Related papers: The geometric mean of two matrices from a computat…

For a large class of statistical systems a geometric mean value of the observables is constrained. These observables are characterized by a power-law statistical distribution.

Geometric mechanics is usually studied in applied mathematics and most introductory texts are hence aimed at a mathematically minded audience. The present note tries to provide the intuition of geometric mechanics and to show the relevance…

Geometric symmetry induces symmetries of function spaces, and the latter yields a clue to global analysis via representation theory. In this note we summarize recent developments on the general theory about how geometric conditions affect…

This paper presents a unified geometric framework for the statistical analysis of a general ill-posed linear inverse model which includes as special cases noisy compressed sensing, sign vector recovery, trace regression, orthogonal matrix…

We provide a comprehensive study of the convergence of the forward-backward algorithm under suitable geometric conditions, such as conditioning or {\L}ojasiewicz properties. These geometrical notions are usually local by nature, and may…

The geometric median, a notion of center for multivariate distributions, has gained recent attention in robust statistics and machine learning. Although conceptually distinct from the mean (i.e., expectation), we demonstrate that both are…

Many combinatorial problems can be formulated as a polynomial optimization problem that can be solved by state-of-the-art methods in real algebraic geometry. In this paper we explain many important methods from real algebraic geometry, we…

In this review article we consider linear regression analysis from a geometric perspective, looking at standard methods and outputs in terms of the lengths of the relevant vectors and the angles between these vectors. We show that standard…

Geodesic metric spaces support a variety of averaging constructions for given finite sets. Computing such averages has generated extensive interest in diverse disciplines. Here we consider the inverse problem of recognizing computationally…

We discuss computability and computational complexity of conformal mappings and their boundary extensions. As applications, we review the state of the art regarding computability and complexity of Julia sets, their invariant measures and…

An arbitrary Mueller matrix can be decomposed into a sum of up to four deterministic Mueller-Jones matrices, with strengths given by the eigenvalues of an associated Hermitian matrix. A geometrical representation of the eigenvalues in terms…

This paper introduces a new metric and mean on the set of positive semidefinite matrices of fixed-rank. The proposed metric is derived from a well-chosen Riemannian quotient geometry that generalizes the reductive geometry of the positive…

This paper lays the foundations for a unified framework for numerically and computationally applying methods drawn from a range of currently distinct geometrical approaches to statistical modelling. In so doing, it extends information…

We review some recent applications of machine learning to algebraic geometry and physics. Since problems in algebraic geometry can typically be reformulated as mappings between tensors, this makes them particularly amenable to supervised…

Online averaged stochastic gradient algorithms are more and more studied since (i) they can deal quickly with large sample taking values in high dimensional spaces, (ii) they enable to treat data sequentially, (iii) they are known to be…

We describe the mathematical properties of pairwise comparisons matrices with coefficients in an arbitrary group. We provide a vocabulary adapted for the description of main algebraic properties of inconsistency maps, describe an example…

We shall give a refinement of the arithmetic-geometric mean inequality.

In this note we present a refinement of the AM-GM inequality, and then we estimate in a special case the typical size of the improvement.

This paper is dedicated to the analysis and detailed study of a procedure to generate both the weighted arithmetic and harmonic means of $n$ positive real numbers. Together with this interpretation, we prove some relevant properties that…

The development of computational techniques in the last decade has made possible to attack some classical problems of algebraic geometry. In this survey, we briefly describe some open problems related to algebraic curves which can be…