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Weak Detection in the Spiked Wigner Model with General Rank [article]

Ji Hyung Jung, Hye Won Chung, Ji Oon Lee
2021 arXiv   pre-print
We study the statistical decision process of detecting the signal from a 'signal+noise' type matrix model with an additive Wigner noise.  ...  We also introduce an algorithm that estimates the rank of the signal when it is not known a priori.  ...  For other results on the rank-1 spiked Wigner model, we refer to [17, 27, 14] and references therein. Main problem: We consider the detection problem in the general spiked Wigner model in (1.1).  ... 
arXiv:2001.05676v3 fatcat:453fvsnfxjheldv35sa2rfar6e

Optimality and sub-optimality of PCA I: Spiked random matrix models

Amelia Perry, Alexander S. Wein, Afonso S. Bandeira, Ankur Moitra
2018 Annals of Statistics  
Such results form the basis of our understanding of when PCA can detect a low-rank signal in the presence of noise.  ...  A central problem of random matrix theory is to understand the eigenvalues of spiked random matrix models, introduced by Johnstone, in which a prominent eigenvector (or "spike") is planted into a random  ...  The authors are indebted to Philippe Rigollet for helpful discussions and for many comments on a draft. We thank the anonymous reviewers for many helpful, detailed comments.  ... 
doi:10.1214/17-aos1625 fatcat:7yrxzyhfszeudhjh4jinvnl6wq

Weak detection in the spiked Wigner model [article]

Hye Won Chung, Ji Oon Lee
2019 arXiv   pre-print
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 establish a central limit theorem for the linear spectral statistics of general rank-one spiked Wigner matrices as an intermediate step.  ...  The idea of the test can be further extended to more general models such as rank-k spiked Wigner matrices with k > 1 or spiked sample covariance matrices.  ... 
arXiv:1809.10827v3 fatcat:qkzujrdx2zhdhpt3ipiip4wbvq

Information-Theoretic Bounds and Phase Transitions in Clustering, Sparse PCA, and Submatrix Localization

Jess Banks, Cristopher Moore, Roman Vershynin, Nicolas Verzelen, Jiaming Xu
2018 IEEE Transactions on Information Theory  
We study the problem of detecting a structured, low-rank signal matrix corrupted with additive Gaussian noise.  ...  This regime is analogous to the conjectured 'hard but detectable' regime for community detection in sparse graphs. J. Banks is with the  ...  SPARSE PCA Consider the following spiked Wigner model, where the underlying signal is a rank-one matrix: X = λ √ n vv † + W , (2) Here, v ∈ R n , λ > 0 and W ∈ R n×n is a Wigner random matrix with W ii  ... 
doi:10.1109/tit.2018.2810020 fatcat:djasi4gkhrb2rk6ochxgec3gwa

Information-theoretic bounds and phase transitions in clustering, sparse PCA, and submatrix localization

Jess Banks, Cristopher Moore, Roman Vershynin, Nicolas Verzelen, Jiaming Xu
2017 2017 IEEE International Symposium on Information Theory (ISIT)  
We study the problem of detecting a structured, low-rank signal matrix corrupted with additive Gaussian noise.  ...  This regime is analogous to the conjectured 'hard but detectable' regime for community detection in sparse graphs. J. Banks is with the  ...  SPARSE PCA Consider the following spiked Wigner model, where the underlying signal is a rank-one matrix: X = λ √ n vv † + W , (2) Here, v ∈ R n , λ > 0 and W ∈ R n×n is a Wigner random matrix with W ii  ... 
doi:10.1109/isit.2017.8006706 dblp:conf/isit/BanksMVVX17 fatcat:6n7hc2hjbrehviy52ddwsvx2ou

Statistical limits of spiked tensor models [article]

Amelia Perry and Alexander S. Wein and Afonso S. Bandeira
2017 arXiv   pre-print
We study the statistical limits of both detecting and estimating a rank-one deformation of a symmetric random Gaussian tensor.  ...  In particular, the large d asymptotics of the threshold location differs between problems with discrete priors versus continuous priors.  ...  We thank Yash Deshpande and Thibault Lesieur for resolving an issue in the replica predictions section of the first version of this paper.  ... 
arXiv:1612.07728v2 fatcat:i5ana3ou3nbpjlf2cgqoxyk5le

Subexponential-Time Algorithms for Sparse PCA [article]

Yunzi Ding, Dmitriy Kunisky, Alexander S. Wein, Afonso S. Bandeira
2020 arXiv   pre-print
We study the computational cost of recovering a unit-norm sparse principal component x ∈R^n planted in a random matrix, in either the Wigner or Wishart spiked model (observing either W + λ xx^ with W drawn  ...  We investigate the precise amount of time required for recovery in the "possible but hard" regime 1/√(n)≪ρ≪ 1 by exploring the power of subexponential-time algorithms, i.e., algorithms running in time  ...  Consider the Wishart model (we will see that the Wigner model shares essentially the same behavior) in the regimeλ = Θ(1) withλ < 1 (so that ordinary PCA fails at weak recovery).  ... 
arXiv:1907.11635v2 fatcat:ftd7pmyxe5ehdkdo2vhtuxneqm

Nonasymptotic Guarantees for Spiked Matrix Recovery with Generative Priors [article]

Jorio Cocola, Paul Hand, Vladislav Voroninski
2020 arXiv   pre-print
We present this analysis in the case of both the Wishart and Wigner spiked matrix models.  ...  In this work, we study an alternative prior where the low-rank component is in the range of a trained generative network.  ...  to control the noise term Λ x HΛ x where: • in the Spiked Wishart Model H = Σ N − Σ; • in the Spiked Wigner Model H = H.  ... 
arXiv:2006.07953v2 fatcat:bj2chmc5uvhk5kgn3wdfymbaiy

Optimality and Sub-optimality of PCA for Spiked Random Matrices and Synchronization [article]

Amelia Perry and Alexander S. Wein and Afonso S. Bandeira and Ankur Moitra
2016 arXiv   pre-print
Such results form the basis of our understanding of when PCA can detect a low-rank signal in the presence of noise.  ...  Our results include: I) For the Gaussian Wigner ensemble, we show that PCA achieves the optimal detection threshold for a variety of benign priors for the spike.  ...  Acknowledgements The authors are indebted to Philippe Rigollet for helpful discussions and for many comments on a draft, and to Amit Singer and his group for discussions about synchronization.  ... 
arXiv:1609.05573v2 fatcat:buxtnl7y2zbkfipyjgl5wgj5rm

Community Detection and Stochastic Block Models

Emmanuel Abbe
2018 Foundations and Trends in Communications and Information Theory  
For example, a spiked Wigner model with observation Y = XX T +Z, where X is an unknown vector and Z is Wigner, can be viewed as a labeled graph where edge (i, j)'s label is given by Y ij = X i X j + Z  ...  The model takes also different interpretations, such as a planted spin-glass model [63] , a sparse-graph code [13, 68] or a low-rank (spiked) random matrix model [123, 154, 162] among others.  ... 
doi:10.1561/0100000067 fatcat:3udcotdugbhwlgrdmqmelqwsty

Fundamental limits for rank-one matrix estimation with groupwise heteroskedasticity [article]

Joshua K. Behne, Galen Reeves
2022 arXiv   pre-print
These results are based on a novel reduction from the low-rank matrix tensor product model (with homogeneous noise) to a rank-one model with heteroskedastic noise.  ...  In particular, we study the problem of estimating a rank-one matrix from Gaussian observations where different blocks of the matrix are observed under different noise levels.  ...  Acknowledgement This work was supported in part by NSF grant CCF-1750362.  ... 
arXiv:2106.11950v2 fatcat:vakodsqyanbm5ftdkfjvvf4cji

Detection of Signal in the Spiked Rectangular Models [article]

Ji Hyung Jung, Hye Won Chung, Ji Oon Lee
2021 arXiv   pre-print
We consider the problem of detecting signals in the rank-one signal-plus-noise data matrix models that generalize the spiked Wishart matrices.  ...  We also propose a hypothesis test to detect the presence of signal with low computational complexity, based on the linear spectral statistics, which minimizes the sum of the Type-I and Type-II errors when  ...  In these models, the signal is in the form of rank-1 mean matrix (spiked Wigner matrix) or rank-1 perturbation of the identity in the covariance matrix (spiked Wishart matrix).  ... 
arXiv:2104.13517v1 fatcat:cedaichtf5fy3c6wq62eecjaeq

Estimating Rank-One Spikes from Heavy-Tailed Noise via Self-Avoiding Walks [article]

Jingqiu Ding, Samuel B.Hopkins, David Steurer
2020 arXiv   pre-print
We study symmetric spiked matrix models with respect to a general class of noise distributions.  ...  Given a rank-1 deformation of a random noise matrix, whose entries are independently distributed with zero mean and unit variance, the goal is to estimate the rank-1 part.  ...  Our main problem is: Problem 1.1 (Generalized spiked Wigner model, recovery).  ... 
arXiv:2008.13735v1 fatcat:scaigqkfg5f3nibw53xqlt2w3a

Non universality of fluctuations of outlier eigenvectors for block diagonal deformations of Wigner matrices [article]

Mireille Capitaine, Catherine Donati-Martin
2018 arXiv   pre-print
In this paper, we investigate the fluctuations of a unit eigenvector associated to an outlier in the spectrum of a spiked N× N complex Deformed Wigner matrices M_N.  ...  Hermitian deterministic matrix A_N=diag(θ, A_N-1), θ∈R has multiplicity one and generates an outlier in the spectrum of M_N.  ...  for spiked covariance matrices in [5] and for low rank deformations of G.U.E. in [23] . 3 In this paper, we consider additive perturbations of Wigner matrices.  ... 
arXiv:1807.07773v1 fatcat:stxsh7qkaffmxo7uw4bgxamome

Spin relaxation of radicals in cryptochrome and its role in avian magnetoreception

Susannah Worster, Daniel R. Kattnig, P. J. Hore
2016 Journal of Chemical Physics  
The maximum rank of Wigner function used to ensure convergence within each potential is given in Table I .  ...  The size of the basis set used limits the precision of the solution, with increasing numbers of higher rank Wigner functions being required to describe the diffusion for larger values of A.  ... 
doi:10.1063/1.4958624 pmid:27448908 fatcat:za5caev6sbcj7lyzsmvdl6ub54
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