We extend Kac-Rice formula to compute expected intersections of random submanifolds.
problem Computing expected intersections of random submanifolds.
method Generalized Kac-Rice formula using measure theory and integration.
result Formula computes expected cardinality of preimages of submanifolds via random maps.
Method analyzes complexity of empirical risk landscapes for generalized linear models.
problem Understanding the complexity of empirical risk landscapes in generalized linear models.
method Kac-Rice method and replicated method from theoretical physics.
result Explicit variational formulas for the number of critical points of empirical risk landscapes.
Study finds the number of modes in Gaussian kernel density estimators scales with sqrt(β log β).
problem Determining the number of clusters in Transformers.
method Used Kac-Rice formula and Edgeworth expansion to prove scaling.
result The expected number of modes scales as Θ(√(β log β)).
Formula for critical points of chi fields on manifolds.
problem Computing critical points of chi fields on general manifolds.
method Semi-analytic formula using Kac-Rice argument and Hessian matrix representation.
result Expression for expected value of critical points in high-threshold limit.
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.
Develops calculus for random submanifolds using zonoids.
problem Calculating properties of random submanifolds defined by zero sets of vector fields.
method Defines zonoid sections and uses them to compute expected volumes and currents.
result Establishes new inequalities and formulas for random submanifolds.
The study of topological properties of random smooth maps, focusing on Kac-Rice formula and Betti numbers.
problem Topological and geometric properties of random smooth maps.
method Developed a general framework for differential geometric and topological issues of smooth Gaussian Random Fields, generalized Kac-Rice formula, applied to Kostlan random polynomials, and proved an original theorem in Differential Topology.
result The Betti numbers of the solution of a system of regular equations cannot decrease under a C0-small perturbation of the equations. We analyze the landscape of empirical risk minimization for high-dimensional models, predicting phase transitions and critical point properties.
problem Understanding the complexity and structure of high-dimensional empirical risk landscapes.
method Using the Kac-Rice formula, we analyze the expected number of critical points and their spectral properties, providing detailed predictions.
result We derive complete topological phase diagrams for the phase retrieval problem, predicting BBP-type transitions and critical point stability.
The paper analyzes local minima in high-dimensional empirical risk minimization.
problem Understanding local minima in high-dimensional data models.
method Using Kac-Rice formula and proportional asymptotics, the paper derives bounds on local minima.
result Sharp asymptotics on estimation and prediction errors are derived.
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.
problem Complexity of random Gaussian polynomials with deterministic spikes on a sphere.
method Variational formulas, Kac-Rice formula, determinant asymptotics of finite-rank perturbation of Gaussian Wigner matrices.
result Identification of a topological phase transition in the complexity function.
Study on variance of Laplace eigenfunctions on manifolds.
problem Investigating the variance of Laplace eigenfunctions on compact manifolds.
method Combining Kac-Rice formula, Wiener-Itô chaos decompositions, and pointwise Weyl law analysis.
result Established a quantitative bound for the fluctuations of nodal volumes, improving existing results.
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∈C3(Rn,R) and 0 is a regular value of…
Study analyzes landscape complexity of empirical loss functions with correlated data.
problem Understanding the complexity of loss landscapes in machine learning with structured data.
method Kac-Rice formula and random matrix theory applied to high-dimensional empirical loss functions.
result Characterizes the average number of critical points in loss functions with structured data.
Consider a d×d matrix M whose rows are independent centered non-degenerate Gaussian vectors ξ1,...,ξd with covariance matrices Σ1,...,Σd. Denote by Ei the location-dispersion ellipsoid of ξi:Ei=x∈Rd:x⊤Σi−1x⩽1. We sh…
New test detects sparse alternatives in Gaussian random fields.
problem Detecting sparse alternatives in Gaussian random fields.
method Ad-hoc Kac Rice formula for second maximum distribution, exact spacing test.
result Exact t-spacing test for high power in detecting sparse alternatives. Non-asymptotic tail bounds for Kostlan-Shub-Smale field on sphere
problem Estimating rank-R symmetric signal tensor from Gaussian observation
method Profile maximum likelihood estimator
result Finite-(k,d) error bound recovers asymptotically optimal rate