Related papers: Kac-Rice fixed point analysis for single- and mult…
We study the energy landscape of a model of a single particle on a random potential, that is, we investigate the topology of level sets of smooth random fields on $\mathbb R^{N}$ of the form $X_N(x) +\frac\mu2 \|x\|^2,$ where $X_{N}$ is a…
We study a broad class of high-dimensional mean-field exchange models, encompassing both noisy and singular dynamics, along with their dual processes. This includes a generalized version of the averaging process as well as some…
This paper develops several average-case reduction techniques to show new hardness results for three central high-dimensional statistics problems, implying a statistical-computational gap induced by robustness, a detection-recovery gap and…
In this paper we lay special stress on analyzing the topological properties of the lattice systems and try to ovoid the conventional ways to calculate the critical points. Only those clusters with finite sizes can execute the self similar…
Our model is a generalized linear programming relaxation of a much studied random K-SAT problem. Specifically, a set of linear constraints C on K variables is fixed. From a pool of n variables, K variables are chosen uniformly at random and…
While matrix variate regression models have been studied in many existing works, classical statistical and computational methods for the analysis of the regression coefficient estimation are highly affected by high dimensional and noisy…
We consider a family of singular maps as an example of a simple model of dynamical systems exhibiting the property of robust chaos on a well defined range of parameters. Critical boundaries separating the region of robust chaos from the…
This paper tackles the problem of robust covariance matrix estimation when the data is incomplete. Classical statistical estimation methodologies are usually built upon the Gaussian assumption, whereas existing robust estimation ones assume…
We analytically determine the number and distribution of fixed points in a canonical model of a chaotic neural network. This distribution reveals that fixed points and dynamics are confined to separate shells in phase space. Furthermore,…
An n dimensional monomial dynamical system over a finite field K is a nonlinear deterministic time discrete dynamical system with the property that each of the n component functions is a monic nonzero monomial function in n variables. In…
The time-evolving precision matrix of a piecewise-constant Gaussian graphical model encodes the dynamic conditional dependency structure of a multivariate time-series. Traditionally, graphical models are estimated under the assumption that…
We study the probability distribution of the area and the number of vertices of random polygons in a convex set $K\subset\mathbb{R}^2$. The novel aspect of our approach is that it yields uniform estimates for all convex sets…
We construct families of rational functions $f \colon \bP^1_k \to \bP^1_k$ of degree $d \geq 2$ over a perfect field $k$ whose associated fixed-point processes fail to be martingales. Conversely, for any normal variety $X \subset…
In this paper, we introduce an iterative process which converges strongly to a common element of sets of solutions of finite family of generalized equilibrium problems, sets of fixed points of finite family of continuous relatively…
We propose a new procedure to monitor and forecast the onset of transitions in high dimensional complex systems. We describe our procedure by an application to the Tangled Nature model of evolutionary ecology. The quasi-stable…
In this paper we study random linear systems with $k$ variables per equation over the finite field GF(2), or equivalently $k$-XOR-CNF formulas. In a previous paper Creignou and Daud\'e proved that the phase transition for the consistency…
To provide generalized solutions if a given problem admits no actual solution is an important task in mathematics and the natural sciences. It has a rich history dating back to the early 19th century when Carl Friedrich Gauss developed the…
We study the limiting distribution of critical points and extrema of random spherical harmonics, in the high energy limit. In particular, we first derive the density functions of extrema and saddles; we then provide analytic expressions for…
The transformation of the free-energy landscape from smooth to hierarchical is one of the richest features of mean-field disordered systems. A well-studied example is the de Almeida-Thouless transition for spin glasses in a magnetic field,…
In classical random matrix theory the Gaussian and chiral Gaussian random matrix models with a source are realized as shifted mean Gaussian, and chiral Gaussian, random matrices with real $(\beta = 1)$, complex ($\beta = 2)$ and real…