arXiv research

Paper proves new theorems about curvature in weighted manifolds.

The paper develops heat kernel comparison theorems and applies them to spectral geometry.

Study compares spectral properties of a specific tensor in geometry.

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

The paper improves spectral ranking methods for diverse comparison graphs.

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…

Study improves understanding of Ricci curvature in manifolds.

BSD is a Bayesian framework for analyzing neural spectral data.

Paper quantifies uncertainty in pairwise comparison models.

Paper establishes statistical inference for pairwise comparison models.

New inequalities for spectral zeta kernels on spheres and manifolds.

Paper tackles ranking items with a semi-random comparison graph and a monotone adversary.

The study compares spectral volumes of manifolds with weakly convex boundaries.

We prove spectral, stochastic and mean curvature estimates for complete $m$-submanifolds $\varphi \colon M \to N$ of $n$-manifolds with a pole $N$ in terms of the comparison isoperimetric ratio $I_{m}$ and the extrinsic radius $r_\varphi\leq \infty$. Our proof holds for the bounded case $r_\varphi< \infty$, recovering …

This paper is concerned with the problem of top-$K$ ranking from pairwise comparisons. Given a collection of $n$ items and a few pairwise comparisons across them, one wishes to identify the set of $K$ items that receive the highest ranks. To tackle this problem, we adopt the logistic parametric model --- the Bradley-Te…

AUASE embeds dynamic networks with stability guarantees for node comparison.

New model for pairwise comparisons without stochastic transitivity.

Using McCann's transportation map, we establish a transport inequality on compact manifolds with positive Ricci curvature. This inequality contains the sharp spectral comparison estimates.

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…

COPT optimizes graph distances via simultaneous optimal transport.

High-dimensional k-sample comparison is a common applied problem. We construct a class of easy-to-implement nonparametric distribution-free tests based on new tools and unexplored connections with spectral graph theory. The test is shown to possess various desirable properties along with a characteristic exploratory fl…

Paper solves long neck problem on odd-dimensional spin manifolds.

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…

Standard kernels such as Matérn or RBF kernels only encode simple monotonic dependencies within the input space. Spectral mixture kernels have been proposed as general-purpose, flexible kernels for learning and discovering more complicated patterns in the data. Spectral mixture kernels have recently been generalized in…

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

GWCA analyzes cross-graph correlations for movie retrieval.

Unified framework for analyzing graph neural operators converging to graph limits.

This study bridges the gap between spatial and spectral GNNs.

Improved rank aggregation via spectral method reduces sample complexity.

Multi-output Gaussian processes (MOGPs) are an extension of Gaussian Processes (GPs) for predicting multiple output variables (also called channels, tasks) simultaneously. In this paper we use the convolution theorem to design a new kernel for MOGPs, by modeling cross channel dependencies through cross convolution of t…

We first show that a Laplace isospectral family of Riemannian orbifolds, satisfying a lower Ricci curvature bound, contains orbifolds with points of only finitely many isotropy types. If we restrict our attention to orbifolds with only isolated singularities, and assume a lower sectional curvature bound, then the numbe…

Under appropriate spectral assumptions we prove two existence results for positive solutions of Lichnerowicz-type equations on complete manifolds. We also give a priori bounds and a comparison result that immediately yields uniqueness for certain classes of solutions. No curvature assumptions are involved in our analys…

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

A new notion of stochastic ordering is introduced to compare multivariate stochastic risk models with respect to extreme portfolio losses. In the framework of multivariate regular variation comparison criteria are derived in terms of ordering conditions on the spectral measures, which allows for analytical or numerical…

In many areas of machine learning, it becomes necessary to find the eigenvector decompositions of large matrices. We discuss two methods for reducing the computational burden of spectral decompositions: the more venerable Nystom extension and a newly introduced algorithm based on random projections. Previous work has c…

Study reveals $1/f$ noise in signals made from nonoverlapping rectangular pulses.

Rank aggregation systems collect ordinal preferences from individuals to produce a global ranking that represents the social preference. Rank-breaking is a common practice to reduce the computational complexity of learning the global ranking. The individual preferences are broken into pairwise comparisons and applied t…

Spectral estimation (SE) aims to identify how the energy of a signal (e.g., a time series) is distributed across different frequencies. This can become particularly challenging when only partial and noisy observations of the signal are available, where current methods fail to handle uncertainty appropriately. In this c…

Hidden Markov Models (HMMs) can be accurately approximated using co-occurrence frequencies of pairs and triples of observations by using a fast spectral method in contrast to the usual slow methods like EM or Gibbs sampling. We provide a new spectral method which significantly reduces the number of model parameters tha…

Hypercube graphs are optimal in spectral rigidity due to Bakry--Émery curvature.

A new metric compares dynamical systems using operator eigenvalues.

We review the spectral analysis and the time-dependent approach of scattering theory for manifolds with asymptotically cylindrical ends. For the spectral analysis, higher order resolvent estimates are obtained via Mourre theory for both short-range and long-range behaviors of the metric and the perturbation at infinity…

This is a review of explicit computations of Connes distance in noncommutative geometry, covering finite dimensional spectral triples, almost-commutative geometries, and spectral triples on the algebra of compact operators. Several applications to physics are covered, like the metric interpretation of the Higgs field, …

Sharp spectral theorems and isoperimetric inequalities for manifolds with nonnegative Ricci curvature.

This paper proposes a new approach to construct high quality space-filling sample designs. First, we propose a novel technique to quantify the space-filling property and optimally trade-off uniformity and randomness in sample designs in arbitrary dimensions. Second, we connect the proposed metric (defined in the spatia…

Study improves the exponential rate of metric difference in Higgs bundles.

We investigate Relational Graph Attention Networks, a class of models that extends non-relational graph attention mechanisms to incorporate relational information, opening up these methods to a wider variety of problems. A thorough evaluation of these models is performed, and comparisons are made against established be…

New method clusters evolving networks using spatio-temporal graph Laplacian.