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.
problem Decomposing spectral peaks in the presence of distortion and interference.
method Optimizing a nonparametric approach using pseudo-symmetric functions with nonincreasing behavior.
result Decomposed spectral peaks show pseudo-orthogonal behavior and power preserving equality.
The study finds arbitrarily small spectral gaps for random hyperbolic surfaces with many cusps.
problem Understanding spectral gaps of random hyperbolic surfaces with many cusps.
method Analysis of moduli spaces of hyperbolic surfaces with Weil-Petersson metric.
result Arbitrarily small spectral gaps are observed as the number of cusps grows slower than the genus.
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.
problem Analyze the resolvent of Hodge Laplacian on manifolds with fibred boundary metrics.
method Develop a 'split' pseudodifferential calculus to handle different asymptotic behaviors.
result Precise asymptotic behavior of resolvent as a fibred boundary pseudodifferential operator.
This paper studies how adding leaves to a tree affects its spectral properties.
problem Investigating the asymptotic behavior of tree spectra under leaf attachment.
method Analyzing the Ricci matrix and its largest eigenvalue for trees with pendant edges added.
result The sequence of largest eigenvalues converges to a limit that depends on local branch data.
New method extrapolates spectral densities from smaller models to larger ones.
problem Limited practical computations for large machine learning models.
method Algebraic spectral curve theory for free decompression.
result Framework enables extrapolation of spectral densities with multiple or multi-modal bulks.
Graph signal processing detects hallucinations in large language models.
problem Detecting factual reasoning from hallucinations in large language models.
method Modeling transformer layers as dynamic graphs, using spectral analysis to define diagnostics.
result Spectral signatures can distinguish different types of hallucinations and achieve high accuracy.
Study spectral properties of sub-Laplacians in Carnot groups.
problem Spectral properties of sub-Laplacians in Carnot groups.
method Proved pure point spectrum and spectral gap; applied to small ball problem and heat content.
result Proved existence of spectral gap and pure point spectrum.
Improved spectral clustering algorithm for better performance.
problem Improving the performance of spectral clustering algorithms.
method Developed a new performance guarantee under a weaker assumption and evaluated using a different spectral embedding map.
result Better performance guarantee under a weaker assumption and evaluation of a new spectral embedding map.
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…
Study contact manifold heat kernels under Riemannian metrics blow-up.
problem Analyze spectral invariants on contact manifolds.
method Examine Hodge Laplacian heat kernel behavior under Riemannian metrics.
result Contact versions of eta-invariant and analytic torsion are topological.
Accelerates optimal transport computation by 10x with spectral insights.
problem Exponential slow-down of convergence in Entropic Optimal Transport as regularization weakens.
method Spectral insights and spectral warm-start strategy to mitigate convergence issues.
result Faster convergence compared to the reference method Sinkhorn algorithm.
Study the spectral flow of Dirac operators on spinor bundles.
problem Understanding the asymptotic behavior of spectral flow for Dirac operators.
method Variation of eta invariant and local index theory technique.
result Established a uniform estimate of the eta invariant for large parameter values.
New method embeds dynamic networks with stability for node behavior.
problem Embed time-evolving node representations with stability.
method Unfolded adjacency spectral embedding for dynamic networks.
result Method satisfies cross-sectional and longitudinal stability.
Gaussian process models simplify neural network behavior for easier understanding.
problem Understanding and predicting the behavior of deep learning systems.
method Constructing surrogate models using Gaussian processes from finite neural networks.
result Surrogate models capture phenomena like spectral bias and predict generalization well.
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.
problem Understanding the training dynamics of deep neural networks.
method Investigates a family of enriched transformation layers with constrained pathways and adaptive corrections.
result Improved robustness, smoother optimization, and scalable depth behavior are achieved through structured representations.
Study the Bochner-Schrödinger operator on symplectic manifolds, proving gap existence and asymptotic kernel behavior.
problem Analyzing the spectrum and asymptotic behavior of the Bochner-Schrödinger operator on symplectic manifolds.
method Rough asymptotic description, existence proof, off-diagonal exponential estimate, complete asymptotic expansion.
result Existence of gaps in the spectrum 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.
problem Estimating spectral gaps for higher-order operators on Cartan-Hadamard manifolds.
method Symmetrization-free proofs based on general functional inequalities.
result Solves a sharp asymptotic problem from Cheng and Yang and answers a question from Kristály.
Unified framework for analyzing graph neural operators converging to graph limits.
problem Analyzing convergence of graph neural operators to graph limits.
method Develops a unified spectral framework for graph neural operators under various graphon assumptions.
result Unified framework enables direct comparison of convergence rates and tradeoffs.
New models explain heavy-tailed behavior in neural networks.
problem Heavy-tailed spectral densities in neural networks.
method High-temperature Marchenko-Pastur (HTMP) ensemble models.
result Heavy-tailed behavior arises from three factors: data structure, training temperature, and eigenvector entropy.
We study the behavior of the degeneration at the second step of the Frölicher spectral sequence of a C∞ 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.
problem Analyzing hot spots on specific geometric structures.
method Examining solutions to the heat equation on warped product manifolds and infinite cones.
result Hot spots behavior on warped product manifolds and infinite cones determined.
Muon optimizer simplifies matrix optimization with spectral orthogonalization.
problem Matrix optimization challenges, especially with large condition numbers.
method Simplified Muon optimizer using spectral orthogonalization of gradients.
result Simplified Muon converges linearly with independent scalar sequences, outperforming gradient descent and Adam.
The paper constructs noncompact hyperbolic surfaces with uniform spectral gaps using random graph models.
problem Building noncompact hyperbolic surfaces with uniform spectral gaps.
method Introduced a random graph model Fχ,n to construct expanding families of graphs, then applied these families to create hyperbolic surfaces. result Explicitly constructed an expanding family of graphs in the critical regime, leading to a sequence of complete, noncompact hyperbolic surfaces with uniformly positive spectral gaps.
A time-varying network reveals community structure in cryptocurrencies.
problem Investing in cryptocurrencies from different communities can diversify risk.
method Dynamic covariate-assisted spectral clustering method.
result Investors can earn 1.08% daily return by diversifying across communities.
Deeper quantum circuits can improve performance on unseen data, contrary to traditional views.
problem Understanding scaling behavior of parameterized quantum circuits and their generalization.
method Gradient-based PQCs, add-one-in perturbation techniques, spectral properties of random matrices.
result Gradient-based PQCs can exhibit improved performance on unseen data as model size increases, displaying double descent behavior.
Study Bergman and spectral kernels for non-compact complex manifolds.
problem Analyze asymptotic behavior of kernels over non-compact complex manifolds.
method Generalize scaling method to study Bergman and spectral kernels.
result Derive leading term of Bergman and spectral kernels under local convergence of Chern curvatures.
Non-Markovian point process shows power-law scaling, similar to nonlinear Markovian process.
problem Understanding the scaling behavior of non-Markovian point processes.
method Analyzed a confined fractional Brownian motion-driven point process and compared it to a nonlinear Markovian process.
result A nonlinear Markovian process can reproduce the power-law scaling behavior of a non-Markovian point process.
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.
problem Data clustering algorithms.
method Fokker-Planck equations on data graphs.
result New theoretical insights on diffusion maps and mean shift dynamics.
This work analyzes a two-stage algorithm for single index models, showing precise asymptotics of gradient descent.
problem Learning single index models with non-convex optimization.
method Spectral initialization followed by gradient descent, with detailed analysis of dynamics and asymptotics.
result Gradient descent converges to long-time fixed points in the large system limit, representing mean field behavior.
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.
problem Understanding AMP dynamics in high-dimensional spiked models.
method Developed a non-asymptotic framework for AMP in spiked matrix estimation.
result Predicted AMP behavior for up to O(polylognn) iterations in Z2 synchronization. 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.
problem Designing effective filters for spectral GCNNs with regularization properties.
method Exploring regularization properties of graph Laplacian and proposing a generalized framework for filter design.
result New filters derived from the framework outperform state-of-the-art techniques in semi-supervised node classification.
Spectral algorithms improve under covariate shift with novel weighted techniques.
problem Improving spectral algorithms' performance under covariate shift.
method Analysis of spectral algorithms in non-parametric regression over RKHS, proposing a weighted spectral algorithm with clipped weights.
result Normalized weighted spectral algorithm achieves optimal capacity-independent convergence rates, and clipped weights can approach optimal capacity-dependent rates.
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…
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.
problem Predicting long-term wave behavior in dam-break floods.
method Solved Saint-Venant equations using Lax-Wendroff scheme, trained RC-ESN with flow depth data.
result RC-ESN model predicts 286 time-steps ahead with RMSE < 0.01, outperforming LSTM.