Sparse Hopfield network reconstruction with ℓ1\ell_{1} regularization

We propose an efficient strategy to infer sparse Hopfield network based on magnetizations and pairwise correlations measured through Glauber samplings. This strategy incorporates the $\ell_{1}$ regularization into the Bethe approximation by a quadratic approximation to the log-likelihood, and is able to further reduce the inference error of the Bethe approximation without the regularization. The optimal regularization parameter is observed to be of the order of $M^{-\nu}$ where $M$ is the number of independent samples. The value of the scaling exponent depends on the performance measure. $\nu\simeq0.5001$ for root mean squared error measure while $\nu\simeq0.2743$ for misclassification rate measure. The efficiency of this strategy is demonstrated for the sparse Hopfield model, but the method is generally applicable to other diluted mean field models. In particular, it is simple in implementation without heavy computational cost.Comment: 9 pages, 3 figures, Eur. Phys. J. B (in press

Generalized incompressible flows, multi-marginal transport and Sinkhorn algorithm

Starting from Brenier's relaxed formulation of the incompressible Euler equation in terms of geodesics in the group of measure-preserving diffeomorphisms, we propose a numerical method based on Sinkhorn's algorithm for the entropic regularization of optimal transport. We also make a detailed comparison of this entropic regularization with the so-called Bredinger entropic interpolation problem. Numerical results in dimension one and two illustrate the feasibility of the method

Capacity Control of ReLU Neural Networks by Basis-path Norm

Recently, path norm was proposed as a new capacity measure for neural networks with Rectified Linear Unit (ReLU) activation function, which takes the rescaling-invariant property of ReLU into account. It has been shown that the generalization error bound in terms of the path norm explains the empirical generalization behaviors of the ReLU neural networks better than that of other capacity measures. Moreover, optimization algorithms which take path norm as the regularization term to the loss function, like Path-SGD, have been shown to achieve better generalization performance. However, the path norm counts the values of all paths, and hence the capacity measure based on path norm could be improperly influenced by the dependency among different paths. It is also known that each path of a ReLU network can be represented by a small group of linearly independent basis paths with multiplication and division operation, which indicates that the generalization behavior of the network only depends on only a few basis paths. Motivated by this, we propose a new norm \emph{Basis-path Norm} based on a group of linearly independent paths to measure the capacity of neural networks more accurately. We establish a generalization error bound based on this basis path norm, and show it explains the generalization behaviors of ReLU networks more accurately than previous capacity measures via extensive experiments. In addition, we develop optimization algorithms which minimize the empirical risk regularized by the basis-path norm. Our experiments on benchmark datasets demonstrate that the proposed regularization method achieves clearly better performance on the test set than the previous regularization approaches