arXiv research

We extend Kac-Rice formula to compute expected intersections of random submanifolds.

Study finds the number of modes in Gaussian kernel density estimators scales with sqrt(β log β).

Formula for critical points of chi fields on manifolds.

We establish a few formulas that compute the volume of the zero-set (or nodal set) of a function on a compact Riemannian manifold as integrals of functionals of the function and its derivatives.

We present a method to obtain the average and the typical value of the number of critical points of the empirical risk landscape for generalized linear estimation problems and variants. This represents a substantial extension of previous applications of the Kac-Rice method since it allows to analyze the critical points…

Develops calculus for random submanifolds using zonoids.

The study of topological properties of random smooth maps, focusing on Kac-Rice formula and Betti numbers.

We analyze the landscape of empirical risk minimization for high-dimensional models, predicting phase transitions and critical point properties.

The paper analyzes local minima in high-dimensional empirical risk minimization.

Non-convex optimization with local search heuristics has been widely used in machine learning, achieving many state-of-art results. It becomes increasingly important to understand why they can work for these NP-hard problems on typical data. The landscape of many objective functions in learning has been conjectured to …

We study rough high-dimensional landscapes in which an increasingly stronger preference for a given configuration emerges. Such energy landscapes arise in glass physics and inference. In particular we focus on random Gaussian functions, and on the spiked-tensor model and generalizations. We thoroughly analyze the stati…

Gradient-based algorithms are effective for many machine learning tasks, but despite ample recent effort and some progress, it often remains unclear why they work in practice in optimising high-dimensional non-convex functions and why they find good minima instead of being trapped in spurious ones. Here we present a qu…

In this work we analyse quantitatively the interplay between the loss landscape and performance of descent algorithms in a prototypical inference problem, the spiked matrix-tensor model. We study a loss function that is the negative log-likelihood of the model. We analyse the number of local minima at a fixed distance …

Study on complexity of random polynomials with deterministic spikes, identifying phase transitions.

Study on variance of Laplace eigenfunctions on manifolds.

We establish formulas that give the intrinsic volumes, or curvature measures, of sublevel sets of functions defined on Riemannian manifolds as integrals of functionals of the function and its derivatives. For instance, in the Euclidean case, if $f \in \mathcal{C}^3(\mathbb{R}^n, \mathbb{R})$ and 0 is a regular value of…

Study analyzes landscape complexity of empirical loss functions with correlated data.

Consider a $d\times d$ matrix $M$ whose rows are independent centered non-degenerate Gaussian vectors $ξ_1,...,ξ_d$ with covariance matrices $Σ_1,...,Σ_d$. Denote by $\mathcal{E}_i$ the location-dispersion ellipsoid of $ξ_i:\mathcal{E}_i={\mathbf{x}\in\mathbb{R}^d : \mathbf{x}^\topΣ_i^{-1} \mathbf{x}\leqslant1}$. We sh…

New test detects sparse alternatives in Gaussian random fields.

Non-asymptotic tail bounds for Kostlan-Shub-Smale field on sphere