Ozan Yildiz | Searcharxiv

arXiv2018

On Geodesically Convex Formulations for the Brascamp-Lieb Constant

We consider two non-convex formulations for computing the optimal constant in the Brascamp-Lieb inequality corresponding to a given datum, and show that they are geodesically log-concave on the manifold of positive definite matrices endowed with the Riemannian metric corresponding to the Hessian of the log-determinant function. The first formulation is present in the work of Lieb and the second is inspired by the work of Bennett et al. Recent works of Garg et al.and Allen-Zhu et al. also imply a geodesically log-concave formulation of the Brascamp-Lieb constant through a reduction to the operator scaling problem. However, the dimension of the arising optimization problem in their reduction depends exponentially on the number of bits needed to describe the Brascamp-Lieb datum. The formulations presented here have dimensions that are polynomial in the bit complexity of the input datum.

cs.DS↗

arXiv2017

Energy Saving Additive Neural Network

In recent years, machine learning techniques based on neural networks for mobile computing become increasingly popular. Classical multi-layer neural networks require matrix multiplications at each stage. Multiplication operation is not an energy efficient operation and consequently it drains the battery of the mobile device. In this paper, we propose a new energy efficient neural network with the universal approximation property over space of Lebesgue integrable functions. This network, called, additive neural network, is very suitable for mobile computing. The neural structure is based on a novel vector product definition, called ef-operator, that permits a multiplier-free implementation. In ef-operation, the "product" of two real numbers is defined as the sum of their absolute values, with the sign determined by the sign of the product of the numbers. This "product" is used to construct a vector product in $R^N$. The vector product induces the $l_1$ norm. The proposed additive neural network successfully solves the XOR problem. The experiments on MNIST dataset show that the classification performances of the proposed additive neural networks are very similar to the corresponding multi-layer perceptron and convolutional neural networks (LeNet).

cs.NE↗