Spectral alignment of correlated Gaussian random matrices

In this paper we analyze a simple method ($EIG1$) for the problem of matrix alignment, consisting in aligning their leading eigenvectors: given $A$ and $B$, we compute $v_1$ and $v'_1$ two leading eigenvectors of $A$ and $B$. The algorithm returns a permutation $\hat{\Pi}$ such that the rank of the coordinate $\hat{\Pi}(i)$ in $v_1$ is the rank of the coordinate $i$ in $v'_1$ (up to the sign of $v'_1$). We consider a model where $A$ belongs to the Gaussian Orthogonal Ensemble (GOE), and $B= \Pi^T (A+\sigma H) \Pi$, where $\Pi$ is a permutation matrix and $H$ is an independent copy of $A$. We show the following 0-1 law: under the condition $\sigma N^{7/6+\epsilon} \to 0$, the $EIG1$ method recovers all but a vanishing part of the underlying permutation $\Pi$. When $\sigma N^{7/6-\epsilon} \to \infty$, this algorithm cannot recover more than $o(N)$ correct matches. This result gives an understanding of the simplest and fastest spectral method for matrix alignment (or complete weighted graph alignment), and involves proof methods and techniques which could be of independent interest.Comment: 29 pages, 4 figure

Spectral alignment of correlated Gaussian random matrices

29 pages, 4 figuresIn this paper we analyze a simple method ($EIG1$) for the problem of matrix alignment, consisting in aligning their leading eigenvectors: given $A$ and $B$, we compute $v_1$ and $v'_1$ two leading eigenvectors of $A$ and $B$. The algorithm returns a permutation $\hat{\Pi}$ such that the rank of the coordinate $\hat{\Pi}(i)$ in $v_1$ is the rank of the coordinate $i$ in $v'_1$ (up to the sign of $v'_1$). We consider a model where $A$ belongs to the Gaussian Orthogonal Ensemble (GOE), and $B= \Pi^T (A+\sigma H) \Pi$, where $\Pi$ is a permutation matrix and $H$ is an independent copy of $A$. We show the following 0-1 law: under the condition $\sigma N^{7/6+\epsilon} \to 0$, the $EIG1$ method recovers all but a vanishing part of the underlying permutation $\Pi$. When $\sigma N^{7/6-\epsilon} \to \infty$, this algorithm cannot recover more than $o(N)$ correct matches. This result gives an understanding of the simplest and fastest spectral method for matrix alignment (or complete weighted graph alignment), and involves proof methods and techniques which could be of independent interest

Kernel spectral clustering of large dimensional data

This article proposes a first analysis of kernel spectral clustering methods in the regime where the dimension $p$ of the data vectors to be clustered and their number $n$ grow large at the same rate. We demonstrate, under a $k$-class Gaussian mixture model, that the normalized Laplacian matrix associated with the kernel matrix asymptotically behaves similar to a so-called spiked random matrix. Some of the isolated eigenvalue-eigenvector pairs in this model are shown to carry the clustering information upon a separability condition classical in spiked matrix models. We evaluate precisely the position of these eigenvalues and the content of the eigenvectors, which unveil important (sometimes quite disruptive) aspects of kernel spectral clustering both from a theoretical and practical standpoints. Our results are then compared to the actual clustering performance of images from the MNIST database, thereby revealing an important match between theory and practice

Free Probability based Capacity Calculation of Multiantenna Gaussian Fading Channels with Cochannel Interference

During the last decade, it has been well understood that communication over multiple antennas can increase linearly the multiplexing capacity gain and provide large spectral efficiency improvements. However, the majority of studies in this area were carried out ignoring cochannel interference. Only a small number of investigations have considered cochannel interference, but even therein simple channel models were employed, assuming identically distributed fading coefficients. In this paper, a generic model for a multi-antenna channel is presented incorporating four impairments, namely additive white Gaussian noise, flat fading, path loss and cochannel interference. Both point-to-point and multiple-access MIMO channels are considered, including the case of cooperating Base Station clusters. The asymptotic capacity limit of this channel is calculated based on an asymptotic free probability approach which exploits the additive and multiplicative free convolution in the R- and S-transform domain respectively, as well as properties of the eta and Stieltjes transform. Numerical results are utilized to verify the accuracy of the derived closed-form expressions and evaluate the effect of the cochannel interference.Comment: 16 pages, 4 figures, 1 tabl