| Speaker | Title | Abstract |
|---|---|---|
| Cucker, Felipe | Eigenvalue computations and random matrices | A long standing open problem in numerical linear algebra is the computation of eigenpairs (eigenvalue–eigenvector). We either know efficient algorithm that fail to be numerically stable or algorithms exhibiting numerical stability but whose complexity analysis (and even convergence!) eludes us. In the talk, we give some advances emphasizing the role played by random matrices on these advances. |
| Bordenave, Charles | Non-backtracking spectrum of random graphs | The non-backtracking matrix of a graph is a non-symmetric matrix on the oriented edge of a graph which has interesting algebraic properties and appears notably in connection with the Ihara zeta function and in some generalizations of Ramanujan graphs. It has also be proposed recently in the context of community detection. In this talk, we will study the largest eigenvalues of this matrix for the Erdos-Renyi graph G(n,c/n) and other simple inhomogeneous random graphs (stochastic block model). This is a joint work with Marc Lelarge and Laurent Massoulié. |
| Merlevede, Florence | On the empirical spectral distribution for matrices with long memory and independent rows | The talk will focus on the empirical eigenvalue distribution of sample covariance matrices. We will show in particular that if the sample covariance matrix is generated by independent copies of a stationary regular sequence then its empirical eigenvalue distribution always has a limiting distribution depending only on the spectral density of the sequence. We characterize this limit in terms of Stieltjes transform via a certain simple equation. No rate of convergence to zero of the covariances is imposed. If the entries of the stationary sequence are functions of independent random variables the result holds without any other additional assumptions. The talk is based on a joint work with M. Peligrad. |
| Collins, Benoit | Random positive maps | In operator algebras (and quantum information theory), positive and completely positive maps play an important role. While completely positive are well understood and characterized, the situation for positive maps is very different. The purpose of this talk is to show that random matrix techniques supply a large class of new positive maps. We will explain how our random constructions can act as powerful entanglement witnesses. |
| Couillet, Romain | Random Matrices and Robust Estimation | This talk will browse through the recent advances of random matrix theory applied to robust statistics, and more precisely to robust covariance matrix estimation. It shall be shown in particular that some widely used classes of robust covariance matrix estimates (such as Maronna's and Tyler's), which take the form of the matrix solution of an implicit equation, behave in the random matrix regime as classical random matrix models more amenable to analysis. Such results allow for the design of novel improved signal processing algorithms that account both for the commensurability of population and sample dimensions and for the impulsiveness in the observations (either due to non-Gaussian observations or to the existence of outliers among the samples). Applications to array processing (and the introduction of the Robust G-MUSIC algorithm) as well as to portfolio optimization based on financial data will be discussed. |
| Salez, Justin | Atoms in the limiting spectrum of diluted random graphs | A decade ago, Khorunzhy, Shcherbina and Vengerovsky established convergence of the empirical spectral distribution of large Erdos-Renyi random graphs with fixed average degree. Yet, only very little is known about the limiting measure. In particular, Ben Arous asked for the precise location of its atoms (Open Problem 14 of the 2010 AIM Workshop on Random Matrices). In this talk, I will present a complete answer to this question. |
| Soshnikov, Alexander | Products of independent elliptic random matrices | For fixed $m>1$, we study the product of m independent $N\times N$ elliptic random matrices as N tends to infinity. Our main result shows that the empirical spectral distribution of the product converges, with probability 1, to the m-th power of the circular law, regardless of the joint distribution of the mirror entries in each matrix. This leads to a new kind of universality phenomenon: the limit law for the product of independent random matrices is independent of the limit laws for the individual matrices themselves. This is a joint work with Sean O'Rourke, David Renfrew, and Van Vu. |
| Adamczak, Radoslaw | Norms of submatrices and entropic uncertainty relations for high dimensional random unitaries | I will show that with high probability random unitary matrices satisfy an almost optimal version of entropic uncertainty principle. As a tool I will present certain uniform estimates for norms of submatrices of a random unitary matrix complemented by corresponding lower bounds. I will conclude with a discussion of possible extensions and open questions. Based on joint work with Rafal Latala, Zbigniew Puchala and Karol Zyczkowski. |
| Zheng, Xinghua | Integrated covariance matrix estimation for high-dimensional diffusion processes in the presence of microstructure noise | We consider estimation of the integrated covariance (ICV) matrices of high-dimensional diffusion processes based on high-frequency data in the presence of microstructure noise. We adopt the pre-averaging approach to deal with microstructure noise, and establish the connection between the underlying ICV matrix and the pre-averaging estimator in terms of their limiting spectral distributions (LSDs). A key element of the argument is a result describing how the LSD of (true) sample covariance matrices depends on that of sample covariance matrices constructed from \emph{noisy} observations. This result enables one to make inferences about the covariance structure of underlying signals based on noisy observations. We further propose an alternative estimator, the pre-averaging time-variation adjusted realized covariance matrix, which possesses two desirable properties: it eliminates the impact of noise, and its LSD depends only on that of the targeting ICV through the standard Marchenko-Pastur equation when the covolatility process satisfies certain structural conditions. Based on joint work with Ningning Xia. |
| Hardy, Adrien | Large complex correlated Wishart matrices: Fluctuations and asymptotic independence at the edges | We study the asymptotic behavior of eigenvalues of large complex correlated Wishart matrices at the edges of the limiting spectrum. In this setting, the support of the limiting eigenvalue distribution may have several connected components. Under mild conditions for the population matrices, we show that for every generic positive edge of that support, there exists an extremal eigenvalue which converges almost surely towards that edge and fluctuates according to the Tracy-Widom law. Moreover, given several generic positive edges, we establish that the associated extremal eigenvalue fluctuations are asymptotically independent. Finally, when the leftmost edge is the origin, we prove that the smallest eigenvalue fluctuates according to the hard-edge Tracy-Widom law. This is a joint work with W. Hachem and J. Najim. |
| Caputo, Pietro | Spectral properties of random Markov matrices | We discuss various models of Markov matrices obtained by normalizing the entries of an i.i.d. random matrix. The spectrum of such matrices conveys information on the convergence to equilibrium of the associated Markov chain. We show the existence of a limiting spectral distribution and investigate its properties for both reversible and non reversible Markov chains. In the sparse regime obtained by taking the original variables to be heavy-tailed we describe the limiting distribution in terms of the spectral measures of suitable randomly weighted trees. Based on joint works with Djalil Chafai, Charles Bordenave, and Daniele Piras. |
| Kozhan, Rostyslav | Eigenvalues of rank one perturbations of $\beta$-ensembles | We propose sparse matrix models for rank one sub-unitary perturbations of $\beta$-ensembles of unitary random matrices and rank one non-Hermitian perturbations of $\beta$-ensembles of Hermitian random matrices. The eigenvalues of these perturbations have interpretation as scattering resonances of open quantum systems in the physics literature. Using these matrix models and theory of orthogonal polynomials we are able to explicitly compute the joint eigenvalue density for any $\beta<\infty$. If time permits, we will discuss the microscopic limit of density of states and show it is universal for a class of sub-unitary CMV random matrices. This is a joint work with Rowan Killip. |
| Nadakuditi, Raj Rao | New applications of random theory | We describe some recent success stories where random matrix theory has enabled new applications: these include new theory and algorithms for transmitting light perfectly through highly scattering or opaque media, for separating foreground and background of videos and in medical MRI datasets. We conclude by highlighting some newly discovered random matrix universality phenomena emerging from scattering theory and semidefinite optimization that suggest intriguing inter-connections and directions for theoretical research. |
| Shcherbina, Mariya | CLT for eigenvalue counting function of orthogonal and symplectic matrix models | While Central Limit Theorems (CLT) for linear eigenvalue statistics of smooth test functions are proven now for many ensembles of random matrices, similar results for the test functions with jumps (e.g. indicators of some intervals) as a rule require much more advanced techniques, and corresponding CLT are known only for a few classical models with gaussian distribution of entries. We present a proof of CLT for the jump test functions for orthogonal and symplectic matrix models with analytic potentials. |
| Bose, Arup | Large sample behaviour of high dimensional autocovariance matrices | We study the large sample behaviour of the sequence of high dimensional sample autocovariance matrices $\{\hat{\Gamma}_{i}\}_{i \geq 0}$ from an infinite dimensional vector linear process. Under suitable conditions on the coefficient matrices and the driving process, we prove that the expected average trace of any polynomial in these matrices converge. In particular, the limiting spectral distribution of any symmetric polynomials of these matrices, including the matrices $\hat{\Gamma}_{i} + \hat{\Gamma}_{i}^{*}$ and $\hat{\Gamma}_{i}\hat{\Gamma}_{i}^{*}$ exist. Our approach is through the algebraic method of free probability in conjunction with the method of moments. Thus, we are able to provide a general description for the limits in terms of some freely independent variables. |
| Yin, Jun | Eigenvector distribution of anisotropic random matrices | As an application of the anisotropic local law (introduced in A. Knowles’s talk), we study the eigenvector distribution of anisotropic random matrices. Examples include sample covariance matrices whose underlying population has nontrivial correlations, and Wigner matrices whose entries have arbitrary mean. It turns out that eigenvectors are asymptotically independent and uniformly distributed on an ellipsoid. A key ingredient of the proof is a new type of self-consistent comparison method. |
| Knowles, Antti | Anisotropic local laws for random matrices | Most of the literature on random matrices focuses on matrix models that are isotropic in the sense that their Green functions are with high probability close to a multiple of the identity matrix. Such models include Wigner matrices, Erdos-Renyi graphs, and random band matrices. I will talk about matrix models that are not isotropic. An important family of examples is provided by sample covariance matrices whose underlying population has nontrivial correlations. I present a new method that yields local laws for such anisotropic models. An application is the proof of the Tracy-Widom-Airy statistics near the soft edges. This result applies in the single-cut and the multi-cut cases. Further applications to the distribution of eigenvectors will be given in the talk of J. Yin. |
| Male, Camille | The spectrum of random graphs in free probability theory | Free probability theory has been introduced by Voiculescu in the 80's for the study of the von Neumann algebras of the free groups. In that context, the notion of freeness plays the role of independence in classical probability. In the 90's, Voiculescu realized that a family of independent random matrices whose eigenvectors are sufficiently uniformly distributed are asymptotically free. This phenomenon is called asymptotic freeness and had a deep impact in operator algebra and probability. Nevertheless, the approach does not work when the eigenvectors are not enough uniformly distributed, for instance for adjacency matrices of random graphs. Yet, the framework of free probability can be enriched to describe these models, thanks to an extended notion of freeness. The aim of this talk is to present the application of this recent method for two important models of random graphs, namely the Erdős-Rényi graphs and the uniform regular graphs. |
| Chen, Yang | Hankel determinant and Painlevé III | We study a Hankel determinant generated by a singularly deformed Laguerre weight, through the multiplication of the standard Laguerre weight by exp(-t/x), x\in R^+, t>0. It transpires that the finite n Hankel determinant which maybe expressed in an integral representation of a particular Painlevé III. In a double scaling scheme, where n tends to infinity and t tends to 0, in a combination, the infinite Hankel determinant has again an integral representation in terms of a C potential and its derivatives. Here the C potential satisfies a second order non-linear ode, which turned out to a Painleve III with lesser parameters. From which we obtain asymptotic expansions of the scaled and in some sense infinite Hankel determinant. |
| Pan, Guangming | The Tracy-Widom law for the largest eigenvalue of F matrix | Let X and Y be p \times n and p \times m independent and real random matrices consisting of i.i.d random variables with mean zero, variance one and subGaussian tail . The distribution of the largest eigenvalue of (XX'+YY')^{-1}XX' is proved to follow the Tracy-Widom law, which generalizes Johnstone's earlier result. |
| Lytova, Anna | On the CLT for linear eigenvalue statistics of the sum of independent matrices of rank one | We consider random matrices of the form $M_{n}=\sum_{\alpha =1}^{m}{\tau _{\alpha }}\mathbf{y}_{\alpha } \mathbf{y}_{\alpha }^T$, where $\mathbf{y}_{\alpha }\in\mathbb{R}^n$ are i.i.d. normalized isotropic random vectors with dependent components and $\tau _{\alpha }$ are real numbers. We find a class of random vectors satisfying some moment conditions such that for any smooth enough test-function $\varphi$ the linear eigenvalue statistics $Tr \varphi(M_n)$ converge in distribution to the Gaussian random variable. We also consider a tensor analog of such matrices, $\mathbf{M}_{n}=\sum_{\alpha =1}^{m}{\tau _{\alpha }}Y_{\alpha }Y_{\alpha }^T$, where now for every $\alpha$ vector $Y_\alpha=\mathbf{y}_{\alpha 1}\otimes...\otimes \mathbf{y}_{\alpha k}$ is a tensor product of $k$ i.i.d. normalized isotropic random vectors, $k$ is fixed, and $m/n^k\rightarrow c\in [0,\infty)$ while $n\rightarrow\infty$. We show, in particular, that the limit of the empirical spectral distribution of $\mathbf{M}_{n}$ coincides with that one for $M_{n}$. |
| Onatski, Alexei | Testing hypotheses about sub- and super-critical spikes in multivariate statistical models | This is a second half of a joint talk with Iain Johnstone. We consider spiked models representing each of the five classes of multivariate statistical problems identified by James (1964). For each of the models, we describe the phase transition of the largest eigenvalue, and derive the asymptotic behavior of the likelihood ratios that correspond to null and alternative hypotheses about sub- and super-critical spikes. We find that the statistical experiment of observing the eigenvalues in the super-critical regime, parameterized by local deviations of the spike from its value under the null, converges to simple Gaussian shift experiment, and therefore, the best test about a single super-critical spike is based on the largest eigenvalue only. Our findings for the sub-critical regime are totally different. In that regime, the experiment of observing the eigenvalues converges to a Gaussian sequence experiment, and no optimal test about a sub- critical spike is available. We derive the asymptotic power envelopes for such tests. |
| Maida, Mylene | Yang Mills, unitary Brownian bridge and discrete orthogonal polynomials | About twenty years ago, the physicists Douglas, Kazakov, Gross and Matytsin, working on Yang-Mills theory, made some predictions on the behavior of the partition function of the models they considered. These results played an important role in the developments of this kind of jauge theories. In this talk, I will explain how some modern tools, and in particular recent results on discrete orthogonal polynomials obtained by K. Liechty and D. Wang, allow us to give a rigourous proof and a better undertsanding of the prediction of physicists. This is joint work with T. Levy. |
| ZHOU, Wenxin | Cramér moderate deviations for studentized two-sample U-statistics with applications | Two-sample U-statistics are widely used in a broad range of applications, including those in the fields of biostatistics and econometrics. In this paper, we establish sharp Cramér type moderate deviation theorems for studentized two-sample U-statistics in a general framework, including the two-sample t-statistic and Studentized Mann- Whitney test statistic as prototypical examples. In particular, a refined moderate deviation theorem with second-order accuracy is established for the two-sample t-statistic. These results provide theoretical guarantees and therefore extend the applicability of the existing statistical methodologies from the one-sample $t$-statistic to more general nonlinear statistics. Applications to two-sample large-scale multiple testing problems with false discovery rate control and the regularized bootstrap method are also discussed. This is a joint work with Qi-Man Shao and Jinyuan Chang. |
| Bai, Zhidong | Order determination of high-dimensional dynamic factor models | Consider the following dynamic factor model: $$ {\mathbf{y}}_t=\sum_{l=0}^q{\mathbf{\Lambda}}_{l}{\mathbf{f}}_{t-l}+{\boldsymbol{\varepsilon}}_t, \ t=1,2,\cdots,T $$ where ${\mathbf{\Lambda}} _l$ is an $N\times k$ loading matrix (of full rank), $\{{\mathbf{f}}_t\}$ are i.i.d. $k$-factors, and ${\boldsymbol{\varepsilon}}_t$ are independent white noises. This model is very important in financial statistics. Now, we assume that $N/T\to c>0$ and we want to estimate the orders $k$ and $q$ separately. Define a random matrix $$ {\mathbf{\Phi}}(\tau)=\frac1{2T}\sum_{t=1}^T({\mathbf{y}}_t{\mathbf{y}}_{t+\tau}^*+{\mathbf{y}}_{t+\tau}{\mathbf{y}}_t^*) $$ where $\tau\ge 0$ is an integer. When there are no factors, the matrix ${\mathbf{\Phi}}(\tau)$ reduces to $$ {\mathbf{M}}_\tau=\frac1{2T}\sum_{t=1}^T({\boldsymbol{\varepsilon}}_t{\boldsymbol{\varepsilon}}_{t+\tau}^*+{\boldsymbol{\varepsilon}}_{t+\tau}{\boldsymbol{\varepsilon}}_t^*). $$ When $\tau=0$, ${\mathbf{M}}_\tau$ reduces to the usual sample covariance matrix whose ESD tends to the well-known M-P law and thus ${\mathbf{\Phi}}(0)$ reduces to the standard spike model and hence the number $k(q+1)$ can be estimated by the number of spiked eigenvalues of ${\mathbf{\Phi}}(0)$. To obtain separate estimates of $k$ and $q$, we have to employ the spectral analysis of ${\mathbf{M}}_\tau$. In our work, we derive the LSD of ${\mathbf{M}}_\tau$ and then establish the strong limits of extreme eigenvalues of ${\mathbf{M}}_\tau$ for some $\tau>0$. Finally, we will establish the spiked model analysis for ${\mathbf{\Phi}}(\tau)$. |
| Pajor, Alain | Interactions between high dimensional geometry and random matrix theory | We shall discuss some results on quantitative estimates of the smallest and largest singular values of random matrices with independent columns. We shall also survey recent results on the approximation of the covariance matrix. |
| Guionnet, Alice | About topological expansions | We discuss large dimensions asymptotics of various models. This is based on joint work with E. Maurel Segala, G. Borot and K. Kozlowski. |
| Johnstone, Iain | James' Five Fold Way and spiked models in multivariate statistics | This is a first half of a joint talk with Alexei Onatski. It is 50 years since James' 1964 paper on the distribution of matrix variates and latent roots, in which he gave a remarkable classification of many of the eigenvalue distribution problems of multivariate statistics. We revisit the classification, now from the viewpoint of high dimensional models and low rank departures from the usual null hypotheses. |
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