The paper proves that most metrics satisfy a strong version of Arnold's conjecture for Laplace eigenvalues.
problem Understanding metrics that satisfy a strong version of Arnold's conjecture for Laplace eigenvalues.
method Using geometric characterizations and perturbation theory, the paper proves the conjecture for most metrics.
result The Strong Arnold Hypothesis is satisfied for all metrics except for a set of infinite codimension.
New estimate for stability eigenvalues of singular minimal hypersurfaces in spheres.
problem Estimating the first stability eigenvalue of singular minimal hypersurfaces in spheres.
method Extending an estimate by J. Simons to the singular setting.
result Any singular minimal hypersurface in Sn+1 has a first stability eigenvalue at most -2n. A test for sparsity in Bayesian networks helps choose algorithms.
problem Selecting appropriate structure discovery algorithms for Bayesian networks.
method Developed a hypothesis test using the largest eigenvalue of the normalized inverse covariance matrix.
result The hypothesis test can determine if a BN has max in-degree greater than 1.
Quantum Racah matrices for R=[2,2] are fully described and evaluated.
problem Calculating Racah matrices for quantum groups U_q(sl_N).
method Eigenvalue hypothesis for most matrices, highest weight method for degenerate cases.
result Complete Racah matrices for |R| ≤ 4, allowing calculation of HOMFLY polynomials.
Improves signal detection in non-Gaussian noise using transformed data.
problem Signal detection in rank-one signal-plus-noise data matrices.
method Pre-transforming matrix entries and using linear spectral statistics for hypothesis testing.
result Sharp phase transition of largest eigenvalues in spiked rectangular matrices.
The second eigenvalue of the Jacobi operator characterizes certain hypersurfaces in manifolds.
problem Characterizing hypersurfaces via the second eigenvalue of the Jacobi operator.
method Analyzing the second eigenvalue of the Jacobi operator on various manifolds.
result The slices of the warped product IimeshSn are the only hypersurfaces saturating a certain inequality involving the second eigenvalue of the Jacobi operator. We prove Cheng's eigenvalue comparison theorems for geodesic balls within the cut locus under weaker geometric hypothesis, and we also show that there are certain geometric rigidity in case of equality of the eigenvalues. This rigidity becomes isometric rigidity under upper sectional curvature bounds or lower Ricci cur…
Variational methods yield formulas for eigenvalues of elliptic operators, with applications to metric evolution.
problem Deriving formulas for eigenvalues of elliptic operators on compact manifolds.
method Variational methods applied to elliptic operators on compact Riemannian manifolds.
result Generic subsets of metrics yield simple spectra of elliptic operators.
This article investigates the correlation structure of the global crude oil market using the daily returns of 71 oil price time series across the world from 1992 to 2012. We identify from the correlation matrix six clusters of time series exhibiting evident geographical traits, which supports Weiner's (1991) regionaliz…
Antonio Ros gave a lower bound for the first eigenvalue λ1 of Δ of a P-manifold (M,g) in terms of the lower bound on the Ricci curvature RicM and asked what happened when this lower bound was achieved. In this paper we look in to this question and show that there are strong implications on the geometry and…
Improved eigenvalue distribution method for financial data.
problem Noise and complexity in financial markets.
method Matrix H theory, hierarchical structure, informational cascade.
result Captures a larger fraction of data variance in financial markets.
Recall that Federer-Fleming defined the notion of flat convergence of submanifolds of Euclidean space to solve the Plateau problem. Here we prove the upper semicontinuity of Neumann eigenvalues of the submanifolds when they converge in the flat sense without losing volume. With an additional condition on the boundaries…
Method identifies causal interactions between time series using extreme eigenvalue variability.
problem Detecting causal interactions between time series.
method Largest eigenvalue of lagged correlation matrices, measuring causal interactions through variability.
result The method outperforms traditional Granger causality tests in detecting structural changes.
Kernel interpolation is inconsistent for norms with smoothness above a constant.
problem Inconsistency of kernel interpolation in reproducing kernel Hilbert spaces.
method Lower bounds for generalization error in Sobolev norms.
result Kernel interpolation is always inconsistent for norms with smoothness above a constant.
Study on the spectrum of drift Laplacian on Ricci expanders.
problem Analyzing the spectrum of the drift Laplacian on Ricci expanders.
method Investigation of discrete spectrum under proper potential function, asymptotic behavior of potential function, and computation of eigenvalues.
result Discrete spectrum of the drift Laplacian on Ricci expanders with bounded Ricci curvature.
We analyze the spectrum of a non-backtracking matrix in a degree-corrected stochastic block model.
problem Characterizing the spectrum of the non-backtracking matrix in a degree-corrected stochastic block model.
method We consider a random graph with two equal-sized clusters and analyze the spectrum of the non-backtracking matrix.
result The leading eigenvalue of the non-backtracking matrix is asymptotic to $ρ= rac{a+b}{2} Φ^{(2)}$ and the second eigenvalue is asymptotic to $μ_2 = rac{a-b}{2} Φ^{(2)}$ under certain conditions.
The paper reviews methods for determining the number of communities in network data.
problem Determining the number of communities in network data.
method Statistical methods for hypothesis testing and clustering in network models.
result SCORE and NCV methods evaluated for clustering in Degree-Corrected Block Models, with NCV facing challenges.
Character expansion expresses extended HOMFLY polynomials through traces of products of finite dimensional R- and Racah mixing matrices. We conjecture that the mixing matrices are expressed entirely in terms of the eigenvalues of the corresponding R-matrices. Even a weaker (and, perhaps, more reliable) version of this …
The paper analyzes how quantization affects the Fisher Information Matrix's dominant eigenvalue.
problem The impact of quantization on the Fisher Information Matrix's dominant eigenvalue.
method The study examines spectral perturbation of the empirical Fisher Information Matrix under in-distribution input and quantized parameter perturbations.
result A bound on the eigenvalue under quantization noise, showing it strictly exceeds the unperturbed value at leading order.
Formula establishes determinant majorization for symmetric matrices.
problem Determining determinant majorization for symmetric matrices.
method Establishes determinant majorization formula for symmetric matrices using invariant Garding-Dirichlet polynomials.
result Formula F(A)N1≥det(A)n1 for symmetric matrices. We analyze cross-correlations between price fluctuations of different stocks using methods of random matrix theory (RMT). Using two large databases, we calculate cross-correlation matrices C of returns constructed from (i) 30-min returns of 1000 US stocks for the 2-yr period 1994--95 (ii) 30-min returns of 881 US stock…
In a complete Riemannian manifold (M,g) if the hessian of a real valued function satisfies some suitable conditions then it restricts the geometry of (M,g). In this paper we characterize all compact rank-1 symmetric spaces, as those Riemannian manifolds (M,g) admitting a real valued function u such that the …
Data structure affects deep learning performance, study finds.
problem Understanding why deep learning performs poorly on typical datasets.
method Analyzed input correlation matrices and network Hessians, developed PAC-Bayes bounds.
result Sloppy eigenspectra in input data correlate with poor deep learning performance.
Community detection in networks is a key exploratory tool with applications in a diverse set of areas, ranging from finding communities in social and biological networks to identifying link farms in the World Wide Web. The problem of finding communities or clusters in a network has received much attention from statisti…
We present a new approach to understanding credit relationships between commercial banks and quoted firms, and with this approach, examine the temporal change in the structure of the Japanese credit network from 1980 to 2005. At each year, the credit network is regarded as a weighted bipartite graph where edges corresp…
We develop a framework for post model selection inference, via marginal screening, in linear regression. At the core of this framework is a result that characterizes the exact distribution of linear functions of the response y, conditional on the model being selected (``condition on selection" framework). This allows…
The paper extends hypothesis testing to non-diagonalizable matrices, improving network statistics inference.
problem Testing on non-diagonalizable matrices for network statistics.
method Generalizes Wald and t-tests to non-symmetric matrices, controlling convergence rates.
result Improved inference on network statistics from directed networks.
Einstein 4-manifolds become conformally Kähler with positive scalar curvature.
problem Characterizing Einstein 4-manifolds with specific curvature properties.
method Combining LeBrun's conformal normalization with weighted divergence equations and first-order identities.
result Einstein metrics with simple largest eigenvalue of self-dual Weyl curvature become conformally Kähler with positive scalar curvature.
The paper proves a stability conjecture for manifolds with zero Euler characteristic.
problem Stability of manifolds with zero Euler characteristic under certain curvature conditions.
method Analyzes manifolds with dimensions 5 or more, proving stability under specific curvature and completeness conditions.
result 2006 Rosenberg's S1-stability holds for manifolds with zero Euler characteristic. Improves detection of low-rank signals from noisy data matrices.
problem Statistical detection of low-rank signals in noisy data matrices.
method Entrywise pre-transforming data matrix for non-Gaussian noise, sharp phase transition thresholds, central limit theorem for linear spectral statistics, hypothesis test.
result Improves detection of low-rank signals from noisy data matrices, generalizing known results.
The paper analyzes the latent geometry of generative diffusion models.
problem The manifold overfitting phenomenon in generative models.
method Statistical physics approach to analyze the spectrum of eigenvalues and singular values of the Jacobian of the score function.
result Three distinct qualitative phases during the generative process: trivial, manifold coverage, and consolidation phases.
New method reconstructs interbank networks enforcing reciprocity to improve stability and risk prediction.
problem Lack of public interbank network data and difficulty in replicating cycles.
method Proposes a new network reconstruction method enforcing sparsity and link reciprocity from aggregate data.
result Adding reciprocity improves prediction of network properties, including largest real eigenvalue and eccentricity of eigenvalues.
Universal algorithm learns unknown distribution for various decision-making problems.
problem Various statistical measures in contextual sequential decision-making.
method Infinite-dimensional functional regression oracle for cumulative distribution functions.
result Utility regret rate bounded by polynomial decay of eigenvalue sequence.
Active local learning uses fewer labels to predict near-optimal functions.
problem Efficiently predicting near-optimal functions with fewer labels.
method Active local learning algorithm for estimating functions with fewer labels.
result Algorithm makes significantly fewer label queries than traditional methods.
Explains eigenvalue and generalized eigenvalue problems with examples.
problem Eigenvalue and generalized eigenvalue problems.
method Introduction and examples from machine learning.
result Solutions to eigenvalue and generalized eigenvalue problems.
The paper compares Steklov and Laplacian eigenvalues on graphs.
problem Understanding the relationship between Steklov and Laplacian eigenvalues on graphs.
method Analyzing eigenvalues and discussing rigidity.
result Obtained Lichnerowicz-type estimates and combinatorial estimates for Steklov eigenvalues.
Paper finds bounds for Steklov eigenvalues on manifolds.
problem Eigenvalue bounds for Steklov eigenvalues on manifolds.
method Eigenvalue comparison theorems and bounds for the first non-zero eigenvalue of the Wentzell eigenvalue problem.
result Established bounds for Steklov eigenvalues and Wentzell eigenvalues.
The paper explores curvature types of surfaces in normed spaces.
problem Investigating curvature types of surfaces in normed spaces.
method Defining curvature types using a normal vector field and eigenvalues of the differential of the Gauss map.
result A compact surface with constant Minkowski Gaussian curvature is a Minkowski sphere.
Study of loops in sums of Laplace eigenfunctions on surfaces.
problem Uniform bound for the number of nested loops in sums of Laplace eigenfunctions.
method Real-analytic category analysis and biharmonic function construction.
result Uniform bound for the number of rooted double nests in terms of surface, root, and spectral cutoff.
A framework for hypothesis testing on attributed graphs using sampling.
problem Statistical testing on graph data, especially large attributed graphs.
method Sampling-based framework with PHASE and PHASEopt for accurate and efficient hypothesis testing.
result PHASE and PHASEopt improve accuracy and efficiency of hypothesis testing in attributed graphs.
Estimates eigenvalue of p-Laplacian on curved spaces.
problem Estimating eigenvalues of p-Laplacian on curved spaces.
method Integral curvature conditions to estimate eigenvalues.
result Various estimates of the first eigenvalue.
The paper explores inequalities between eigenvalues on Riemannian manifolds.
problem Investigating relationships between eigenvalues on Riemannian manifolds.
method Constructing gradient estimates for a first eigenfunction to derive inequalities.
result Obtained some relationships between weighted p-Laplacian first eigenvalues. Study on generalization for data-dependent hypothesis sets.
problem Understanding generalization in hypothesis sets dependent on data.
method Learning guarantee based on transductive Rademacher complexity and hypothesis set stability.
result Generalization bound for data-dependent hypothesis sets.
The study improves bounds for Laplace eigenvalues in Kaehler manifolds.
problem Improving bounds for Laplace eigenvalues in Kaehler manifolds.
method Generalizing classical inequalities to higher eigenvalues and applying to analytic varieties.
result Proves inequalities for Laplace eigenvalues of Kaehler manifolds and analytic varieties.
Paper extends Steklov eigenvalue estimates to higher values.
problem Estimating higher Steklov eigenvalues.
method Generalization of Raulot-Savo's first eigenvalue estimate.
result Estimates for higher Steklov eigenvalues established.
The paper connects Steklov eigenvalues and Laplacian eigenvalues on Riemannian manifolds.
problem Understanding the relationship between Steklov and Laplacian eigenvalues on Riemannian manifolds.
method Using sectional curvature conditions, the paper proves mutual control between Steklov and Laplacian eigenvalues.
result Derives a Weyl-type upper bound for Steklov eigenvalues.
Eigenvalues of Steklov eigenproblems change predictably with boundary tweaks.
problem Understanding how Steklov eigenvalues respond to boundary changes.
method Analyzing smooth boundary perturbations of Steklov eigenvalues.
result Steklov eigenvalues are generically simple under such perturbations.
Study compares eigenvalues on spherically symmetric manifolds to Euclidean balls.
problem Comparing eigenvalues on spherically symmetric manifolds to Euclidean balls.
method Examines Dirichlet Laplace eigenvalues on balls of spherically symmetric manifolds and Euclidean space.
result Eigenvalues on spherically symmetric manifolds are smaller for small radii, but larger for hyperbolic spaces.