arXiv research

New methods rank players using covariates and comparisons, outperforming existing algorithms.

We describe a seriation algorithm for ranking a set of items given pairwise comparisons between these items. Intuitively, the algorithm assigns similar rankings to items that compare similarly with all others. It does so by constructing a similarity matrix from pairwise comparisons, using seriation methods to reorder t…

Partial convexification improves tractability of low-rank spectral optimization problems.

Spectral ranking methods are improved against semi-random graph sampling.

Deep networks learn clean structure before memorizing corrupted labels, leaving a spectral signature in gradient centered scatter.

In this note, we present a new way to associate a spectral triple to the noncommutative $C^*$-algebra $C^*(Λ)$ of a strongly connected finite higher-rank graph $Λ$. We generalize a spectral triple of Consani and Marcolli from Cuntz-Krieger algebras to higher-rank graph $C^*$-algebras $C^*(Λ)$, and we prove that these s…

The paper improves spectral ranking methods for diverse comparison graphs.

This paper explores the preference-based top-$K$ rank aggregation problem. Suppose that a collection of items is repeatedly compared in pairs, and one wishes to recover a consistent ordering that emphasizes the top-$K$ ranked items, based on partially revealed preferences. We focus on the Bradley-Terry-Luce (BTL) model…

New spectral methods improve matrix estimation in RL with low-rank structure.

The study characterizes wobbly rank-2 bundles on Riemann surfaces using spectral curves.

Paper improves robust spectral clustering for noisy data.

Gradient descent with small random init mimics spectral methods for low-rank matrix recovery.

Improved rank aggregation via spectral method reduces sample complexity.

We introduce a new parameterization method for deep learning layers using spectral tensor train decomposition.

Spectral gradient methods outperform Euclidean in certain deep learning scenarios.

The paper examines how gradient descent stabilizes low-rank matrix factorization in noisy conditions.

Study Higgs bundles on curves with punctures, extending spectral correspondence.

Solves the Wiegold problem by showing free products of left-orderable groups have normal rank > 1.

Study spectral flow on a warped cylinder with special boundary conditions.

Paper improves MVSC using tensor low-rank modeling.

Study critical exponents for L^p-cohomology of higher rank Lie groups and manifolds.

New algorithm learns low-rank matrices with linear number of samples.

A well-known conjecture of Rasmussen states that for any knot $K$ in $S^{3}$, the rank of the reduced Khovanov homology of $K$ is greater than or equal to the rank of the reduced knot Floer homology of $K$. This rank inequality is supposed to arise as the result of a spectral sequence from Khovanov homology to knot Flo…

New method improves matrix completion accuracy, especially in noisy data.

Unified view of spectral networks linking geometry and gauge theory.

Constructs coordinate systems from spectral curve sheaves.

New method for initializing low-rank neural networks improves performance.

Let $E_{k}^{F}(D)$ be the spectral sequence induced by the oriented cube of resolutions on knot Floer homology. We prove that $E_{2}^{F}(D)$ is a triply graded link invariant whose graded Euler characteristic is the HOMFLY-PT polynomial and that the higher pages are link invariants. By construction, the spectral sequen…

Optimizes ranking of top-k players from partial comparison data.

SON-NMF estimates nonnegative rank on-the-fly for NMF.

Develops methods to estimate high rank tensors from noisy data.

Study on harmonic metrics for rank 3 Higgs bundles in Hitchin section.

Consider the problem of estimating a low-rank matrix when its entries are perturbed by Gaussian noise. If the empirical distribution of the entries of the spikes is known, optimal estimators that exploit this knowledge can substantially outperform simple spectral approaches. Recent work characterizes the asymptotic acc…

The spectral $k$-support norm enjoys good estimation properties in low rank matrix learning problems, empirically outperforming the trace norm. Its unit ball is the convex hull of rank $k$ matrices with unit Frobenius norm. In this paper we generalize the norm to the spectral $(k,p)$-support norm, whose additional para…

We explore the top-$K$ rank aggregation problem. Suppose a collection of items is compared in pairs repeatedly, and we aim to recover a consistent ordering that focuses on the top-$K$ ranked items based on partially revealed preference information. We investigate the Bradley-Terry-Luce model in which one ranks items ac…

We accelerate the power method for strong low-rank approximation using fast sketching.

The paper improves tensor completion bounds using spectral gap.

New method ranks sectors and countries using local and aggregate I-O data.

For a 2-periodic link $\tilde L$ in the thickened annulus and its quotient link $L$, we exhibit a spectral sequence with $E^1 \cong AKh(\tilde L) \otimes_{\mathbb{F}_2} \mathbb{F}_2[θ, θ^{-1}] \rightrightarrows E^\infty \cong AKh(L) \otimes_{\mathbb{F}_2} \mathbb{F}_2[θ, θ^{-1}].$ This spectral sequence splits along qu…

This paper proposes a new Nystrom-based clustering algorithm for large-scale data.

The paper classifies links with low rank knot Floer and Khovanov homologies.

We show that the spectral norm of a random $n_1\times n_2\times \cdots \times n_K$ tensor (or higher-order array) scales as $O\left(\sqrt{(\sum_{k=1}^{K}n_k)\log(K)}\right)$ under some sub-Gaussian assumption on the entries. The proof is based on a covering number argument. Since the spectral norm is dual to the tensor…

This paper improves spectral clustering for large datasets using the Nystrom method.

Batch normalization prevents rank collapse in deep networks, improving training stability.

This study analyzes why attention layers in neural networks can cause signal loss and proposes a solution.

Given a graphical model (GM), computing its partition function is the most essential inference task, but it is computationally intractable in general. To address the issue, iterative approximation algorithms exploring certain local structure/consistency of GM have been investigated as popular choices in practice. Howev…

Paper introduces a novel framework for recognizing dynamic ranking structures in preference-based data.

The study reveals the spectral structure of attention layers and its implications for generalization.