arXiv research

We perform a parallel analysis of the spectral density of (i) the logarithm of price and (ii) the daily number of trades of a set of stocks traded in the New York Stock Exchange. The stocks are selected to be representative of a wide range of stock capitalization. The observed spectral densities show a different power-…

A robust method for decomposing spectral peaks robust to distortion and interference.

The study finds arbitrarily small spectral gaps for random hyperbolic surfaces with many cusps.

Empirical analysis of the foreign exchange market is conducted based on methods to quantify similarities among multi-dimensional time series with spectral distances introduced in [A.-H. Sato, Physica A, 382 (2007) 258--270]. As a result it is found that the similarities among currency pairs fluctuate with the rotation …

We introduce the notion of spectral flow along a periodic semi-Riemannian geodesic, as a suitable substitute of the Morse index in the Riemannian case. We study the growth of the spectral flow along a closed geodesic under iteration, determining its asymptotic behavior.

Study low energy resolvent behavior on fibred boundary metrics.

This paper studies how adding leaves to a tree affects its spectral properties.

New method extrapolates spectral densities from smaller models to larger ones.

Graph signal processing detects hallucinations in large language models.

Study spectral properties of sub-Laplacians in Carnot groups.

Spectral algorithms are graph partitioning algorithms that partition a node set of a graph into groups by using a spectral embedding map. Clustering techniques based on the algorithms are referred to as spectral clustering and are widely used in data analysis. To gain a better understanding of why spectral clustering i…

This note summarizes results that were obtained by the author in his habilitation thesis (arXiv:1607.08792) concerning the development of a spectral theory for simply periodic, 2-dimensional, complex-valued solutions of the sinh-Gordon equation. Spectral data for such solutions are defined for periodic Cauchy data on a…

Let Y be a compact, oriented 3-manifold with a contact form a. For any Dirac operator D, we study the asymptotic behavior of the spectral flow between D and D+cl(-ira) as r very large. If a is the Thurston-Winkelnkemper contact form whose monodromy is the product of Dehn twists along disjoint circles, we prove that the…

We formulate a precise conjecture about the universal behavior near the diagonal of the spectral function of the Laplacian of a smooth compact Riemann manifold. We prove this conjecture when the manifold and the metric are real analytic, and we also present an alternate proof when the manifold is the round sphere.

In this paper we study the asymptotic behavior of second-order uniformly elliptic operators on weighted Riemannian manifolds. They naturally emerge when studying spectral properties of the Laplace-Beltrami operator on families of manifolds with rapidly oscillating metrics. We appeal to the notion of H-convergence intro…

Accelerates optimal transport computation by 10x with spectral insights.

Study the spectral flow of Dirac operators on spinor bundles.

New method embeds dynamic networks with stability for node behavior.

Gaussian process models simplify neural network behavior for easier understanding.

To date, most studies on spam have focused only on the spamming phase of the spam cycle and have ignored the harvesting phase, which consists of the mass acquisition of email addresses. It has been observed that spammers conceal their identity to a lesser degree in the harvesting phase, so it may be possible to gain ne…

The paper explores how structured representations influence learning dynamics in neural networks.

Study the Bochner-Schrödinger operator on symplectic manifolds, proving gap existence and asymptotic kernel behavior.

When analyzing weighted networks using spectral embedding, a judicious transformation of the edge weights may produce better results. To formalize this idea, we consider the asymptotic behavior of spectral embedding for different edge-weight representations, under a generic low rank model. We measure the quality of dif…

This paper provides a full controlled version of algebraic $K$-theory. This includes a rich array of assembly maps; the controlled assembly isomorphism theorem identifying the controlled group with homology; and the stability theorem describing the behavior of the inverse limit as the control parameter goes to 0. There…

In this paper, we give an easy proof of the main results of Andrews and Clutterbuck's paper [J. Amer. Math. Soc. 24 (2011), no. 3, 899--916], which gives both a sharp lower bound for the spectral gap of a Schröinger operator and a sharp modulus of concavity for the logarithm of the corresponding first eigenfunction. We…

Sharp spectral gap estimates for higher-order operators on hyperbolic spaces.

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

New models explain heavy-tailed behavior in neural networks.

We study the behavior of the degeneration at the second step of the Frölicher spectral sequence of a $\mathcal{C}^{\infty}$ family of compact complex manifolds. Using techniques from deformation theory and adapting them to pseudo-differential operators we prove a result \textit{à la Kodaira-Spencer} for the dimension o…

Study hot spots on warped product manifolds and infinite cones.

Muon optimizer simplifies matrix optimization with spectral orthogonalization.

The paper constructs noncompact hyperbolic surfaces with uniform spectral gaps using random graph models.

A time-varying network reveals community structure in cryptocurrencies.

Non-Markovian point process shows power-law scaling, similar to nonlinear Markovian process.

Study Bergman and spectral kernels for non-compact complex manifolds.

We study the limiting behavior of eigenfunctions/eigenvalues of the Laplacian of a family of Riemannian metrics that degenerates on a hypersurface. Our results generalize earlier work concerning the degeneration of hyperbolic surfaces.

Interpolates mean shift and spectral clustering on graphs.

This work analyzes a two-stage algorithm for single index models, showing precise asymptotics of gradient descent.

A conformal immersion of a 2-torus into the 4-sphere is characterized by an auxiliary Riemann surface, its spectral curve. This complex curve encodes the monodromies of a certain Dirac type operator on a quaternionic line bundle associated to the immersion. The paper provides a detailed description of the geometry and …

New framework predicts AMP behavior in spiked models for finite iterations.

We analyze the Bombay stock exchange (BSE) price index over the period of last 12 years. Keeping in mind the large fluctuations in last few years, we carefully find out the transient, non-statistical and locally structured variations. For that purpose, we make use of Daubechies wavelet and characterize the fractal beha…

Developed a framework for designing filters in spectral GCNNs with improved performance.

Spectral algorithms improve under covariate shift with novel weighted techniques.

A recent line of work has uncovered a new form of data poisoning: so-called \emph{backdoor} attacks. These attacks are particularly dangerous because they do not affect a network's behavior on typical, benign data. Rather, the network only deviates from its expected output when triggered by a perturbation planted by an…

We study the behavior of the heat kernel of the Hodge Laplacian on a contact manifold endowed with a family of Riemannian metrics that blow-up the directions transverse to the contact distribution. We apply this to analyze the behavior of global spectral invariants such as the eta-invariant and the determinant of the L…

I review the milestones of the mathematical work of Krzysztof P. Wojciechowski. This will at the same time be a tour of Analysis and Geometry of Boundary Value Problems. Starting in the 80s I will discuss the spectral flow and the general linear conjugation problem, the Calderon projector and the topology of space of e…

Machine learning predicts dam-break flood wave behavior accurately.

In this article, associated with each lattice $T\subseteq \mathbb{Z}^n$ the concept of a harmonic-counting measure $ν_T$ on a sphere $S^{n-1}$ is introduced and it is applied to determine the asymptotic behavior of the eigenfunctions of the Laplace-Beltrami operator on a lens space. In fact, the asymptotic behavior of …