arXiv research

Optimal spectral method found for inhomogeneous spiked Wigner model.

Researchers found the Wigner derivative and its inverse are equal for spherical tetrahedra.

Many statistical methods for network data parameterize the edge-probability by attributing latent traits to the vertices such as block structure and assume exchangeability in the sense of the Aldous-Hoover representation theorem. Empirical studies of networks indicate that many real-world networks have a power-law dist…

In this paper we study global properties of the Wigner caustic of parameterized closed planar curves. We find new results on its geometry and singular points. In particular, we consider the Wigner caustic of rosettes, i.e. regular closed parameterized curves with non-vanishing curvature. We present a decomposition of a…

A novel method computes Wigner kernels for atomic environments, achieving state-of-the-art accuracy.

Consistent model selection for spiked Wigner model via AIC-type criteria.

We show that, for each alpha in the interval (-1,1), the only Riemannian metrics on the space of positive definite matrices for which the alpha and -alpha-connections are mutually dual are matrix multiples fo the Wigner-Yanase-Dyson metric. If we further impose that the metric be monotone, then this set is reduced to s…

Study detects signals in spiked Wigner models using log likelihood ratio.

Wigner's theorem asserts that an isometric (probability conserving) transformation on a quantum state space must be generated by a Hamiltonian that is Hermitian. It is shown that when the Hermiticity condition on the Hamiltonian is relaxed, we obtain the following complex generalisation of Wigner's theorem: a holomorph…

Recently it has become popular to learn sparse Gaussian graphical models (GGMs) by imposing l1 or group l1,2 penalties on the elements of the precision matrix. Thispenalized likelihood approach results in a tractable convex optimization problem. In this paper, we reinterpret these results as performing MAP estimation u…

Study on eigenvalue distribution of correlated time series deforming the semi-circle law.

We examine the recovery of block sparse signals and extend the framework in two important directions; one by exploiting signals' intra-block correlation and the other by generalizing signals' block structure. We propose two families of algorithms based on the framework of block sparse Bayesian learning (BSBL). One fami…

Extends Wigner's representation to study super hyperbolic geometry.

Study provides selective inference method for latent block models.

New model for detecting communities in weighted bipartite networks.

In this paper we study singular points of the Wigner caustic and affine $λ$--equidistants of planar curves based on shapes of these curves. We generalize the Blaschke-Süss theorem on the existence of antipodal pairs of a convex curve.

We consider the weak detection problem in a rank-one spiked Wigner data matrix where the signal-to-noise ratio is small so that reliable detection is impossible. We propose a hypothesis test on the presence of the signal by utilizing the linear spectral statistics of the data matrix. The test is data-driven and does no…

Training large machine learning (ML) models with many variables or parameters can take a long time if one employs sequential procedures even with stochastic updates. A natural solution is to turn to distributed computing on a cluster; however, naive, unstructured parallelization of ML algorithms does not usually lead t…

Study optimal algorithms for recovering signals through inhomogeneous low-rank channels.

PPM improves graph matching for correlated Gaussian Wigner models with high probability.

Affine $λ$-equidistants of convex polygons with parallel opposite sides have applications to isoperimetric inequalities.

On the manifold of positive definite matrices, we investigate the existence of pairs of flat affine connections, dual with respect to a given monotone metric. The connections are defined either using the $α$-embeddings and finding the duals with respect to the metric, or by means of contrast functionals. We show that i…

Extends particle classification to curved space-times using groupoids.

Study of correlated Wigner matrices with BBP transitions.

We study the statistical decision process of detecting the signal from a `signal+noise' type matrix model with an additive Wigner noise. We propose a hypothesis test based on the linear spectral statistics of the data matrix, which does not depend on the distribution of the signal or the noise. The test is optimal unde…

Paper develops new method for detecting latent structure in large symmetric data matrices.

We present an original and novel method based on random matrix approach that enables to distinguish the respective role of temporal autocorrelations inside given time series and cross correlations between various time series. The proposed algorithm is based on properties of Wigner eigenspectrum of random matrices inste…

New matrix ensembles better match deep neural network spectral densities.

Study asymptotics of unitary matrix elements in quantum mechanics.

The classical isoperimetric inequality in the Euclidean plane $\mathbb{R}^2$ states that for a simple closed curve $M$ of the length $L_{M}$, enclosing a region of the area $A_{M}$, one gets \begin{align*} L_{M}^2\geqslant 4πA_{M}. \end{align*} In this paper we present the improved isoperimetric inequality, which state…

We prove that either the images of the mapping class groups by quantum representations are not isomorphic to higher rank lattices or else the kernels have a large number of normal generators. Further we show that the images of the mapping class groups have nontrivial 2-cohomology, at least for small levels. For this pu…

Study on complexity of random polynomials with deterministic spikes, identifying phase transitions.

New estimators reduce computation for Kendall's tau and conditional Kendall's tau matrices under structural assumptions.

Two spectral algorithms for community detection in graphs with covariates are compared.

The paper sets thresholds for testing correlation in hypergraphs, distinguishing between independent and correlated states.

Optimizes network sampling for efficient community detection.

We study the fundamental limits of detecting the presence of an additive rank-one perturbation, or spike, to a Wigner matrix. When the spike comes from a prior that is i.i.d. across coordinates, we prove that the log-likelihood ratio of the spiked model against the non-spiked one is asymptotically normal below a certai…

We analyze a new spectral graph matching algorithm, GRAph Matching by Pairwise eigen-Alignments (GRAMPA), for recovering the latent vertex correspondence between two unlabeled, edge-correlated weighted graphs. Extending the exact recovery guarantees established in the companion paper for Gaussian weights, in this work,…

Paper studies vertex correspondence recovery in correlated graphs with node features.

Algorithm detects and estimates correlated signals in spiked matrices.

Community detection is an important task in network analysis, in which we aim to learn a network partition that groups together vertices with similar community-level connectivity patterns. By finding such groups of vertices with similar structural roles, we extract a compact representation of the network's large-scale …

Paper tackles robust graph matching in dense graphs with AMP type algorithm.

Simplified Ricci curvature for spherical fluid dynamics models.

Clusterpath estimator simplifies graphical model interpretation for large datasets.

Learning the "blocking" structure is a central challenge for high dimensional data (e.g., gene expression data). Recently, a sparse singular value decomposition (SVD) has been used as a biclustering tool to achieve this goal. However, this model ignores the structural information between variables (e.g., gene interacti…

Proposes BMME for optimizing nonsmooth nonconvex problems with block structure.

Paper proposes new methods for improving interatomic potentials.

The manipulation of LIBOR by a group of banks became one of the major blows to the remaining confidence in financial industry. Yet, despite an enormous amount of popular literature on the subject, rigorous time-series studies are few. In my paper, I discuss the following hypothesis. Namely, if we should assume for a st…