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annotate pylearn/algorithms/pca_online_estimator.py @ 1438:8f15ef656598

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author Frederic Bastien <nouiz@nouiz.org>
date Fri, 25 Feb 2011 16:37:48 -0500
parents cedb48a300fc
children c584d8f8f280
rev   line source
1409
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1 import numpy
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2 from scipy import linalg
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3
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4 # Todo:
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5 # - complete docstring (explain arguments, pseudo code)
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6 # - Consider case with discount = 1.0
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7 # - reevaluation when not at the end of a minibatch
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8
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9 class PcaOnlineEstimator(object):
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10 """Online estimation of the leading eigen values/vectors of the covariance
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11 of some samples.
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12
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13 Maintains a moving (with discount) low rank (n_eigen) estimate of the
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14 covariance matrix of some observations. New observations are accumulated
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15 until the batch is complete, at which point the low rank estimate is
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16 reevaluated.
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17
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18 Example:
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19
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20 pca_esti = pca_online_estimator.PcaOnlineEstimator(dimension_of_the_samples)
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21
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22 for i in range(number_of_samples):
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23 pca_esti.observe(samples[i])
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24
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25 [eigvals, eigvecs] = pca_esti.getLeadingEigen()
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26
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27 """
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28
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29
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30 def __init__(self, n_dim, n_eigen = 10, minibatch_size = 25, gamma = 0.999, regularizer = 1e-6, centering = True):
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31 # dimension of the observations
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32 self.n_dim = n_dim
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33 # rank of the low-rank estimate
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34 self.n_eigen = n_eigen
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35 # how many observations between reevaluations of the low rank estimate
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36 self.minibatch_size = minibatch_size
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37 # the discount factor in the moving estimate
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38 self.gamma = gamma
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39 # regularizer of the covariance estimate
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40 self.regularizer = regularizer
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41 # wether we center the observations or not: obtain leading eigen of
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42 # covariance (centering = True) vs second moment (centering = False)
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43 self.centering = centering
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44
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45 # Total number of observations: to compute the normalizer for the mean and
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46 # the covariance.
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47 self.n_observations = 0
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48 # Index in the current minibatch
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49 self.minibatch_index = 0
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50
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51 # Matrix containing on its *rows*:
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52 # - the current unnormalized eigen vector estimates
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53 # - the observations since the last reevaluation
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54 self.Xt = numpy.zeros([self.n_eigen + self.minibatch_size, self.n_dim])
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55
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56 # The discounted sum of the observations.
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57 self.x_sum = numpy.zeros([self.n_dim])
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58
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59 # The Gram matrix of the observations, ie Xt Xt' (since Xt is rowwise)
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60 self.G = numpy.zeros([self.n_eigen + self.minibatch_size, self.n_eigen + self.minibatch_size])
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61 for i in range(self.n_eigen):
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62 self.G[i,i] = self.regularizer
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63
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64 # I don't think it's worth "allocating" these 3 next (though they need to be
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65 # declared). I don't know how to do in place operations...
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66
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67 # Hold the results of the eigendecomposition of the Gram matrix G
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68 # (eigen vectors on columns of V).
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69 self.d = numpy.zeros([self.n_eigen + self.minibatch_size])
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70 self.V = numpy.zeros([self.n_eigen + self.minibatch_size, self.n_eigen + self.minibatch_size])
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71
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72 # Holds the unnormalized eigenvectors of the covariance matrix before
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73 # they're copied back to Xt.
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74 self.Ut = numpy.zeros([self.n_eigen, self.n_dim])
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75
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76
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77 def observe(self, x):
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78 assert(numpy.size(x) == self.n_dim)
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79
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80 self.n_observations += 1
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81
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82 # Add the *non-centered* observation to Xt.
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83 row = self.n_eigen + self.minibatch_index
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84 self.Xt[row] = x
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85
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86 # Update the discounted sum of the observations.
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87 self.x_sum *= self.gamma
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88 self.x_sum += x
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89
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90 # To get the mean, we must normalize the sum by:
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91 # /gamma^(n_observations-1) + /gamma^(n_observations-2) + ... + 1
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92 normalizer = (1.0 - pow(self.gamma, self.n_observations)) /(1.0 - self.gamma);
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93 #print "normalizer: ", normalizer
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94
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95 # Now center the observation.
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96 # We will lose the first observation as it is the only one in the mean.
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97 if self.centering:
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98 self.Xt[row] -= self.x_sum / normalizer
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99
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100 # Multiply the observation by the discount compensator. Basically
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101 # we make this observation look "younger" than the previous ones. The actual
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102 # discount is applied in the reevaluation (and when solving the equations in
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103 # the case of TONGA) by multiplying every direction with the same aging factor.
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104 rn = pow(self.gamma, -0.5*(self.minibatch_index+1));
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105 self.Xt[row] *= rn
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106
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107 # Update the Gram Matrix.
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108 # The column.
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109 self.G[:row+1,row] = numpy.dot( self.Xt[:row+1,:], self.Xt[row,:].transpose() )
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110 # The symetric row.
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111 # There are row+1 values, but the diag doesn't need to get copied.
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112 self.G[row,:row] = self.G[:row,row].transpose()
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113
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114 self.minibatch_index += 1
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115
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116 if self.minibatch_index == self.minibatch_size:
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117 self.reevaluate()
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118
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119
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120 def reevaluate(self):
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121 # TODO do the modifications to handle when this is not true.
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122 assert(self.minibatch_index == self.minibatch_size);
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123
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124 # Regularize - not necessary but in case
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125 for i in range(self.n_eigen + self.minibatch_size):
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126 self.G[i,i] += self.regularizer
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127
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128 # The Gram matrix is up to date. Get its low rank eigendecomposition.
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129 # NOTE: the eigenvalues are in ASCENDING order and the vectors are on
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130 # the columns.
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131 # With scipy 0.7, you can ask for only some eigenvalues (the n_eigen top
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132 # ones) but it doesn't look loke it for scipy 0.6.
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133 self.d, self.V = linalg.eigh(self.G) #, overwrite_a=True)
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134
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135 # Convert the n_eigen LAST eigenvectors of the Gram matrix contained in V
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136 # into *unnormalized* eigenvectors U of the covariance (unnormalized wrt
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137 # the eigen values, not the moving average).
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138 self.Ut = numpy.dot(self.V[:,-self.n_eigen:].transpose(), self.Xt)
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139
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140 # Take into account the discount factor.
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141 # Here, minibatch index is minibatch_size. We age everyone. Because of the
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142 # previous multiplications to make some observations "younger" we multiply
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143 # everyone by the same factor.
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144 # TODO VERIFY THIS!
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145 rn = pow(self.gamma, -0.5*(self.minibatch_index+1))
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146 inv_rn2 = 1.0/(rn*rn)
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147 self.Ut *= 1.0/rn
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148 self.d *= inv_rn2;
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149
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150 #print "*** Reevaluate! ***"
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151 #normalizer = (1.0 - pow(self.gamma, self.n_observations)) /(1.0 - self.gamma)
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152 #print "normalizer: ", normalizer
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153 #print self.d / normalizer
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154 #print self.Ut # unnormalized eigen vectors (wrt eigenvalues AND moving average).
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155
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156 # Update Xt, G and minibatch_index
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157 self.Xt[:self.n_eigen,:] = self.Ut
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158
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159 for i in range(self.n_eigen):
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160 self.G[i,i] = self.d[-self.n_eigen+i]
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161
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162 self.minibatch_index = 0
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163
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164 # Returns a copy of the current estimate of the eigen values and vectors
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165 # (normalized vectors on rows), normalized by the discounted number of observations.
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166 def getLeadingEigen(self):
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167 # We subtract self.minibatch_index in case this call is not right after a reevaluate call.
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168 normalizer = (1.0 - pow(self.gamma, self.n_observations - self.minibatch_index)) /(1.0 - self.gamma)
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169
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170 eigvals = self.d[-self.n_eigen:] / normalizer
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171 eigvecs = numpy.zeros([self.n_eigen, self.n_dim])
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172 for i in range(self.n_eigen):
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173 eigvecs[i] = self.Ut[-self.n_eigen+i] / numpy.sqrt(numpy.dot(self.Ut[-self.n_eigen+i], self.Ut[-self.n_eigen+i]))
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174
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175 return [eigvals, eigvecs]
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176