CS35: Edge and spectrum of heterogeneous ensembles

date: 7/14/2025, time: 16:00-17:30, room: IM WS

Organizer: Arijit Chakrabarty (Indian Statistical Institute)

Chair: Dipankar Bandyopadhyay (Virginia Commonwealth University)

Free denoising

Kamil Szpojankowski (Warsaw University of Technology)

We study the free analogue of the classical statistical denoising problem. Let \(a\) and \(b\) be free selfadjoint noncommutative random variables in a tracial \(W^*\)-probability space, where \(a\) represents the signal and \(b\) the noise. Our goal is to compute the conditional expectation of \(a\) (or more generally \(f(a)\)) given \(a+b\). To do this, we introduce a coupling between the distributions of \(a\) and \(a+b\) and show that it is absolutely continuous with respect to the product measure of the marginal distributions. A similar approach applies to the multiplicative case. More generally, we consider the conditional expectation of \(a\) given \(P(a,b)\), for any noncommutative polynomial \(P\), and show that it can be addressed using the c-freeness framework of Bożejko, Leinert, and Speicher. We conclude with applications of our results to matrix denoising.

Bibliography

\([1]\) M. Fevrier, A. Nica, K. Szpojankowski. (2024). Free denoising via overlap measures and c-freeness techniques. arXiv.org:2412.20792.

FLUCTUATIONS OF TOP EIGENVALUES OF CORRELATED GAUSSIAN MATRICES

Bodhisatta Das (Indian Statistical Institute, Kolkata)

We shall describe the fluctuation of the eigenvalues of a special type of symmetric random matrix with entries from a Gaussian process. It is shown in \([2]\), that the empirical spectral distribution of a symmetric random matrix with independent on and off-diagonal entries satisfying some moment conditions, converges to Wigner’s semicircle law. Let \(\{X_{i,j}:i,j\in\mathop{\mathrm{\mathbb{Z}}}\}\) be a real stationary Gaussian process with zero mean and positive variance. The LSD of the matrix \(A_N(i,j)=X_{i,j}+X_{j,i}\) were studied by \([1]\). We shall describe the fluctuation of the top eigenvalues of this matrix. This is a joint work with Arijit Chakrabarty.

Bibliography

\([1]\) A. Chakrabarty, R. S. Hazra, and D. Sarkar. From random matrices to long range dependence. Random Matrices: Theory and Applications, 05(02):1650008, 2016. doi:\ 10.1142/S2010326316500088. URL https://doi.org/10.1142/S2010326316500088

\([2]\) E. P. Wigner. On the distribution of the roots of certain symmetric matrices. Annals of Mathematics, 67(2):325–327, 1958. ISSN 0003486X, 19398980.
URL http://www.jstor.org/stable/1970008

Geostatistical modeling of positive definite matrices, with an application to diffusion tensor imaging

Dipankar Bandyopadhyay (Virginia Commonwealth University)

Geostatistical modeling for continuous point-referenced data has been extensively applied to neuroimaging, because it produces efficient and valid statistical inference. However, diffusion tensor imaging (DTI), a neuroimaging characterizing the brain’s anatomical structure, produces voxel-level positive definite (p.d) matrices. Current geostatistical modeling has not been extended to p.d matrices as introducing spatial dependence among positive definite matrices properly is non-trivial. In this paper, we use the spatial Wishart process, a spatial stochastic process (random field), where each p.d matrix-variate response marginally follows a Wishart distribution, and the spatial dependence is induced by latent Gaussian processes. This process is valid on an uncountable collection of spatial locations and is almost surely continuous, leading to a flexible way of modeling spatial dependence. Motivated by a DTI dataset of cocaine users, we propose a spatial matrix-variate regression model based on the spatial Wishart Process (sWP). However, a shortcoming of the sWP is the lack of a closed-form density function. Hence, we propose some approximations to obtain a feasible Cholesky decomposition model, and show that the Cholesky decomposition model is asymptotically equivalent to the sWP model. A local likelihood approximation technique is employed to achieve fast computation. Simulation studies and application to the motivating DTI data demonstrate that the Cholesky decomposition process model produces reliable inference, and improved performance, compared to existing alternatives.