Fisher vectors with sklearn

fisher_vector.py

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.

    Parameters
    ----------
    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.

    Returns
    -------
    fv: array_like, shape (K + 2 * D * K, )
        Fisher vector (derivatives with respect to the mixing weights, means
        and variances) of the given descriptors.

    Reference
    ---------
    J. Krapac, J. Verbeek, F. Jurie.  Modeling Spatial Layout with Fisher
    Vectors for Image Categorization.  In ICCV, 2011.
    http://hal.inria.fr/docs/00/61/94/03/PDF/final.r1.pdf

    """
    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 = np.dot(Q.T, xx) / N
    Q_xx_2 = np.dot(Q.T, 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')
    gmm.fit(xx_tr)

    fv = fisher_vector(xx_te, gmm)
    pdb.set_trace()


if __name__ == '__main__':
    main()

@danoneata ,thanks for your explanation . Now i understand how this equation is formed. But why inv(gmm.covars) is not used in d_mu equation according to equation 16?

But why inv(gmm.covars) is not used in d_mu equation according to equation 16?

Yes, that's a valid observation, @khizerali! The reason is similar to what I've previously explained when motivating the missing 0.5 factor in d_sigma – since I'm standardizing the Fisher vectors, any linear transformation on a given dimension will be canceled away (and in this case, the inverse covariance represents a constant scaling applied to the d_mu component of all Fisher vectors). Practically, avoiding the multiplication with gmm.covars_ ** -1 doesn't change the results, but saves some computation. In fact, now I notice that for the same reason, I could have also avoided subtracting the GMM weights when computing d_pi.

Thanks for the implementation.
I had a question about your implementation. I would be grateful if you could reply.

I just realized there is no Fisher information matrix in your implementation. However, In the paper "Fisher Kernels on Visual Vocabularies for Image Categorization" authors mentioned:
To normalize the dynamic range of the different dimensions of the gradient vectors, we need to compute the diagonal of the Fisher information matrix F.

So isn't this degrading performance of FV if you are not using Fisher information matrix F?

Hello @sobhanhemati You are right, my code doesn't include the normalization with the Fisher information matrix. In practice, we usually approximate this normalization by standardizing the Fisher vectors (scaling to zero mean and unit variance); the implementation will look something along these lines:

from sklearn.preprocessing import StandardScaler
fvs = np.vstack([fisher_vector(get_descs(img), gmm) for img in imgs])
scaler = StandardScaler()
fvs = scaler.fit(fvs).transform(fvs)

Standardizing the Fisher vectors corresponds to using a diagonal approximation of the sample covariance matrix of the Fisher vectors. For more information please check section 3.5 in (Krapac et al., 2011) and for an empirical evaluation of the performance see page 9 (approximate FIM vs. empirical FIM) in (Sanchez et al., 2013); the latter report the following accuracies for image classification (so the higher the values the better):

• 61.8% for analytical diagonal approximation;
• 60.6% for empirical diagonal approximation (the approach mentioned above);
• 59.8% for using the identity matrix as the Fisher information matrix (that is, performing no normalization).

Hope this helps!

P.S.: If the dimensionality of your data allows, you can also estimate the full sample covariance matrix (which is equivalent to whitening the Fisher vectors).

Thank you for clarification.
Do you have any implementation of the analytical diagonal approximation so that I can add that the current implementation?
It seems that analytical diagonal approximation works about 1 percent better :))
Thank you in advance

@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.).

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

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