Related papers: Geometric Approaches to Matrix Normalization and G…

Computing pseudospectra of non-normal matrices is essential for understanding the stability and transient behavior of dynamical systems. Such analysis is critical in applications including fluid dynamics, control systems, and differential…

This work analyzes the solution trajectory of gradient-based algorithms via a novel basis function decomposition. We show that, although solution trajectories of gradient-based algorithms may vary depending on the learning task, they behave…

For a simple and connected graph, several lower and upper bounds of graph invariants expressed in terms of the eigenvalues of the normalized Laplacian matrix have been proposed in literature. In this paper, through a unified approach based…

We present theoretical results on the convergence of \emph{non-convex} accelerated gradient descent in matrix factorization models with $\ell_2$-norm loss. The purpose of this work is to study the effects of acceleration in non-convex…

Matrix factorization is a popular approach for large-scale matrix completion. The optimization formulation based on matrix factorization can be solved very efficiently by standard algorithms in practice. However, due to the non-convexity…

In this paper, we mainly study the trace norm of the adjacency matrix of a graph, also known as the energy of graph. We give the maximum trace norms for the graph and its complement. In fact, the above problem is stated and solved in a more…

In this paper we form relations for the determination of the elements of the E\"otv\"os matrix of the Earth's normal gravity field. In addition a relation between the Gauss curvature of the normal equipotential surface and the Gauss…

We consider the problem of approximating a function by an element of a nonlinear manifold which admits a differentiable parametrization, typical examples being neural networks with differentiable activation functions or tensor networks.…

In 1978 Gutman introduced the energy of a graph as the sum of the absolute values of graph eigenvalues, and ever since then graph energy has been intensively studied. Since graph energy is the trace norm of the adjacency matrix, matrix…

Natural gradient descent is an optimization method traditionally motivated from the perspective of information geometry, and works well for many applications as an alternative to stochastic gradient descent. In this paper we critically…

Generating collision-free and smooth motions remains a central challenge in robotic manipulation, particularly in cluttered environments and narrow passages where feasible regions are highly constrained and fragmented. We propose a…

In dually flat manifolds, there is a deep connection between gradient flows and pregeodesics. This was one of the many important contributions of Amari to information geometry. In this paper, we extend the study of this relationship to…

Green's functions are highly useful in analyzing the dynamical behavior of polynomials in their escaping set. The aim of this paper is to construct an analogue of Green's functions for planar quasiregular mappings of degree two and constant…

While there has been a significant amount of work studying gradient descent techniques for non-convex optimization problems over the last few years, all existing results establish either local convergence with good rates or global…

Any procedure applied to data, and any quantity derived from data, is required to respect the nature and symmetries of the data. This axiom applies to refinement procedures and multiresolution transforms as well as to more basic operations…

We consider the dynamics of gradient descent (GD) in overparameterized single hidden layer neural networks with a squared loss function. Recently, it has been shown that, under some conditions, the parameter values obtained using GD achieve…

In this short note we make a few remarks on a class of generalized incidence matrices whose matroids do not depend on the orientation of the underlying graph and natural commutative algebras associated to such matrices.

We develop a geometric convergence theory for neural-network optimization within the minimizing movement scheme (MMS) framework. Reformulating each neural MMS step as a minimization over the set of increments in a Hilbert space, we show…

Several researchers have recently explored various graph parameters that can or cannot be characterized by the spectrum of a matrix associated with a graph. In this paper we show that several NP-hard zero forcing numbers are not…

The study of singular perturbations of the Dirichlet energy is at the core of the phenomenological-description paradigm in soft condensed matter. Being able to pass to the limit plays a crucial role in the understanding of the…