--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/pylearn/algorithms/pca_online_estimator.py	Mon Jan 31 12:23:20 2011 -0500
@@ -0,0 +1,191 @@
+# Copyright 2009 PA Manzagol (manzagop AT iro DOT umontreal DOT ca)
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+# You may obtain a copy of the License at
+#
+#      http://www.apache.org/licenses/LICENSE-2.0
+#
+# Unless required by applicable law or agreed to in writing, software
+# distributed under the License is distributed on an "AS IS" BASIS,
+# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+# See the License for the specific language governing permissions and
+# limitations under the License.
+#
+
+import numpy
+from scipy import linalg
+
+# Todo:
+#   - complete docstring (explain arguments, pseudo code)
+#   - Consider case with discount = 1.0
+#   - reevaluation when not at the end of a minibatch
+
+class PcaOnlineEstimator(object):
+  """Online estimation of the leading eigen values/vectors of the covariance
+  of some samples.
+
+  Maintains a moving (with discount) low rank (n_eigen) estimate of the
+  covariance matrix of some observations. New observations are accumulated
+  until the batch is complete, at which point the low rank estimate is 
+  reevaluated.
+
+  Example:
+
+    pca_esti = pca_online_estimator.PcaOnlineEstimator(dimension_of_the_samples)
+
+    for i in range(number_of_samples):
+      pca_esti.observe(samples[i])
+
+    [eigvals, eigvecs] = pca_esti.getLeadingEigen()
+
+  """
+
+
+  def __init__(self, n_dim, n_eigen = 10, minibatch_size = 25, gamma = 0.999, regularizer = 1e-6, centering = True):
+    # dimension of the observations
+    self.n_dim = n_dim 
+    # rank of the low-rank estimate
+    self.n_eigen = n_eigen
+    # how many observations between reevaluations of the low rank estimate
+    self.minibatch_size = minibatch_size
+    # the discount factor in the moving estimate
+    self.gamma = gamma
+    # regularizer of the covariance estimate
+    self.regularizer = regularizer
+    # wether we center the observations or not: obtain leading eigen of
+    # covariance (centering = True) vs second moment (centering = False)
+    self.centering = centering
+
+    # Total number of observations: to compute the normalizer for the mean and
+    # the covariance.
+    self.n_observations = 0
+    # Index in the current minibatch
+    self.minibatch_index = 0
+
+    # Matrix containing on its *rows*:
+    # - the current unnormalized eigen vector estimates
+    # - the observations since the last reevaluation
+    self.Xt = numpy.zeros([self.n_eigen + self.minibatch_size, self.n_dim])
+
+    # The discounted sum of the observations.
+    self.x_sum = numpy.zeros([self.n_dim])
+
+    # The Gram matrix of the observations, ie Xt Xt' (since Xt is rowwise)
+    self.G = numpy.zeros([self.n_eigen + self.minibatch_size, self.n_eigen + self.minibatch_size])
+    for i in range(self.n_eigen):
+      self.G[i,i] = self.regularizer
+
+    # I don't think it's worth "allocating" these 3 next (though they need to be
+    # declared). I don't know how to do in place operations... 
+
+    # Hold the results of the eigendecomposition of the Gram matrix G
+    # (eigen vectors on columns of V).
+    self.d = numpy.zeros([self.n_eigen + self.minibatch_size])
+    self.V = numpy.zeros([self.n_eigen + self.minibatch_size, self.n_eigen + self.minibatch_size])
+
+    # Holds the unnormalized eigenvectors of the covariance matrix before
+    # they're copied back to Xt.
+    self.Ut = numpy.zeros([self.n_eigen, self.n_dim])
+
+
+  def observe(self, x):
+    assert(numpy.size(x) == self.n_dim)
+
+    self.n_observations += 1
+
+    # Add the *non-centered* observation to Xt.
+    row = self.n_eigen + self.minibatch_index
+    self.Xt[row] = x
+
+    # Update the discounted sum of the observations.
+    self.x_sum *= self.gamma
+    self.x_sum += x
+
+    # To get the mean, we must normalize the sum by:
+    # /gamma^(n_observations-1) + /gamma^(n_observations-2) + ... + 1
+    normalizer = (1.0 - pow(self.gamma, self.n_observations)) /(1.0 - self.gamma);
+    #print "normalizer: ", normalizer
+
+    # Now center the observation.
+    # We will lose the first observation as it is the only one in the mean.
+    if self.centering:
+      self.Xt[row] -= self.x_sum / normalizer 
+
+    # Multiply the observation by the discount compensator. Basically
+    # we make this observation look "younger" than the previous ones. The actual
+    # discount is applied in the reevaluation (and when solving the equations in
+    # the case of TONGA) by multiplying every direction with the same aging factor.
+    rn = pow(self.gamma, -0.5*(self.minibatch_index+1));
+    self.Xt[row] *= rn
+
+    # Update the Gram Matrix.
+    # The column.
+    self.G[:row+1,row] = numpy.dot( self.Xt[:row+1,:], self.Xt[row,:].transpose() )
+    # The symetric row.
+    # There are row+1 values, but the diag doesn't need to get copied.
+    self.G[row,:row] = self.G[:row,row].transpose()
+
+    self.minibatch_index += 1
+
+    if self.minibatch_index == self.minibatch_size:
+      self.reevaluate()
+
+
+  def reevaluate(self):
+    # TODO do the modifications to handle when this is not true.
+    assert(self.minibatch_index == self.minibatch_size);
+
+    # Regularize - not necessary but in case
+    for i in range(self.n_eigen + self.minibatch_size):
+      self.G[i,i] += self.regularizer
+
+    # The Gram matrix is up to date. Get its low rank eigendecomposition.
+    # NOTE: the eigenvalues are in ASCENDING order and the vectors are on
+    # the columns. 
+    # With scipy 0.7, you can ask for only some eigenvalues (the n_eigen top
+    # ones) but it doesn't look loke it for scipy 0.6.
+    self.d, self.V = linalg.eigh(self.G) #, overwrite_a=True)
+
+    # Convert the n_eigen LAST eigenvectors of the Gram matrix contained in V
+    # into *unnormalized* eigenvectors U of the covariance (unnormalized wrt 
+    # the eigen values, not the moving average).
+    self.Ut = numpy.dot(self.V[:,-self.n_eigen:].transpose(), self.Xt)
+
+    # Take into account the discount factor.
+    # Here, minibatch index is minibatch_size. We age everyone. Because of the
+    # previous multiplications to make some observations "younger" we multiply
+    # everyone by the same factor.
+    # TODO VERIFY THIS!
+    rn = pow(self.gamma, -0.5*(self.minibatch_index+1))
+    inv_rn2 = 1.0/(rn*rn)
+    self.Ut *= 1.0/rn
+    self.d *= inv_rn2;
+
+    #print "*** Reevaluate! ***"
+    #normalizer = (1.0 - pow(self.gamma, self.n_observations)) /(1.0 - self.gamma)
+    #print "normalizer: ", normalizer
+    #print self.d / normalizer
+    #print self.Ut # unnormalized eigen vectors (wrt eigenvalues AND moving average).
+
+    # Update Xt, G and minibatch_index
+    self.Xt[:self.n_eigen,:] = self.Ut
+
+    for i in range(self.n_eigen):
+      self.G[i,i] = self.d[-self.n_eigen+i]
+
+    self.minibatch_index = 0
+
+  # Returns a copy of the current estimate of the eigen values and vectors
+  # (normalized vectors on rows), normalized by the discounted number of observations.
+  def getLeadingEigen(self):
+    # We subtract self.minibatch_index in case this call is not right after a reevaluate call.
+    normalizer = (1.0 - pow(self.gamma, self.n_observations - self.minibatch_index)) /(1.0 - self.gamma)
+
+    eigvals = self.d[-self.n_eigen:] / normalizer
+    eigvecs = numpy.zeros([self.n_eigen, self.n_dim])
+    for i in range(self.n_eigen):
+      eigvecs[i] = self.Ut[-self.n_eigen+i] / numpy.sqrt(numpy.dot(self.Ut[-self.n_eigen+i], self.Ut[-self.n_eigen+i]))
+
+    return [eigvals, eigvecs]
+