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Rare events can potentially occur in many applications. When manifested as opportunities to be exploited, risks to be ameliorated, or certain features to be extracted, such events become of paramount significance. Due to their sporadic…
Effective machine learning models can automatically learn useful information from a large quantity of data and provide decisions in a high accuracy. These models may, however, lead to unfair predictions in certain sense among the population…
Unsupervised feature extraction algorithms form one of the most important building blocks in machine learning systems. These algorithms are often adapted to the event-based domain to perform online learning in neuromorphic hardware.…
We establish a maximum principle for a two-point function in order to analyze the convexity of level sets of harmonic functions. We show that this can be used to prove a strict convexity result involving the smallest principal curvature of…
Some techniques for the study of intermittency by means of wavelet transforms, are presented on an example of synthetic turbulent signal. Several features of the turbulent field, that cannot be probed looking at standard structure function…
For a probability distribution $P$ on an at most countable alphabet $\mathcal A$, this article gives finite sample bounds for the expected occupancy counts $\mathbb E K_{n,r}$ and probabilities $\mathbb E M_{n,r}$. Both upper and lower…
The aim of this paper is to present an original approach that takes advantage from the geometric features of strictly convex functions to tackle the problem of finding the minimum from another perspective. The general idea is that near the…
This thesis establishes new quantitative records in several problems of incidence geometry and growth. After the necessary background in Chapters 1, 2 and 3, the following results are proven. Chapter 4 gives new results in the incidence…
This paper investigates a function of macroscopic variables known as the singular potential, building on previous work by Ball and Majumdar. The singular potential is a function of the admissible statistical averages of probability…
We illustrate the potential applications in machine learning of the Christoffel function, or more precisely, its empirical counterpart associated with a counting measure uniformly supported on a finite set of points. Firstly, we provide a…
Influence diagnostics such as influence functions and approximate maximum influence perturbations are popular in machine learning and in AI domain applications. Influence diagnostics are powerful statistical tools to identify influential…
We present a new finite-sample analysis of M-estimators of locations in $\mathbb{R}^d$ using the tool of the influence function. In particular, we show that the deviations of an M-estimator can be controlled thanks to its influence function…
Motivated by applications to the study of depth functions for tree-indexed random variables generated by point processes, we describe functional limit theorems for the intensity measure of point processes. Specifically, we establish uniform…
An approach to modelling random sets with locally finite perimeter as random elements in the corresponding subspace of $L^1$ functions is suggested. A Crofton formula for flat sections of the perimeter is shown. Finally, random processes of…
We consider threshold phenomenons in the context of weighted $\ell^2$-spaces. Our main result is a summable Baire category version of K\"orner's topological Ivashev-Musatov Theorem, which is proved to be optimal from several aspects.
The Lov\'{a}sz Local Lemma (LLL) says that, given a set of bad events that depend on the values of some random variables and where each event happens with probability at most $p$ and depends on at most $d$ other events, there is an…
We prove Pach-Sharir type incidence theorems for a class of curves in R^n and surfaces in R^3, which we call pseudoflats. In particular, our results apply to a wide class of generic irreducible real algebraic sets of bounded degree.
This paper deals with functions that defined in metric spaces and valued in complete paranormed vector spaces or valued in Banach spaces, and obtains some necessary and sufficient conditions for weak convergence of finite measures.
The aim of this paper is to establish an abstract theory based on the so-called fractional-maximal distribution functions (FMDs). From the rough ideas introduced in~\cite{AM2007}, we develop and prove some abstract results related to the…
Exceedance refers to instances where a dynamic process surpasses given thresholds, e.g., the occurrence of a heat wave. We propose a novel exceedance framework for functional data, where each observed random trajectory is transformed into…