arXiv research

Paper proposes a method to reduce hallucinations in diffusion models using Laplacian score sharpening.

Proposes a symmetric graph autoencoder for unsupervised learning.

SpecAE detects anomalies in attributed networks by projecting them into a tailored space.

In the first part, we derive a sharp gradient estimate for the log of Dirichlet heat kernel and Poisson heat kernel on domains, and a sharpened local Li-Yau gradient estimate that matches the global one. In the second part, without explicit curvature assumptions, we prove a global upper bound for the fundamental soluti…

Proposes a flexible deep learning model for complex distributions.

The goal of the paper is to sharpen and generalise bounds involving the Cheeger's isoperimetric constant $h$ and the first eigenvalue $λ_{1}$ of the Laplacian. A celebrated lower bound of $λ_{1}$ in terms of $h$, $λ_{1}\geq h^{2}/4$, was proved by Cheeger in 1970 for smooth Riemannian manifolds. An upper bound on $λ_{1…

Self-improvement refines language models by verifying their own outputs.

Study inverse problems with measure samples, improving estimator calibration and recovery.

New approach ties loss curvature to model performance in deep learning.

Graph convolutions can enhance high frequencies, leading to over-sharpening.

Study sharpens threshold for matching correlated graphs without labels.

New theorem shows curvature concentration depends linearly on volume ratio.

This paper addresses Cheeger and Gromoll's question of which vector bundles admit a complete metric of nonnegative curvature, and relates their question to the issue of which sphere bundles admit a metric of positive curvature. We show that any vector bundle which admits a metric of nonnegative curvature must admit a c…

The paper sharpens inequalities in hyperbolic spaces.

Detecting edge correlation between two graphs sharpens a threshold based on densest subgraph.

We study the Gassner representation of the pure braid group $P_n$ by considering its restriction to a free subgroup $F$. The kernel of the restriction is shown to lie in the subgroup $[Γ^3 F,Γ^2 F]$, sharpening a result of Lipschutz.

We give tight concentration bounds for mixtures of martingales that are simultaneously uniform over (a) mixture distributions, in a PAC-Bayes sense; and (b) all finite times. These bounds are proved in terms of the martingale variance, extending classical Bernstein inequalities, and sharpening and simplifying prior wor…

We sharpen the construction of representation space in the paper "Principal Series Representations of Infinite Dimensional Lie Groups II: Construction of Induced Representations". We show that the principal series representation spaces constructed there, are completions of spaces of sections of Hilbert bundles rather t…

We give a new lower bound for the first gap $λ_2 - λ_1$ of the Dirichlet eigenvalues of the Schr{ö}dinger operator on a bounded convex domain $Ω$ in R$^n$ or S$^n$ and greatly sharpens the previous estimates. The new bound is explicit and computable.

The paper develops $L^2$-Hodge theory on almost Kähler manifolds and proves the Hopf conjecture.

Cohen et al. (2021) show GD trajectories align on a bifurcation diagram.

In this work, complete constant mean curvature 1 (CMC-1) surfaces in hyperbolic 3-space with total absolute curvature at most 4 pi are classified. This classification suggests that the Cohn-Vossen inequality can be sharpened for surfaces with odd numbers of ends, and a proof of this is given.

We further sharpen higher type adjunction inequalities of P. Ozsváth and Z. Szabó on a 4-manifold $M$ with a nonzero Seiberg-Witten invariant for a Spin$^c$ structure $\frak{s}$, when an embedded surface $Σ\subset M$ satisfies $[Σ]\cdot [Σ]\geq 0$ and $|\langle [Σ],c_1(\frak{s})\rangle|+[Σ]\cdot [Σ]\geq 2b_1(M).$

Improved Metropolized HMC mixing time with multi-step gradients.

Adversarial training makes logistic regression weight loss landscapes sharper.

From concentration inequalities for the suprema of Gaussian or Rademacher processes an inequality is derived. It is applied to sharpen existing and to derive novel bounds on the empirical Rademacher complexities of unit balls in various norms appearing in the context of structured sparsity and multitask dictionary lear…

Defines vector Laplacian on statistical manifolds.

Colding and Minicozzi have shown that an embedded minimal disk $0\inΣ\subset B_R$ in $\Real^3$ with large curvature at 0 looks like a helicoid on the scale of $R$. Near 0, this can be sharpened: on the scale of $|A|^{-1}(0)$, $Σ$ is close, in a Lipschitz sense, to a piece of a helicoid. We use surfaces constructed by C…

NM-PPG optimizes adaptive feature acquisition in POMDPs for better predictions.

Given a data matrix $X \in R^{n\times d}$ and a response vector $y \in R^{n}$, suppose $n>d$, it costs $O(n d^2)$ time and $O(n d)$ space to solve the least squares regression (LSR) problem. When $n$ and $d$ are both large, exactly solving the LSR problem is very expensive. When $n \gg d$, one feasible approach to spee…

Quandle cocycle invariants form a powerful and well developed tool in knot theory. This paper treats their variations - namely, positive and twisted quandle cocycle invariants, and shadow invariants. We interpret the former as particular cases of the latter. As an application, several constructions from the shadow worl…

BIG Laplacians bridge combinatorial and Hodge Laplacians for discrete data.

The study analyzes sharpness dynamics in neural networks, revealing mechanisms and conditions.

In this paper we first give a one-move version of Markov's braid theorem for knot isotopy in $S^3$ that sharpens the classical theorem. Then a relative version of Markov's theorem concerning a fixed braided portion in the knot. We also prove an analogue of Markov's theorem for knot isotopy in knot complements. Finally …

Paper introduces magnetic Hodge Laplacian for differential forms.

A consequence of the Cabling Conjecture of Gonzalez-Acuña and Short is that Dehn surgery on a knot in $S^3$ cannot produce a manifold with more than two connected summands. In the event that some Dehn surgery produces a manifold with three or more connected summands, then the surgery parameter is bounded in terms of th…

Weight decay stabilizes training dynamics by slowing progressive sharpening.

The paper extends Laplacian spectra approximations to vector bundles.

Proves Laplacian and Lichnerowicz Laplacian are sectorial in weighted Hölder spaces.

Survey of Laplacian-based methods for data dimensionality reduction and embedding.

The graph Laplacian plays key roles in information processing of relational data, and has analogies with the Laplacian in differential geometry. In this paper, we generalize the analogy between graph Laplacian and differential geometry to the hypergraph setting, and propose a novel hypergraph $p$-Laplacian. Unlike the …

The paper sets up eigenvalue comparison theorems for specific Laplacians on manifolds.

Extended bounds on small eigenvalues for pseudo-Laplacians on hyperbolic surfaces.

Paper investigates rigidity phenomena for weighted Ricci curvature bounds with Laplacian comparison theorem.

Let T be the nilpotent group of 4 x 4 real upper triangular matrices. In this note we show that the Euler equations of certain left-invariant riemannian metrics on T have a horseshoe. We also show, with the aid of a numerical computation of a Melnikov-type integral, that the Euler equations of the sub-riemannian Carnot…

Study Hodge Laplacians for manifold data, improving error bounds.

Root Laplacian Eigenmaps help in spectral embedding of graphs.

New perspective on G2-structures flow from DeTurck Laplacian.