained_variance_ratio_) # doctest: +ELLIPSIS
[ 0.99244... 0.00755...]
>>> print(pca.singular_values_) # doctest: +ELLIPSIS
[ 6.30061... 0.54980...]
References
----------
.. [Halko2009] `Finding structure with randomness: Stochastic algorithms
for constructing approximate matrix decompositions Halko, et al., 2009
(arXiv:909)`
.. [MRT] `A randomized algorithm for the decomposition of matrices
Per-Gunnar Martinsson, Vladimir Rokhlin and Mark Tygert`
"""
def __init__(self,
n_components=None,
copy=True,
iterated_power=2,
whiten=False,
random_state=None):
self.n_components = n_components
self.copy = copy
self.iterated_power = iterated_power
self.whiten = whiten
self.random_state = random_state
def fit(self, X, y=None):
"""Fit the model with X by extracting the first principal components.
Parameters
----------
X : array-like, shape (n_samples, n_features)
Training data, where n_samples in the number of samples
and n_features is the number of features.
y : Ignored.
Returns
-------
self : object
Returns the instance itself.
"""
self._fit(check_array(X))
return self
def _fit(self, X):
"""Fit the model to the data 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
-------
X : ndarray, shape (n_samples, n_features)
The input data, copied, centered and whitened when requested.
"""
random_state = check_random_state(self.random_state)
X = np.atleast_2d(as_float_array(X, copy=self.copy))
n_samples = X.shape[0]
# Center data
self.mean_ = np.mean(X, axis=0)
X -= self.mean_
if self.n_components is None:
n_components = X.shape[1]
else:
n_components = self.n_components
U, S, V = randomized_svd(X,
n_components,
n_iter=self.iterated_power,
random_state=random_state)
self.explained_variance_ = exp_var = (S**2) / (n_samples - 1)
full_var = np.var(X, ddof=1, axis=0).sum()
self.explained_variance_ratio_ = exp_var / full_var
self.singular_values_ = S # Store the singular values.
if self.whiten:
self.components_ = V / S[:, np.newaxis] * math.sqrt(n_samples)
else:
self.components_ = V
return X
def transform(self, X):
"""Apply dimensionality reduction on X.
X is projected on the first principal components previous extracted
from a training set.
Parameters
----------
X : array-like, shape (n_samples, n_features)
New data, where n_samples in the number of samples
and n_features is the number of features.
Returns
-------
X_new : array-like, shape (n_samples, n_components)
"""
check_is_fitted(self, 'mean_')
X = check_array(X)
if self.mean_ is not None:
X = X - self.mean_
X = np.dot(X, self.components_.T)
return X
def fit_transform(self, X, y=None):
"""Fit the model with X and apply the dimensionality reduction on X.
Parameters
----------
X : array-like, shape (n_samples, n_features)
New data, where n_samples in the number of samples
and n_features is the number of features.
y : Ignored.
Returns
-------
X_new : array-like, shape (n_samples, n_components)
"""
X = check_array(X)
X = self._fit(X)
return np.dot(X, self.components_.T)
def inverse_transform(self, X):
"""Transform data back to its original space.
Returns an array X_original whose transform would be X.
Parameters
----------
X : array-like, shape (n_samples, n_components)
New data, where n_samples in the number of samples
and n_components is the number of components.
Returns
-------
X_original array-like, shape (n_samples, n_features)
Notes
-----
If whitening is enabled, inverse_transform does not compute the
exact inverse operation of transform.
"""
check_is_fitted(self, 'mean_')
X_original = np.dot(X, self.components_)
if self.mean_ is not None:
X_original = X_original + self.mean_
return X_original
class MiniBatchPCA(IncrementalPCA):
""" A modified version of IncrementalPCA to effectively
support multi-processing (but not work)
Original Author: Kyle Kastner <[email protected]>
Giorgio Patrini
License: BSD 3 clause
Incremental principal components analysis (IPCA).
Linear dimensionality reduction using Singular Value Decomposition of
centered data, keeping only the most significant singular vectors to
project the data to a lower dimensional space.
Depending on the size of the input data, this algorithm can be much more
memory efficient than a PCA.
This algorithm has constant memory complexity, on the order
of ``batch_size``, enabling use of np.memmap files without loading the
entire file into memory.
The computational overhead of each SVD is
``O(batch_size * n_features ** 2)``, but only 2 * batch_size samples
remain in memory at a time. There will be ``n_samples / batch_size`` SVD
computations to get the principal components, versus 1 large SVD of
complexity ``O(n_samples * n_features ** 2)`` for PCA.
Read more in the :ref:`User Guide <IncrementalPCA>`.
Parameters
----------
n_components : int or None, (default=None)
Number of components to keep. If ``n_components `` is ``None``,
then ``n_components`` is set to ``min(n_samples, n_features)``.
batch_size : int or None, (default=None)
The number of samples to use for each batch. Only used when calling
``fit``. If ``batch_size`` is ``None``, then ``batch_size``
is inferred from the data and set to ``5 * n_features``, to provide a
balance between approximation accuracy and memory consumption.
copy : bool, (default=True)
If False, X will be overwritten. ``copy=False`` can be used to
save memory but is unsafe for general use.
whiten : bool, optional
When True (False by default) the ``components_`` vectors are divided
by ``n_samples`` times ``components_`` to ensure uncorrelated outputs
with unit component-wise variances.
Whitening will remove some information from the transformed signal
(the relative variance scales of the components) but can sometimes
improve the predictive accuracy of the downstream estimators by
making data respect some hard-wired assumptions.
Attributes
----------
components_ : array, shape (n_components, n_features)
Components with maximum variance.
explained_variance_ : array, shape (n_components,)
Variance explained by each of the selected components.
explained_variance_ratio_ : array, shape (n_components,)
Percentage of variance explained by each of the selected components.
If all components are stored, the sum of explained variances is equal
to 1.0
mean_ : array, shape (n_features,)
Per-feature empirical mean, aggregate over calls to ``partial_fit``.
var_ : array, shape (n_features,)
Per-feature empirical variance, aggregate over calls to
``partial_fit``.
noise_variance_ : float
The estimated noise covariance following the Probabilistic PCA model
from Tipping and Bishop 1999. See "Pattern Recognition and
Machine Learning" by C. Bishop, 12.2.1 p. 574 or
http://www.miketipping.com/papers/met-mppca.pdf.
n_components_ : int
The estimated number of components. Relevant when
``n_components=None``.
n_samples_seen_ : int
The number of samples processed by the estimator. Will be reset on
new calls to fit, but increments across ``partial_fit`` calls.
Notes
-----
Implements the incremental PCA model from:
`D. Ross, J. Lim, R. Lin, M. Yang, Incremental Learning for Robust Visual
Tracking, International Journal of Computer Vision, Volume 77, Issue 1-3,
pp. 125-141, May 2008.`
See http://www.cs.toronto.edu/~dross/ivt/RossLimLinYang_ijcv.pdf
This model is an extension of the Sequential Karhunen-Loeve Transform from:
`A. Levy and M. Lindenbaum, Sequential Karhunen-Loeve Basis Extraction and
its Application to Images, IEEE Transactions on Image Processing, Volume 9,
Number 8, pp. 1371-1374, August 2000.`
See http://www.cs.technion.ac.il/~mic/doc/skl-ip.pdf
We have specifically abstained from an optimization used by authors of both
papers, a QR decomposition used in specific situations to reduce the
algorithmic complexity of the SVD. The source for this technique is
`Matrix Computations, Third Edition, G. Holub and C. Van Loan, Chapter 5,
section 5.4.4, pp 252-253.`. This technique has been omitted because it is
advantageous only when decomposing a matrix with ``n_samples`` (rows)
>= 5/3 * ``n_features`` (columns), and hurts the readability of the
implemented algorithm. This would be a good opportunity for future
optimization, if it is deemed necessary.
For `multiprocessing`, you can do parallelized `partial_fit` or `transform`
but you cannot do `partial_fit` in one process and `transform` in the others.
Application
-----------
In detail, in order for PCA to work well, informally we require that
(i) The features have approximately zero mean, and
(ii) The different features have similar variances to each other.
With natural images, (ii) is already satisfied even without variance
normalization, and so we won’t perform any variance normalization.
(If you are training on audio data—say, on spectrograms—or on text data—say,
bag-of-word vectors—we will usually not perform variance normalization
either.)
By using PCA, we aim for:
(i) the features are less correlated with each other, and
(ii) the features all have the same variance.
Original link: http://ufldl.stanford.edu/tutorial/unsupervised/PCAWhitening/
References
----------
D. Ross, J. Lim, R. Lin, M. Yang. Incremental Learning for Robust Visual
Tracking, International Journal of Computer Vision, Volume 77,
Issue 1-3, pp. 125-141, May 2008.
G. Golub and C. Van Loan. Matrix Computations, Third Edition, Chapter 5,
Section 5.4.4, pp. 252-253.
See also
--------
PCA
RandomizedPCA
KernelPCA
SparsePCA
TruncatedSVD
"""
def __init__(self,
n_components=None,
whiten=False,
copy=True,
batch_size=None):
super(MiniBatchPCA, self).__init__(n_components=n_components,
whiten=whiten,
copy=copy,
batch_size=batch_size)
# some statistics
self.n_samples_seen_ = 0
self.mean_ = .0
self.var_ = .0
self.components_ = None
# if nb_samples < nb_components, then the mini batch is cached until
# we have enough samples
self._cache_batches = []
self._nb_cached_samples = 0
@property
def is_fitted(self):
return self.components_ is not None
# ==================== Training ==================== #
def fit(self, X, y=None):
"""Fit the model with X, using minibatches of size batch_size.
Parameters
----------
X: array-like, shape (n_samples, n_features)
Training data, where n_samples is the number of samples and
n_features is the number of features.
y: Passthrough for ``Pipeline`` compatibility.
Returns
-------
self: object
Returns the instance itself.
"""
X = check_array(X, copy=self.copy, dtype=[np.float64, np.float32])
n_samples, n_features = X.shape
if self.batch_size is None:
batch_size = 12 * n_features
else:
batch_size = self.batch_size
for batch in gen_batches(n_samples, batch_size):
x = X[batch]
self.partial_fit(x, check_input=False)
return self
def partial_fit(self, X, y=None, check_input=True):
"""Incremental fit with X. All of X is processed as a single batch.
Parameters
----------
X: array-like, shape (n_samples, n_features)
Training data, where n_samples is the number of samples and
n_features is the number of features.
Returns
-------
self: object
Returns the instance itself.
"""
# ====== check the samples and cahces ====== #
if check_input:
X = check_array(X, copy=self.copy, dtype=[np.float64, np.float32])
n_samples, n_features = X.shape
# check number of components
if self.n_components is None:
self.n_components_ = n_features
elif not 1 <= self.n_components <= n_features:
raise ValueError("n_components=%r invalid for n_features=%d, need "
"more rows than columns for IncrementalPCA "
"processing" % (self.n_components, n_features))
else:
self.n_components_ = self.n_components
# check the cache
if n_samples < n_features or self._nb_cached_samples > 0:
self._cache_batches.append(X)
self._nb_cached_samples += n_samples
# not enough samples yet
if self._nb_cached_samples < n_features:
return
else: # group mini batch into big batch
X = np.concatenate(self._cache_batches, axis=0)