hadamard (or complete $cat(0)$) spaces are complete, non-positive curvature, metric spaces. here, we prove a nonlinear ergodic theorem for continuous non-expansive semigroup in these spaces as well as a strong convergence theorem for the commutative case. our results extend the standard non-linear ergodic theorems for non-expansive maps on real hilbert spaces, to non-expansive maps on had...

In this work we propose a generalization of the Hadamard product between two matrices to a tensor-valued, multi-linear product between k matrices for any k ≥ 1. A multi-linear dual operator to the generalized Hadamard product is presented. It is a natural generalization of the Diag x operator, that maps a vector x ∈ R into the diagonal matrix with x on its main diagonal. Defining an action of t...

Hadamard matrices with a subjacent algebraic structure have been deeply studied as well as the links with other topics in algebraic combinatorics [1]. An important and pioneering paper about this subject is [5], where it is introduced the concept of Hadamard group. In addition, we find beautiful equivalences between Hadamard groups, 2-cocyclic matrices and relative difference sets [4], [7]. Fro...

In this paper we introduce the concept of geometrically quasiconvex functions on the co-ordinates and establish some Hermite-Hadamard type integral inequalities for functions defined on rectangles in the plane. Some  inequalities for product of two geometrically quasiconvex functions on the co-ordinates are considered.

Convolutional neural networks (CNNs) are currently state-of-the-art for various classification tasks, but are computationally expensive. Propagating through the convolutional layers is very slow, as each kernel in each layer must sequentially calculate many dot products for a single forward and backward propagation which equates to O(N2n2) per kernel per layer where the inputs are N×N arrays an...

Convolutional neural networks (CNNs) are currently state-of-the-art for various classification tasks, but are computationally expensive. Propagating through the convolutional layers is very slow, as each kernel in each layer must sequentially calculate many dot products for a single forward and backward propagation which equates to O(N2n2) per kernel per layer where the inputs are N×N arrays an...

6 The convolution layer 13 6.1 What is a convolution? . . . . . . . . . . . . . . . . . . . . . . . . 13 6.2 Why to convolve? . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 6.3 Convolution as matrix product . . . . . . . . . . . . . . . . . . . 18 6.4 The Kronecker product . . . . . . . . . . . . . . . . . . . . . . . 20 6.5 Backward propagation: update the parameters . . . . . . . . ...