arXiv research

We prove sharp criteria on the behavior of radial curvature for the existence of asymptotically flat or hyperbolic Riemannian manifolds with prescribed sets of eigenvalues embedded in the spectrum of the Laplacian. In particular, we construct such manifolds with dense embedded point spectrum and sharp curvature bounds.

Researchers decompose curvature to confirm Hopf conjecture and prove new rigidity theorems.

The paper finds lower bounds for the first eigenvalue of p-Laplacian in specific manifolds.

Let F be a riemannian flow on a closed manifold M. We study the behavior of the first eigenvalues of the Hodge Laplacian acting on differential forms under adiabatic collapsing of the flow. We show that the number of small eigenvalues is related to the basic cohomology of F, and give spectral criteria for the vanishing…

We study the stability of partitions involving two or more phases in convex domains under the assumption of at most two-phase contact, thus excluding in particular triple junctions. We present a detailed derivation of the second variation formula with particular attention to the boundary terms, and then study the sign …

We show that an embedded minimal annulus $Σ^2 \subset B^3$ which intersects $\partial B^3$ orthogonally and is invariant under reflection through the coordinate planes is the critical catenoid. The proof uses nodal domain arguments and a characterization, due to Fraser and Schoen, of the critical catenoid as the unique…

A new method cleans and analyzes stock return correlation matrices.

Paper separates financial time series into fast and slow components.

We establish inequalities for the eigenvalues of Schrödinger operators on compact submanifolds (possibly with nonempty boundary) of Euclidean spaces, of spheres, and of real, complex and quaternionic projective spaces, which are related to inequalities for the Laplacian on Euclidean domains due to Payne, Pólya, and Wei…

In this paper, we consider the eigen-solutions of $-Δu+ Vu=λu$, where $Δ$ is the Laplacian on a non-compact complete Riemannian manifold. We develop Kato's methods on manifold and establish the growth of the eigen-solutions as $r$ goes to infinity based on the asymptotical behaviors of $Δr$ and $V(x)$, where $r=r(x)$ i…

Study radial processes in sub-Riemannian Brownian motions, proving stochastic completeness and eigenvalue estimates.

Consistent model selection for spiked Wigner model via AIC-type criteria.

High-dimensional models become unstable when sample size falls below a critical level, leading to a phase transition.

This paper develops a weighted $L^2$-method for the (half) Dirac equation. For Dirac bundles over closed Riemann surfaces, we give a sufficient condition for the solvability of the (half) Dirac equation in terms of a curvature integral. Applying this to the Dolbeault-Dirac operator, we establish an automatic transversa…

New insights into hidden minima in neural networks.

For Machine Learning (ML) classification problem, where a vector of $\mathbf{x}$--observations (values of attributes) is mapped to a single $y$ value (class label), a generalized Radon--Nikodym type of solution is proposed. Quantum--mechanics --like probability states $ψ^2(\mathbf{x})$ are considered and "Cluster Cente…

Recently ({\em Class. Quant. Grav.} {\bf 20} 625-664) the concept of {\em causal mapping} between spacetimes --essentially equivalent in this context to the {\em chronological map} one in abstract chronological spaces--, and the related notion of {\em causal structure}, have been introduced as new tools to study causal…

We consider the problem of optimal portfolio selection under forward investment performance criteria in an incomplete market. Given multiple traded assets, the prices of which depend on multiple observable stochastic factors, we construct a large class of forward performance processes with power-utility initial data, a…

We give some general criteria of being a homeomorphism for continuous mappings of topological manifolds, as well as criteria of being a diffeomorphism for smooth mappings of smooth manifolds. As an illustration, we apply these criteria to the problems arising in two- and three-dimensional grid generation.

EigenVI uses orthogonal function expansions for efficient variational inference.

The study reveals flaws in pruning criteria and proposes a new assumption for better filter selection.

New method finds optimal hyperparameters for multiple tasks and criteria.

New criteria for Heegaard splittings ensure strong irreducibility and finite Goeritz groups.

Multi-criteria recommender systems have been increasingly valuable for helping consumers identify the most relevant items based on different dimensions of user experiences. However, previously proposed multi-criteria models did not take into account latent embeddings generated from user reviews, which capture latent se…

Develops scenario theory for multi-criteria decision making.

We consider the problem of identifying patterns in a data set that exhibit anomalous behavior, often referred to as anomaly detection. In most anomaly detection algorithms, the dissimilarity between data samples is calculated by a single criterion, such as Euclidean distance. However, in many cases there may not exist …

The paper evaluates criteria for selecting cryptocurrencies based on historical data.

This paper is a tutorial for eigenvalue and generalized eigenvalue problems. We first introduce eigenvalue problem, eigen-decomposition (spectral decomposition), and generalized eigenvalue problem. Then, we mention the optimization problems which yield to the eigenvalue and generalized eigenvalue problems. We also prov…

We present criteria for establishing a triangulation of a manifold. Given a manifold M, a simplicial complex A, and a map H from the underlying space of A to M, our criteria are presented in local coordinate charts for M, and ensure that H is a homeomorphism. These criteria do not require a differentiable structure, or…

The paper compares Steklov and Laplacian eigenvalues on graphs.

When sufficient labeled data are available, classical criteria based on Receiver Operating Characteristic (ROC) or Precision-Recall (PR) curves can be used to compare the performance of un-supervised anomaly detection algorithms. However , in many situations, few or no data are labeled. This calls for alternative crite…

The paper analyzes performance criteria for competing fund managers in Ito-diffusion markets.

Paper defines numerical criteria to test handlebody link irreducibility.

We shall give useful criteria of lips, beaks and swallowtail singularities of smooth map from the plane into the plane. As an application of criteria, we will discuss the singularities of Cauchy problem of single conservation law.

A new method for multi-criteria recommender systems using graph attention networks.

New framework for resilient bi-criteria optimization under noisy feedback.

New algorithms minimize risk in MNL bandits, achieving near-optimal performance.

A game-theoretic approach to multi-criteria ranking from ordinal data.

Stress, edge crossings, and crossing angles play an important role in the quality and readability of graph drawings. Most standard graph drawing algorithms optimize one of these criteria which may lead to layouts that are deficient in other criteria. We introduce an optimization framework, Stress-Plus-X (SPX), that sim…

Recent work on fairness in machine learning has focused on various statistical discrimination criteria and how they trade off. Most of these criteria are observational: They depend only on the joint distribution of predictor, protected attribute, features, and outcome. While convenient to work with, observational crite…

Criteria found for graph drawings on surfaces.

Fairness in machine learning has predominantly been studied in static classification settings without concern for how decisions change the underlying population over time. Conventional wisdom suggests that fairness criteria promote the long-term well-being of those groups they aim to protect. We study how static fairne…

In this paper, two interesting eigenvalue comparison theorems for the first non-zero Steklov eigenvalue of the Laplacian have been established for manifolds with radial sectional curvature bounded from above. Besides, sharper bounds for the first non-zero eigenvalue of the Wentzell eigenvalue problem of the weighted La…

The paper introduces risk consistency properties for credit ratings.

Many machine learning frameworks, such as resource-allocating networks, kernel-based methods, Gaussian processes, and radial-basis-function networks, require a sparsification scheme in order to address the online learning paradigm. For this purpose, several online sparsification criteria have been proposed to restrict …

Study tackles criterion collapse in learning criteria, showing conditions for loss minimization.

The paper explores inequalities between eigenvalues on Riemannian manifolds.

Simple criteria for codimension two surface singularities.