Spectral analysis of the Gram matrix of mixture models

This text is devoted to the asymptotic study of some spectral properties of the Gram matrix $W^{\sf T} W$ built upon a collection $w_1, \ldots, w_n\in \mathbb{R}^p$ of random vectors (the columns of $W$), as both the number $n$ of observations and the dimension $p$ of the observations tend to infinity and are of similar order of magnitude. The random vectors $w_1, \ldots, w_n$ are independent observations, each of them belonging to one of $k$ classes $\mathcal{C}_1,\ldots, \mathcal{C}_k$. The observations of each class $\mathcal{C}_a$ ($1\le a\le k$) are characterized by their distribution $\mathcal{N}(0, p^{-1}C_a)$, where $C_1, \ldots, C_k$ are some non negative definite $p\times p$ matrices. The cardinality $n_a$ of class $\mathcal{C}_a$ and the dimension $p$ of the observations are such that $\frac{n_a}{n}$ ($1\le a\le k$) and $\frac{p}{n}$ stay bounded away from $0$ and $+\infty$. We provide deterministic equivalents to the empirical spectral distribution of $W^{\sf T}W$ and to the matrix entries of its resolvent (as well as of the resolvent of $WW^{\sf T}$). These deterministic equivalents are defined thanks to the solutions of a fixed-point system. Besides, we prove that $W^{\sf T} W$ has asymptotically no eigenvalues outside the bulk of its spectrum, defined thanks to these deterministic equivalents. These results are directly used in our companion paper "Kernel spectral clustering of large dimensional data", which is devoted to the analysis of the spectral clustering algorithm in large dimensions. They also find applications in various other fields such as wireless communications where functionals of the aforementioned resolvents allow one to assess the communication performance across multi-user multi-antenna channels.Comment: 25 pages, 1 figure. The results of this paper are directly used in our companion paper "Kernel spectral clustering of large dimensional data", which is devoted to the analysis of the spectral clustering algorithm in large dimensions. To appear in ESAIM Probab. Statis

Self-consistent tomography of the state-measurement Gram matrix

State and measurement tomography make assumptions about the experimental states or measurements. These assumptions are often not justified because state preparation and measurement errors are unavoidable in practice. Here we describe how the Gram matrix associated with the states and measurement operators can be estimated via semidefinite programming if the states and the measurements are so called globally completable. This is for instance the case if the unknown measurements are known to be projective and non-degenerate. The computed Gram matrix determines the states, and the measurement operators uniquely up to simultaneous rotations in the space of Hermitian matrices. We prove the reliability of the proposed method in the limit of a large number of independent measurement repetitions.Comment: We have completely rewritten the first version because new results from arXiv:1209.6499 allowed to significantly clearify the first submission. We have added some reference

Kernel matrix regression

We address the problem of filling missing entries in a kernel Gram matrix, given a related full Gram matrix. We attack this problem from the viewpoint of regression, assuming that the two kernel matrices can be considered as explanatory variables and response variables, respectively. We propose a variant of the regression model based on the underlying features in the reproducing kernel Hilbert space by modifying the idea of kernel canonical correlation analysis, and we estimate the missing entries by fitting this model to the existing samples. We obtain promising experimental results on gene network inference and protein 3D structure prediction from genomic datasets. We also discuss the relationship with the em-algorithm based on information geometry