相关论文: Indexability, concentration, and VC theory

Galaxy cluster number counts are an important probe to constrain cosmological parameters. One of the main ingredients of the analysis, along with accurate estimates of the clusters' masses, is the selection function, and in particular the…

Understanding self-supervised learning is important but challenging. Previous theoretical works study the role of pretraining losses, and view neural networks as general black boxes. However, the recent work of Saunshi et al. argues that…

In flocking models, the collective motion of self-driven individuals leads to the formation of complex spatiotemporal patterns. The Standard Vicsek Model (SVM) considers individuals that tend to adopt the direction of movement of their…

Hierarchical clustering is a popular unsupervised data analysis method. For many real-world applications, we would like to exploit prior information about the data that imposes constraints on the clustering hierarchy, and is not captured by…

We consider the density of states measure of the Fibonacci Hamiltonian and show that, for small values of the coupling constant $V$, this measure is exact-dimensional and the almost everywhere value $d_V$ of the local scaling exponent is a…

Building on the inequalities for homogeneous tetrahedral polynomials in independent Gaussian variables due to R. Lata{\l}a we provide a concentration inequality for non-necessarily Lipschitz functions $f\colon \R^n \to \R$ with bounded…

The ability to achieve coordinated behavior --engineered or emergent-- on networked systems has attracted widespread interest over several fields. This has led to remarkable advances on the development of a theoretical understanding of the…

We extend some recent results on the Hausdorff convergence of level-sets for total variation regularized linear inverse problems. Dimensions higher than two and measurements in Banach spaces are considered. We investigate the relation…

In this paper we consider well-posedness properties of vector optimization problems with objective function $f: X \to Y$ where $X$ and $Y$ are Banach spaces and $Y$ is partially ordered by a closed convex pointed cone with nonempty…

We consider the problem of visually explaining similarity models, i.e., explaining why a model predicts two images to be similar in addition to producing a scalar score. While much recent work in visual model interpretability has focused on…

A central question in the field of inhomogeneous attractors has been to relate the dimension of an inhomogeneous attractor to the condensation set and associated homogeneous attractor. This has been achieved only in specific settings, with…

We consider models of Bayesian inference of signals with vectorial components of finite dimensionality. We show that, under a proper perturbation, these models are replica symmetric in the sense that the overlap matrix concentrates. The…

Score-based diffusion models have demonstrated outstanding empirical performance in machine learning and artificial intelligence, particularly in generating high-quality new samples from complex probability distributions. Improving the…

The fundamental result of Li, Long, and Srinivasan on approximations of set systems has become a key tool across several communities such as learning theory, algorithms, computational geometry, combinatorics and data analysis. The goal of…

This paper develops a unified framework for measuring concentration in weighted systems embedded in networks of interactions. While traditional indices such as the Herfindahl-Hirschman Index capture dispersion in weights, they neglect the…

We prove concentration inequalities for functions of independent random variables {under} sub-gaussian and sub-exponential conditions. The utility of the inequalities is demonstrated by an extension of the now classical method of Rademacher…

We study the multifractal analysis of self-similar measures arising from random homogeneous iterated function systems. Under the assumption of the uniform strong separation condition, we see that this analysis parallels that of the…

This paper develops systematic approaches to obtain $f$-divergence inequalities, dealing with pairs of probability measures defined on arbitrary alphabets. Functional domination is one such approach, where special emphasis is placed on…

Clustering algorithms have long been the topic of research, representing the more popular side of unsupervised learning. Since clustering analysis is one of the best ways to find some clarity and structure within raw data, this paper…

A measure for the complexity of a differentiable function f(x) on an interval is introduced. It is based on approximations of the function by piecewise constant functions. The measure takes into account the quality of the approximation and…