Related papers: On a convergence property of a geometrical algorit…

We approach the well-studied problem of supervised group invariant and equivariant machine learning from the point of view of geometric topology. We propose a novel approach using a pre-processing step, which involves projecting the input…

In this chapter, we identify fundamental geometric structures that underlie the problems of sampling, optimisation, inference and adaptive decision-making. Based on this identification, we derive algorithms that exploit these geometric…

A tutorial introduction to projective geometric algebra (PGA), a modern, coordinate-free framework for doing euclidean geometry. PGA features: uniform representation of points, lines, and planes; robust, parallel-safe join and meet…

Motivated by a wide variety of applications, ranging from stochastic optimization to dimension reduction through variable selection, the problem of estimating gradients accurately is of crucial importance in statistics and learning theory.…

A general class of Newton algorithms on Gra{\ss}mann and Lagrange-Gra{\ss}mann manifolds is introduced, that depends on an arbitrary pair of local coordinates. Local quadratic convergence of the algorithm is shown under a suitable condition…

Several recent works have explored stochastic gradient methods for variational inference that exploit the geometry of the variational-parameter space. However, the theoretical properties of these methods are not well-understood and these…

We study computing geometric problems on uncertain points. An uncertain point is a point that does not have a fixed location, but rather is described by a probability distribution. When these probability distributions are restricted to a…

A recursive estimator of the conditional geometric median in Hilbert spaces is studied. It is based on a stochastic gradient algorithm whose aim is to minimize a weighted L1 criterion and is consequently well adapted for robust online…

Nonparametric estimation of a mixing density based on observations from the corresponding mixture is a challenging statistical problem. This paper surveys the literature on a fast, recursive estimator based on the predictive recursion…

In this paper we discuss various connections between geometric discrepancy measures, such as discrepancy with respect to convex sets (and convex sets with smooth boundary in particular), and applications to numerical analysis and…

We consider the problem of estimating the support of a measure from a finite, independent, sample. The estimators which are considered are constructed based on the empirical Christoffel function. Such estimators have been proposed for the…

Stochastic gradient algorithms are more and more studied since they can deal efficiently and online with large samples in high dimensional spaces. In this paper, we first establish a Central Limit Theorem for these estimates as well as for…

In this paper, we propose and analyze a trust-region model-based algorithm for solving unconstrained stochastic optimization problems. Our framework utilizes random models of an objective function $f(x)$, obtained from stochastic…

We present a study on connection errors in networks of linear features and methods of error detection. We model networks with special connection specifications as networks with hierarchically connected features and define errors considering…

We consider the following general scheduling problem: The input consists of n jobs, each with an arbitrary release time, size, and a monotone function specifying the cost incurred when the job is completed at a particular time. The…

We study the approximate nearest neighbour method for cost-sensitive classification on low-dimensional manifolds embedded within a high-dimensional feature space. We determine the minimax learning rates for distributions on a smooth…

Learning-to-optimize leverages machine learning to accelerate optimization algorithms. While empirical results show tremendous improvements compared to classical optimization algorithms, theoretical guarantees are mostly lacking, such that…

This tutorial provides an exposition of a flexible geometric framework for high dimensional estimation problems with constraints. The tutorial develops geometric intuition about high dimensional sets, justifies it with some results of…

In this paper, we define the geometric median of a probability measure on a Riemannian manifold, give its characterization and a natural condition to ensure its uniqueness. In order to calculate the median in practical cases, we also…

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…