Introduces spectral-domain Wasserstein distance and Gelbrich bound for elliptical processes.
problem Estimating distances and bounds for elliptical stochastic processes.
method Defines spectral-domain W2 Wasserstein distance and Gelbrich bound. result Develops new spectral-domain bounds for non-elliptical processes.
Study spectral distances on compact RCD spaces.
problem Understanding spectral convergence in RCD spaces.
method Established relationships between different spectral convergences and constructed a spectral approximation map.
result Found canonical spectral approximation map for RCD spaces.
Estimates manifold distances using graph Laplacian, proving consistency.
problem Estimating distances in compact Riemannian manifolds.
method Graph Laplacian estimates of the Laplace-Beltrami operator, bounding errors.
result Proof of consistency for manifold distances.
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 analyze convergence of Fermat distances and their application in clustering.
problem Understanding convergence properties of Fermat distances on Riemannian manifolds.
method Geometric and statistical arguments in percolation theory, leveraging novel arguments for non-uniform densities and curved domains.
result Discrete, sample-based Fermat distances converge to their continuum analogues with a precise rate dependent on intrinsic dimensionality.
Survey of spectral, probabilistic, and deep metric learning methods.
problem Developing effective distance metrics for various machine learning tasks.
method Divided into spectral, probabilistic, and deep approaches, covering various techniques and their applications.
result Comprehensive overview of metric learning methods, including new developments and applications.
This is a review of explicit computations of Connes distance in noncommutative geometry, covering finite dimensional spectral triples, almost-commutative geometries, and spectral triples on the algebra of compact operators. Several applications to physics are covered, like the metric interpretation of the Higgs field, …
Model tracks structural changes in Brownian particle configurations on a sphere.
problem Tracking structural changes in Brownian particle configurations on a sphere.
method Introduces Frustrated Distance Matrix (FDM) model for dynamic distance matrices on S^2.
result Preserves static BBS template with dynamics as redistributed spectral mass.
Sub-Riemannian spectral distance defined using eigenfunctions of sub-Laplacian
problem Sub-Riemannian geometry and eigenvalues of sub-Laplacian
method Embedding manifold into Hilbert space using eigenfunctions
result Defined spectral distance between sub-Riemannian manifolds
We define a class of Euclidean distances on weighted graphs, enabling to perform thermodynamic soft graph clustering. The class can be constructed form the "raw coordinates" encountered in spectral clustering, and can be extended by means of higher-dimensional embeddings (Schoenberg transformations). Geographical flow …
The paper shows how to recover true node positions from a graph or similarity matrix.
problem Recovering true distances and positions from a graph or similarity matrix.
method Two steps: matrix factorisation followed by nonlinear dimension reduction.
result Nonlinear dimension reduction can recover latent positions close to a manifold where geodesic distance is encoded.
Geodesic rays and chordal distances link algebraic and geometric properties of positive metrics.
problem Understanding the geometry of the space of positive metrics at infinity.
method Using Monge-Ampère equations and test configurations, algebraic descriptions of geodesic rays and chordal distances are derived.
result The Mabuchi chordal distance between geodesic rays associated with ample test configurations equals the spectral distance between their filtrations.
Graph curvature measured by inverse resistance distance.
problem Defining and analyzing curvature in graphs.
method Defining curvature via inverse resistance distance and proving properties.
result Graphs with positive curvature have controlled diameter and spectral properties.
Proves Hölder-type inequality for Lagrangians' distance.
problem Understanding the symplectic geometry of Lagrangians.
method Developed methods from previous works to establish the inequality.
result Established a Hölder-type inequality for the Hausdorff distance between Lagrangians.
CAST improves spectral clustering for multi-scale data by integrating reachability similarity.
problem Applying spectral clustering to multi-scale data where clusters vary in size and density.
method CAST integrates reachability similarity with distance-based similarity to derive a coefficient matrix, then applies trace Lasso regularization.
result CAST provides excellent performance and robustness across various multi-scale data test cases.
New method improves speech synthesis quality.
problem Efficiently train parallel speech synthesis models.
method Spectral energy distance for implicit generative models.
result State-of-the-art generation quality achieved.
Unified framework for hyperbolic embeddings from mixed data types.
problem Computing hyperbolic embeddings from noisy metric and non-metric data.
method Semidefinite programming and spectral factorization methods.
result Efficient computation of hyperbolic embeddings from arbitrary data.
We propose a spectral clustering method based on local principal components analysis (PCA). After performing local PCA in selected neighborhoods, the algorithm builds a nearest neighbor graph weighted according to a discrepancy between the principal subspaces in the neighborhoods, and then applies spectral clustering. …
Study predicts U.S. county COVID-19 growth using demographic and social distancing data.
problem Predicting county-level COVID-19 growth during the pandemic.
method Spectral clustering, correlation matrix, demographic features, social distancing scores, LSTM model.
result Effective prediction of future county growth using demographic and social distancing data.
In the context of clustering, we consider a generative model in a Euclidean ambient space with clusters of different shapes, dimensions, sizes and densities. In an asymptotic setting where the number of points becomes large, we obtain theoretical guaranties for a few emblematic methods based on pairwise distances: a si…
Spectral clustering improves accuracy and efficiency for clustering discrete distributions.
problem Inaccurate clustering of discrete distributions using traditional methods.
method Spectral clustering combined with distribution affinity measures (MMD, Wasserstein distance) and linear optimal transport.
result Spectral clustering outperforms traditional methods in accuracy and efficiency.
Researchers compute the full spectrum of Laplace operator on distance spheres in symmetric spaces.
problem Computing the full Laplace spectrum on distance spheres in symmetric spaces.
method Lie-theoretic methods to explicitly compute the spectrum.
result Unified formula for the full spectrum of Laplace operator on distance spheres in symmetric spaces of rank one.
Random geometric graphs are a popular choice for a latent points generative model for networks. Their definition is based on a sample of n points X1,X2,⋯,Xn on the Euclidean sphere~Sd−1 which represents the latent positions of nodes of the network. The connection probabilities between the node…
SNGP improves single-model deep uncertainty by enhancing distance-awareness.
problem Improving uncertainty estimation in deep learning models, especially for real-time applications.
method SNGP improves distance-awareness of DNNs through spectral normalization and Gaussian process layers.
result SNGP outperforms other single-model approaches in prediction, calibration, and out-of-domain detection.
We propose the Wasserstein-Fourier (WF) distance to measure the (dis)similarity between time series by quantifying the displacement of their energy across frequencies. The WF distance operates by calculating the Wasserstein distance between the (normalised) power spectral densities (NPSD) of time series. Yet this ratio…
Let M be a connected, noncompact, complete Riemannian manifold, consider the operator $L=\DD +\nn V$ for some V∈C2(M) with exp[V] integrable w.r.t. the Riemannian volume element. This paper studies the existence of the spectral gap of L. As a consequence of the main result, let $\rr$ be the distance functi…
To a closed Riemannian manifold, we associate a set of (special values of) a family of Dirichlet series, indexed by functions on the manifold. We study the meaning of equality of two such families of spectral Dirichlet series under pullback along a map. This allows us to give a spectral characterization of when a smoot…
This paper establishes the consistency of spectral approaches to data clustering. We consider clustering of point clouds obtained as samples of a ground-truth measure. A graph representing the point cloud is obtained by assigning weights to edges based on the distance between the points they connect. We investigate the…
A new clustering method improves recovery guarantees by re-embedding data.
problem Improving recovery guarantees in clustering algorithms.
method Chaining four techniques: leapfrog distances, multidimensional scaling, spectral methods, and sum-of-norms clustering.
result Re-embedding data improves recovery guarantees of clustering.
The problem of filtering information from large correlation matrices is of great importance in many applications. We have recently proposed the use of the Kullback-Leibler distance to measure the performance of filtering algorithms in recovering the underlying correlation matrix when the variables are described by a mu…
A new clustering algorithm considers data smoothness for better performance.
problem Clustering multi-scale data with varying cluster densities.
method Divide objects into tiny clusters, cluster centers form smooth graphs.
result Significantly outperforms state-of-the-art clustering algorithms.
COPT optimizes graph distances via simultaneous optimal transport.
problem Learning graph representations unsupervisedly.
method Simultaneous optimization of dual transport plans between vertices and graph signals.
result COPT preserves spectral information and outperforms state-of-the-art methods.
SNGP improves DNNs' uncertainty estimation with minimal changes.
problem Uncertainty estimation in deep learning models for real-time applications.
method Formalizing uncertainty as a minimax problem, SNGP adds weight normalization and replaces the output layer with a Gaussian process.
result SNGP outperforms other single-model approaches in uncertainty estimation across vision and language tasks.
The increasing access to brain signal data using electroencephalography creates new opportunities to study electrophysiological brain activity and perform ambulatory diagnoses of neuronal diseases. This work proposes a pairwise distance learning approach for Schizophrenia classification relying on the spectral properti…
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.
New algorithm learns low-rank matrices with linear number of samples.
problem Learning low-rank matrices efficiently in latent-variable applications.
method Proposed algorithm that uses linear number of samples in high dimension.
result Learning kimesk, rank-r, matrices requires $Ω( rac{kr}{ε^2})$ samples. I-BBS identifies latent sub-manifolds from distance matrices, robust to noise.
problem Identifying latent sub-manifolds from distance matrices in high-dimensional spaces.
method Coordinate-free inference using random distance matrix theory and generative noise models.
result Recovering latent geometry from integer-stable signatures of eigenvalues.
We show that the Kullback-Leibler distance is a good measure of the statistical uncertainty of correlation matrices estimated by using a finite set of data. For correlation matrices of multivariate Gaussian variables we analytically determine the expected values of the Kullback-Leibler distance of a sample correlation …
The article deals with intrinsic metrics, Dirac operators and spectral triples induced by regular Dirichlet and resistance forms. We show, in particular, that if a local resistance form is given and the space is compact in resistance metric, then the intrinsic metric yields a geodesic space. Given a regular Dirichlet f…
We consider the optimal prediction problem of stopping a spectrally negative Lévy process as close as possible to a given distance b≥0 from its ultimate supremum, under a squared error penalty function. Under some mild conditions, the solution is fully and explicitly characterised in terms of scale functions. We…
New theorem improves spectral gap for sampling from mixture distributions.
problem Sampling from multimodal distributions with simulated tempering.
method Introduced a decomposition theorem for the restricted spectral gap of simulated tempering.
result Lower bound on the restricted spectral gap for mixture distributions.
Develops DSD for analyzing multiscale biological networks.
problem Analyzing multiscale structure in biological networks.
method Data-driven diffusion process with multitemporal analysis.
result Parameter-free inference of intrinsic data structure.
Random walks on hyperbolic spaces follow predictable large deviation principles.
problem Understanding the behavior of random walks on hyperbolic spaces.
method Large deviation principles for displacement and translation distances.
result Translation and displacement distances satisfy large deviation principles with the same rate function.
High-frequency financial data of the foreign exchange market (EUR/CHF, EUR/GBP, EUR/JPY, EUR/NOK, EUR/SEK, EUR/USD, NZD/USD, USD/CAD, USD/CHF, USD/JPY, USD/NOK, and USD/SEK) are analyzed by utilizing the Kullback-Leibler divergence between two normalized spectrograms of the tick frequency and the generalized Jensen-Sha…
High-dimensional curved diffusions show abrupt convergence at a critical time.
problem Understanding abrupt convergence in high-dimensional curved diffusions.
method Functional inequalities and spectral rigidity.
result Abrupt convergence (cutoff) occurs in high dimensions, linked to spectral rigidity.
Study extends bounds on sample covariance matrices with general dependence.
problem Quantitative bounds on sample covariance matrices with i.i.d. columns.
method Extends previous work on deterministic equivalent to rectangular random matrices with general dependence structure.
result Proves quantitative bounds involving dimensions and spectral parameter, including closer proximity to real positive semi-line.
Spectral clustering is one of the most widely used techniques for extracting the underlying global structure of a data set. Compressed sensing and matrix completion have emerged as prevailing methods for efficiently recovering sparse and partially observed signals respectively. We combine the distance preserving measur…
Unified analysis of multilabel Fisher discriminants with improved dimensionality and robustness.
problem Improving discriminant analysis for multilabel classification with enhanced dimensionality and robustness.
method Unified theoretical analysis of multilabel Fisher discriminants with algebraic and statistical guarantees.
result Unified characterization of multilabel Fisher objectives and their equivalence under orthogonality constraints.