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Fisher vectors with sklearn
import numpy as np
import pdb
from sklearn.datasets import make_classification
from sklearn.mixture import GaussianMixture as GMM
def fisher_vector(xx, gmm):
"""Computes the Fisher vector on a set of descriptors.
xx: array_like, shape (N, D) or (D, )
The set of descriptors
gmm: instance of sklearn mixture.GMM object
Gauassian mixture model of the descriptors.
fv: array_like, shape (K + 2 * D * K, )
Fisher vector (derivatives with respect to the mixing weights, means
and variances) of the given descriptors.
J. Krapac, J. Verbeek, F. Jurie. Modeling Spatial Layout with Fisher
Vectors for Image Categorization. In ICCV, 2011.
xx = np.atleast_2d(xx)
N = xx.shape[0]
# Compute posterior probabilities.
Q = gmm.predict_proba(xx) # NxK
# Compute the sufficient statistics of descriptors.
Q_sum = np.sum(Q, 0)[:, np.newaxis] / N
Q_xx =, xx) / N
Q_xx_2 =, xx ** 2) / N
# Compute derivatives with respect to mixing weights, means and variances.
d_pi = Q_sum.squeeze() - gmm.weights_
d_mu = Q_xx - Q_sum * gmm.means_
d_sigma = (
- Q_xx_2
- Q_sum * gmm.means_ ** 2
+ Q_sum * gmm.covariances_
+ 2 * Q_xx * gmm.means_)
# Merge derivatives into a vector.
return np.hstack((d_pi, d_mu.flatten(), d_sigma.flatten()))
def main():
# Short demo.
K = 64
N = 1000
xx, _ = make_classification(n_samples=N)
xx_tr, xx_te = xx[: -100], xx[-100: ]
gmm = GMM(n_components=K, covariance_type='diag')
fv = fisher_vector(xx_te, gmm)
if __name__ == '__main__':
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danoneata commented Nov 25, 2020

@sobhanhemati Equations (16–18) from (Sanchez et al., 2013) provide the Fisher vectors that include the analytical approximation; hence, you can modify the computation of d_pi, d_mu, d_sigma in the gist above as follows:

    # at line 43
    s = np.sqrt(gmm.weights_)[:, np.newaxis]
    d_pi = (Q_sum.squeeze() - gmm.weights_) / s.squeeze()
    d_mu = (Q_xx - Q_sum * gmm.means_) * np.sqrt(gmm.covariances_) ** -1 / s
    d_sigma = - (
        - Q_xx_2
        - Q_sum * gmm.means_ ** 2
        + Q_sum * gmm.covariances_
        + 2 * Q_xx * gmm.means_) / (s * np.sqrt(2))

Note that I haven't tested this implementation, so you might want to double check it. And I would suggest to try both methods for estimating the diagonal Fisher information matrix and see which one works better for you — Sanchez et al. mention in their paper:

Note that we do not claim that this difference is significant nor that the closed-form approximation is superior to the empirical one in general.

Finally, do not forget to L2 and power normalise the Fisher vectors — these transformations yield much more substantial improvements (about 6-7% points each) than the choice of the approximation for the Fisher information matrix (see Table 1 from Sanchez et al.).

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Thank you so much for the comprehensive answer.
I really appreciate that.

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Hai I'm beginner so i don't know working of fisher vector encoding. Please help to understand this

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