arXiv research

New proof shows certain 3D spaces are essentially like infinite space.

Spectral Clustering is a popular technique to split data points into groups, especially for complex datasets. The algorithms in the Spectral Clustering family typically consist of multiple separate stages (such as similarity matrix construction, low-dimensional embedding, and K-Means clustering as post processing), whi…

New bounds on Bartnik mass for surfaces with non-negative first eigenvalue.

Study improves understanding of Ricci curvature in manifolds.

Geodesic balls with non-negative Ricci curvature have a sharp lower bound on their first Dirichlet eigenvalue.

A novel method relaxes binary constraints to non-negative spheres for multi-matching and clustering.

The study proves inequalities on curved spaces without global curvature bounds.

New rigidity theorem for sharp spectral gap in nonnegatively curved spaces.

A recent theoretical analysis shows the equivalence between non-negative matrix factorization (NMF) and spectral clustering based approach to subspace clustering. As NMF and many of its variants are essentially linear, we introduce a nonlinear NMF with explicit orthogonality and derive general kernel-based orthogonal m…

Unified framework for structured principal subspace estimation with bounds and rates.

We derive a lower bound to the spectral threshold of the Dirichlet Laplacian in tubular neighbourhoods of constant radius about complete surfaces. This lower bound is given by the lowest eigenvalue of a one-dimensional operator depending on the radius and principal curvatures of the reference surface. Moreover, we show…

Let $(M,g)$ be a complete non-compact Riemannian manifold. We consider operators of the form $Δ_g + V$, where $Δ_g$ is the non-negative Laplacian associated with the metric $g$, and $V$ a locally integrable function. Let $ρ: (\hat{M},\hat{g}) \to (M,g)$ be a Riemannian covering, with Laplacian $Δ_{\hat{g}}$ and potenti…

We present a method based on the orthogonal symmetric non-negative matrix tri-factorization of the normalized Laplacian matrix for community detection in complex networks. While the exact factorization of a given order may not exist and is NP hard to compute, we obtain an approximate factorization by solving an optimiz…

In hyperspectral images, some spectral bands suffer from low signal-to-noise ratio due to noisy acquisition and atmospheric effects, thus requiring robust techniques for the unmixing problem. This paper presents a robust supervised spectral unmixing approach for hyperspectral images. The robustness is achieved by writi…

Simple proof that stable minimal hypersurfaces in R^4 are hyperplanes.

The past decade has seen substantial work on the use of non-negative matrix factorization and its probabilistic counterparts for audio source separation. Although able to capture audio spectral structure well, these models neglect the non-stationarity and temporal dynamics that are important properties of audio. The re…

Let $P$ be a non-negative self-adjoint Laplace type operator acting on sections of a hermitian vector bundle over a closed Riemannian manifold. In this paper we review the close relations between various $P$-related coefficients such as the mollified spectral counting coefficients, the heat trace coefficients, the reso…

In this paper, we give a sharp spectral characterization of conformally compact Einstein manifolds with conformal infinity of positive Yamabe type in dimension $n+1>3$. More precisely, we prove that the largest real scattering pole of a conformally compact Einstein manifold $(X,g)$ is less than $\ndemi -1$ if and only …

Let $(M,g)$ be a complete non-compact Riemannian surface. We consider operators of the form $Δ+ aK + W$, where $Δ$ is the non-negative Laplacian, $K$ the Gaussian curvature, $W$ a locally integrable function, and $a$ a positive real number. Assuming that the positive part of $W$ is integrable, we address the question "…

Paper generalizes Bakry-Émery calculus for curvature and applies to Markov chains.

Efficient private matrix analysis algorithms for recent variants.

We show that for every non-negative integer n there is a real n-dimensional family of minimal Lagrangian tori in CP^2, and hence of special Lagrangian cones in C^3 whose link is a torus. The proof utilises the fact that such tori arise from integrable systems, and can be described using algebro-geometric (spectral curv…

The paper proves criticality criteria and spectral splitting theorems for manifolds with Ricci bounds.

One of the longstanding open problems in spectral graph clustering (SGC) is the so-called model order selection problem: automated selection of the correct number of clusters. This is equivalent to the problem of finding the number of connected components or communities in an undirected graph. We propose automated mode…

The (constrained) minimization of a ratio of set functions is a problem frequently occurring in clustering and community detection. As these optimization problems are typically NP-hard, one uses convex or spectral relaxations in practice. While these relaxations can be solved globally optimally, they are often too loos…

Many spectral unmixing methods rely on the non-negative decomposition of spectral data onto a dictionary of spectral templates. In particular, state-of-the-art music transcription systems decompose the spectrogram of the input signal onto a dictionary of representative note spectra. The typical measures of fit used to …

A new algorithm solves nonnegative least squares faster with nonnegative data.

Spectral embedding is a procedure which can be used to obtain vector representations of the nodes of a graph. This paper proposes a generalisation of the latent position network model known as the random dot product graph, to allow interpretation of those vector representations as latent position estimates. The general…

This paper speeds up spectral clustering for large graphs by dilating their eigenspectrum.

Simple AMP algorithm robust to adversarial corruption.

In this work we study permutation synchronisation for the challenging case of partial permutations, which plays an important role for the problem of matching multiple objects (e.g. images or shapes). The term synchronisation refers to the property that the set of pairwise matchings is cycle-consistent, i.e. in the full…

High-dimensional curved diffusions show abrupt convergence at a critical time.

Model place cells as spatial embeddings for efficient path planning and cognitive map construction.

DeepMP improves non-negative sparse recovery performance.

We study both function theoretic and spectral properties on complete noncompact smooth metric measure space $(M,g,e^{-f}dv)$ with nonnegative Bakry-Émery Ricci curvature. Among other things, we derive a gradient estimate for positive $f$-harmonic functions and obtain as a consequence the strong Liouville property under…

Sharp inequality in spaces with non-negative Ricci curvature.

In most sampling algorithms, including Hamiltonian Monte Carlo, transition rates between states correspond to the probability of making a transition in a single time step, and are constrained to be less than or equal to 1. We derive a Hamiltonian Monte Carlo algorithm using a continuous time Markov jump process, and ar…

Non-negative curvature affects Markov chains' mixing and expansion properties.

Formal manifolds with non-negative Ricci curvature have formal covers.

In this paper we study non-negatively curved and rationally elliptic GKM$_4$ manifolds and orbifolds. We show that their rational cohomology rings are isomorphic to the rational cohomology of certain model orbifolds. These models are quotients of isometric actions of finite groups on non-negatively curved torus orbifol…

Study Kähler metrics on complex tori with almost non-negative scalar curvature.

Sharp inequality for submanifolds in manifolds with non-negative Ricci curvature.

The study examines symmetries in spaces with positive or non-negative curvature.

The paper proves conjectures and classifies metrics on 3D manifolds.

We study the asymptotic behavior of the difference $Δρ^{X, Y}_α:= ρ_α(X + Y) - ρ_α(X)$ as $α\rightarrow 1$, where $ρ_α$ is a risk measure equipped with a confidence level parameter $0 < α< 1$, and where $X$ and $Y$ are non-negative random variables whose tail probability functions are regularly varying. The case where …

Survey on rigidity and almost rigidity of Green functions in non-negative Ricci curvature spaces.

We study spaces and moduli spaces of Riemannian metrics with non-negative Ricci or non-negative sectional curvature on closed and open manifolds. We construct, in particular, the first classes of manifolds for which these moduli spaces have non-trivial rational homotopy, homology and cohomology groups. We also show tha…

Study on rigidity with non-negative intermediate curvature on low-dimensional manifolds.