arXiv research

New bound for neural networks with full-rank weights, independent of network width.

Sharp bounds derived for test error of finite-rank kernel ridge regression.

The paper analyzes how low-rank layers in neural networks improve generalization.

The paper improves transformer generalization bounds using rank-dependent covering number bounds.

Paper finds a lower bound for estimating low-rank matrices in logistic regression.

The study finds a limit on subgroup complexity in hyperbolic 3-manifold groups.

This paper studies the problem of finding the exact ranking from noisy comparisons. A comparison over a set of $m$ items produces a noisy outcome about the most preferred item, and reveals some information about the ranking. By repeatedly and adaptively choosing items to compare, we want to fully rank the items with a …

Learning to rank is a supervised learning problem where the output space is the space of rankings but the supervision space is the space of relevance scores. We make theoretical contributions to the learning to rank problem both in the online and batch settings. First, we propose a perceptron-like algorithm for learnin…

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

We construct a geometric decomposition for the convex core of a thick hyperbolic 3-manifold M with bounded rank. Corollaries include upper bounds in terms of rank and injectivity radius on the Heegaard genus of M and on the radius of any embedded ball in the convex core of M.

New algorithms improve RPCA for large matrices with upper rank bounds.

CRS model improves ranking data modeling with theoretical guarantees.

Recently, fundamental conditions on the sampling patterns have been obtained for finite completability of low-rank matrices or tensors given the corresponding ranks. In this paper, we consider the scenario where the rank is not given and we aim to approximate the unknown rank based on the location of sampled entries an…

Characterizes Kähler-hyperbolicity of bounded symmetric domains based on rank and genus.

The study proves stabilizing of ascending chains in specific groups.

Sharp isoperimetric inequalities for Neumann eigenvalues in symmetric spaces.

To a Seifert matrix of a knot K one can associate a matrix w(K) with entries in the rational function field, Q(t). The Murasugi, Milnor, and Levine-Tristram knot signatures, all of which provide bounds on the 4-genus of a knot, are determined by w(K). More generally, the minimal rank of a representative of the class re…

An algorithm tackles low-rank linear bandit problems with improved regret bounds.

This paper explores the adaptive (active) PAC (probably approximately correct) top-$k$ ranking (i.e., top-$k$ item selection) and total ranking problems from $l$-wise ($l\geq 2$) comparisons under the multinomial logit (MNL) model. By adaptively choosing sets to query and observing the noisy output of the most favored …

The paper tackles pure exploration in multi-armed bandits with low rank structure using oblivious sampling.

We use a local argument to prove if an $r$-dimensional torus acts isometrically and effectively on a connected $n$-dimensional manifold which has positive $k^\mathrm{th}$-intermediate Ricci curvature at some point, then $r \leq \lfloor \frac{n+k}{2} \rfloor$. This symmetry rank bound generalizes those established by Gr…

Paper tackles underranking in group-fair ranking systems, proving a trade-off and presenting an algorithm.

The paper tackles learning true rankings from noisy, incomplete data.

The problem of bipartite ranking, where instances are labeled positive or negative and the goal is to learn a scoring function that minimizes the probability of mis-ranking a pair of positive and negative instances (or equivalently, that maximizes the area under the ROC curve), has been widely studied in recent years. …

Paper proposes a new method to separate low rank and sparse matrices without bias.

This paper studies the problem of inferring a global preference based on the partial rankings provided by many users over different subsets of items according to the Plackett-Luce model. A question of particular interest is how to optimally assign items to users for ranking and how many item assignments are needed to a…

Optimal rank-adaptive matrix estimation from linear measurements.

In this paper, we develop a relative error bound for nuclear norm regularized matrix completion, with the focus on the completion of full-rank matrices. Under the assumption that the top eigenspaces of the target matrix are incoherent, we derive a relative upper bound for recovering the best low-rank approximation of t…

This paper improves entropy bounds for ranking time-series complexity.

The study assesses low-rank approximations in Gaussian Process regression.

New bound explains why high-rank neural nets generalize well.

The study assesses low-rank approximations in Gaussian Process regression.

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

New algorithm for weighted low rank approximation with provable guarantees.

We show that codimension one dimensional Jacobian of the barycentric straightening map is uniformly bounded for most of the higher rank symmetric spaces. As a consequence, we prove that the locally finite simplicial volume of most $\mathbb Q$-rank $1$ locally symmetric spaces is positive, which has been open for many y…

Algorithm for low-rank matrix bandits with heavy-tailed rewards, achieving nearly optimal regret bound.

New method solves matrix completion problems to certifiable optimality.

Unified error analysis for low-rank approximation improves data assimilation performance.

Free groups' automorphisms have bounded orbits.

We prove the meridional rank conjecture for twisted links and arborescent links associated to bipartite trees with even weights. These links are substantial generalizations of pretzels and two-bridge links, respectively. Lower bounds on meridional rank are obtained via Coxeter quotients of the groups of link complement…

Paper bounds the minimal rank for kernel ridge regression approximations.

The paper establishes theoretical foundations for low-rank knowledge distillation in LLMs.

Improved rank aggregation via spectral method reduces sample complexity.

The paper designs tests for comparing ranked preference data and finds significant differences.

Let M be a complete Riemannian manifold whose sectional curvature is bounded above by 1. We say that M has positive spherical rank if along every geodesic one hits a conjugate point at t=π. The following theorem is then proved: If M is a complete, simply connected Riemannian manifold with upper curvature bound 1 and po…

If Pi: M -> B is an onto smooth maximal rank map between complete Riemannian manifolds M and B with bounded geometry, we prove sufficient conditions for M to be roughly isometric to the Riemannian product FxB, where F is a fiber of M.

Paper develops RGN method for estimating low-rank tensors from noisy measurements.

We provide new approximation guarantees for greedy low rank matrix estimation under standard assumptions of restricted strong convexity and smoothness. Our novel analysis also uncovers previously unknown connections between the low rank estimation and combinatorial optimization, so much so that our bounds are reminisce…