![]() ![]() You can also play with Deep Sets and the variants for enforcing the permutation invariance in the neural network itself.ĭigression: Using a “Deep Sets” model, results where in between the vanilla MLP (worst results) and the pure permutation invariant approach using the representative (best results): To make it simple, we enforce the permutation invariance using the representative trick. the same model which is permutation invariant. ![]() Now we can benchmark the performance of a vanilla Multi-Layer Perceptron (MLP) vs. The trick to find this representative is to use the permutation defined by sorting the rows (or columns) according to the sum of their coefficients (probability of ties being negligible).Įnough introduction. The second one returns a correlation matrix which can be considered as the representative of the equivalence class. The first function generates several equivalent correlation matrices from a given one by applying the list of permutations given in parameter. The following two functions are used to permute rows and columns of a given correlation matrix. zeros (( k, k )) next_corr = prev_corr next_corr = 1 next_corr = q next_corr = q prev_corr = next_corr return next_corr dot ( sqrtm ( prev_corr ), w ) next_corr = np. dot ( r, theta ) # set q = prev_corr**(1/2) w q = np. sqrt ( y ) # sample a unit vector theta uniformly # from the unit ball surface B^(k-1) v = randn ( k - 1 ) theta = v / np. ones ( 1 )) for k in range ( 2, d ): # sample y = r^2 from a beta distribution # with alpha_1 = (k-1)/2 and alpha_2 = (d-k)/2 y = beta (( k - 1 ) / 2, ( d - k ) / 2 ) r = np. Now, let’s illustrate all that with some code:įrom numpy.random import beta from numpy.random import randn from scipy.linalg import sqrtm from numpy.random import seed def sample_unif_correlmat ( dimension ): d = dimension + 1 prev_corr = np. Since the factorial function grows amazingly fast, it prevents a brute force approach, i.e. One explanation is that the standard neural networks are not data efficient: For 1 point (input) of dimension n, there exist n! equivalent points (inputs). TL DR When permutation invariance matters, standard neural networks can underperform by orders of magnitude special architectures designed to deal with permutation invariance. Given a matrix (known to be a member of one of the equivalence classes), output the index (an integer) of the class. Classification task:Ĭonsider a finite number of equivalence classes for the relation “there exists a permutation of the rows and columns of the matrix such that they are equal” which are indexed by an integer. Given a set of coefficients (the upper diagonal of a correlation matrix), output the sum of its values. This example comes from my recent attempts at developping a Generative Adversarial Network (GAN) to sample realistic financial correlation matrices ( blog 1, blog 2, blog 3): I need a discriminator network which can be able to classify whether a given correlation matrix is realistic or not. I illustrate the point on a concrete example: A regression and a classification tasks whose inputs are the coefficients of a correlation matrix. In this blog, we highlight the limitations of a naive approach which puts too much faith in standard neural network architectures to solve any problems. For some tasks, this is an unwanted behaviour. That is changing the order of their inputs may yield to different outputs as illustrated above. Standard neural networks are not permutation invariant. Permutation invariance in Neural networks ![]()
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