arXiv research

Deep networks with a wide layer ensure sublevel set connectivity.

Extends clustering method to cost-based hierarchies.

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…

This paper shows that every sublevel set of the loss function of a class of deep over-parameterized neural nets with piecewise linear activation functions is connected and unbounded. This implies that the loss has no bad local valleys and all of its global minima are connected within a unique and potentially very large…

Study of Willmore energy on sphere sublevel sets and flow singularities.

The paper studies convexity of products of squared Euclidean distances.

We develop an algorithm for minimizing a function using $n$ batched function value measurements at each of $T$ rounds by using classifiers to identify a function's sublevel set. We show that sufficiently accurate classifiers can achieve linear convergence rates, and show that the convergence rate is tied to the difficu…

We define a new notion---the sub-index of a critical point of a distance function. We show how sub-index affects the homotopy type of sublevel sets of distance functions.

New method robustifies topological data analysis against outliers.

Characterizes complex Hessian equations for bounded energy functions.

Given a finite set of points in $\mathbb R^n$ and a radius parameter, we study the Čech, Delaunay-Čech, Delaunay (or Alpha), and Wrap complexes in the light of generalized discrete Morse theory. Establishing the Čech and Delaunay complexes as sublevel sets of generalized discrete Morse functions, we prove that the four…

New Morse theory for shapes at distances.

We mainly study 3-dimensional complete gradient Ricci solitons with positive sectional curvature, whose scalar curvature attains its maximum at some point. In section 2, we estimate the area growth of level sets and the volume growth of sublevel sets of a Ricci potential. In section 3, we show that the scalar curvature…

Study sharp decay of capacity for subharmonic functions on compact Hermitian manifolds.

The paper develops finite knot theory using ropelength-filtered Reidemeister graphs.

Uniform bounds derived for fully non-linear equations.

Given a compact four-dimensional Riemannian manifold $(M, g)$ with boundary, we study the problem of existence of Riemannian metrics on $M$ conformal to $g$ with prescribed $Q$-curvature in the interior $\mathring{M}$ of $M$, and zero $T$-curvature and mean curvature on the boundary $\partial M$ of $M$. This geometric …

Paper analyzes nonconvex bandit problems with improved adaptive methods.

In this article, we solve the strong openness conjecture on the multiplier ideal sheaves for the plurisubharmonic functions posed by Demailly. We prove two conjectures about the growth of the volumes of the sublevel sets of plurisubharmonic functions related to the complex singularity exponents and quasi-plurisubharmon…

The study analyzes the evolution of Gaussian measures under a specific gradient flow.

Outlier detection methods have become increasingly relevant in recent years due to increased security concerns and because of its vast application to different fields. Recently, Pauwels and Lasserre (2016) noticed that the sublevel sets of the inverse Christoffel function accurately depict the shape of a cloud of data …

Classical Morse theory proceeds by considering sublevel sets $f^{-1}(-\infty, a]$ of a Morse function $f: M \to R$, where $M$ is a smooth finite-dimensional manifold. In this paper, we study the topology of the level sets $f^{-1}(a)$ and give conditions under which the topology of $f^{-1}(a)$ changes when passing a cri…

The paper uses MDM theory to analyze multifiltering functions on simplicial complexes.

Extends Morse-Forman theory to vector-valued functions for multiparameter persistence.

This paper was motivated by work of Arnold where he explains how to count "snakes", i.e. Morse functions on the real axis with prescribed behavior at infinity. This leads immediately to a count of excellent Morse functions on the circle, where following Thom's terminology, excellent means that no two critical points li…

Two of the authors have defined the class $WDC(M)$ as the class of all subsets of a smooth manifold $M$ that may be expressed in local coordinates as certain sublevel sets of DC (differences of convex) functions. If $M$ is Riemanian and $G$ is a group of isometries acting transitively on the sphere bundle $SM$, we def…

We introduce a natural definition of $L^p$-convergence of maps, $p \ge 1$, in the case where the domain is a convergent sequence of measured metric space with respect to the measured Gromov-Hausdorff topology and the target is a Gromov-Hausdorff convergent sequence. With the $L^p$-convergence, we establish a theory of …

We extend the Weil-Petersson metric to a projective variety with continuous local potentials.

New methods connect low-loss points on neural network surfaces.

New acquisition functions improve Bernoulli LSE.

New rigidity result for hyperbolic surfaces based on curve lengths.

MOPI optimizes flexible set-valued mappings to achieve superior shape adaptivity in conformal prediction.

Let $X$ be a compact Kähler manifold and $\om$ a smooth closed form of bidegree $(1,1)$ which is nonnegative and big. We study the classes ${\mathcal E}_χ(X,\om)$ of $\om$-plurisubharmonic functions of finite weighted Monge-Ampère energy. When the weight $χ$ has fast growth at infinity, the corresponding functions are …

New property ensures neural networks generalize well with limited data.

We solve the optimization of two-layer ReLU networks using convex math.

New theorem shows gaps in magnetic Schrödinger operator spectra for large coupling.

Study semicontinuity of capacity in non-smooth spaces using intrinsic flat convergence.

ERM performs well in feature learning with minimal feature maps.

This paper calibrates Gaussian process predictive distributions for Bayesian optimization to improve sampling decisions.

Develops robust MDPs for unknown disturbances with performance guarantees.

Improves conditions for mode connectivity in deep neural networks.

Paper uses TDA for automated Parkinson's disease classification and severity assessment.

In this paper, a new theory is developed for first-order stochastic convex optimization, showing that the global convergence rate is sufficiently quantified by a local growth rate of the objective function in a neighborhood of the optimal solutions. In particular, if the objective function $F(\mathbf w)$ in the $ε$-sub…

In this paper, we study the efficiency of a {\bf R}estarted {\bf S}ub{\bf G}radient (RSG) method that periodically restarts the standard subgradient method (SG). We show that, when applied to a broad class of convex optimization problems, RSG method can find an $ε$-optimal solution with a lower complexity than the SG m…

TRAiL is a linear bandit algorithm that ensures optimal regret and guarantees inference quality.

New method for optimization on Hadamard manifolds with curvature-independent guarantees.

A new depth measure and median defined on Hadamard manifolds.

Persistence diagrams from random matrices follow RMT universality, offering a new spectral diagnostic.