arXiv subjects

Philippe Loubaton

Publications and source records attributed to Philippe Loubaton.

At least 19 recordsLinked to original sources

arXiv2025

Correlation tests and sample spectral coherence matrix in the high-dimensional regime

It is established that the linear spectral statistics (LSS) of the smoothed periodogram estimate of the spectral coherence matrix of a complex Gaussian high-dimensional times series (yn) n$\in$Z with independent components satisfy at each frequency a central limit theorem in the asymptotic regime where the sample size N , the dimension M of the observation, and the smoothing span B both converge towards +$\infty$ in such a way that M = O(N $\alpha$ ) for $\alpha$ < 1 and M B $\rightarrow$ c, c $\in$ (0, 1). It is deduced that two recentered and renormalized versions of the LSS, one based on an average in the frequency domain and the other one based on a sum of squares also in the frequency domain, and both evaluated over a well-chosen frequency grid, also verify a central limit theorem. These two statistics are proposed to test with controlled asymptotic level the hypothesis that the components of y are independent. Numerical simulations assess the performance of the two tests.

math.ST↗

arXiv2021

Large random matrix approach for testing independence of a large number of Gaussian time series

The asymptotic behaviour of Linear Spectral Statistics (LSS) of the smoothed periodogram estimator of the spectral coherency matrix of a complex Gaussian high-dimensional time series $(\y_n)_{n \in \mathbb{Z}}$ with independent components is studied under the asymptotic regime where the sample size $N$ converges towards $+\infty$ while the dimension $M$ of $\y$ and the smoothing span of the estimator grow to infinity at the same rate in such a way that $\frac{M}{N} \rightarrow 0$. It is established that, at each frequency, the estimated spectral coherency matrix is close from the sample covariance matrix of an independent identically $\mathcal{N}_{\mathbb{C}}(0,\I_M)$ distributed sequence, and that its empirical eigenvalue distribution converges towards the Marcenko-Pastur distribution. This allows to conclude that each LSS has a deterministic behaviour that can be evaluated explicitly. Using concentration inequalities, it is shown that the order of magnitude of the supremum over the frequencies of the deviation of each LSS from its deterministic approximation is of the order of $\frac{1}{M} + \frac{\sqrt{M}}{N}+ (\frac{M}{N})^{3}$ where $N$ is the sample size. Numerical simulations supports our results.

cs.IT↗

arXiv2021

Properties of linear spectral statistics of frequency-smoothed estimated spectral coherence matrix of high-dimensional Gaussian time series

math.ST↗

arXiv2021

On the largest singular values of certain large random matrices with application to the estimation of the minimal dimension of the state-space representations of high-dimensional time series

This paper is devoted to the estimation of the minimal dimension P of the state-space realizations of a high-dimensional time series y, defined as a noisy version (the noise is white and Gaussian) of a useful signal with low rank rational spectral density, in the high-dimensional asymptotic regime where the number of available samples N and the dimension of the time series M converge towards infinity at the same rate. In the classical low-dimensional regime, P is estimated as the number of significant singular values of the empirical autocovariance matrix between the past and the future of y, or as the number of significant estimated canonical correlation coefficients between the past and the future of y. Generalizing large random matrix methods developed in the past to analyze classical spiked models, the behaviour of the above singular values and canonical correlation coefficients is studied in the high-dimensional regime. It is proved that they are smaller than certain thresholds depending on the statistics of the noise, except a finite number of outliers that are due to the useful signal. The number of singular values of the sample autocovariance matrix above the threshold is evaluated, is shown to be almost independent from P in general, and cannot therefore be used to estimate P accurately. In contrast, the number s of canonical correlation coefficients larger than the corresponding threshold is shown to be less than or equal to P, and explicit conditions under which it is equal to P are provided. Under the corresponding assumptions, s is thus a consistent estimate of P in the high-dimensional regime. The core of the paper is the development of the necessary large random matrix tools.

cs.IT↗

arXiv2021

On the asymptotic distribution of the maximum sample spectral coherence of Gaussian time series in the high dimensional regime

We investigate the asymptotic distribution of the maximum of a frequency smoothed estimate of the spectral coherence of a M-variate complex Gaussian time series with mutually independent components when the dimension M and the number of samples N both converge to infinity. If B denotes the smoothing span of the underlying smoothed periodogram estimator, a type I extreme value limiting distribution is obtained under the rate assumptions M N $\rightarrow$ 0 and M B $\rightarrow$ c $\in$ (0, +$\infty$). This result is then exploited to build a statistic with controlled asymptotic level for testing independence between the M components of the observed time series. Numerical simulations support our results.

math.ST↗

arXiv2021

On the detection of low-rank signal in the presence of spatially uncorrelated noise: a frequency domain approach

This paper analyzes the detection of a M-dimensional useful signal modeled as the output of a M xK MIMO filter driven by a K-dimensional white Gaussian noise, and corrupted by a M-dimensional Gaussian noise with mutually uncorrelated components. The study is focused on frequency domain test statistics based on the eigenvalues of an estimate of the spectral coherence matrix (SCM), obtained as a renormalization of the frequency-smoothed periodogram of the observed signal. If N denotes the sample size and B the smoothing span, it is proved that in the high-dimensional regime where M, B, N converge to infinity while K remains fixed, the SCM behaves as a certain correlated Wishart matrix. Exploiting well-known results on the behaviour of the eigenvalues of such matrices, it is deduced that the standard tests based on linear spectral statistics of the SCM fail to detect the presence of the useful signal in the high-dimensional regime. A new test based on the SCM, which is proved to be consistent, is also proposed, and its statistical performance is evaluated through numerical simulations.

cs.IT↗

arXiv2021

On the asymptotic behaviour of the eigenvalue distribution of block correlation matrices of high-dimensional time series

We consider linear spectral statistics built from the block-normalized correlation matrix of a set of $M$ mutually independent scalar time series. This matrix is composed of $M \times M$ blocks that contain the sample cross correlation between pairs of time series. In particular, each block has size $L \times L$ and contains the sample cross-correlation measured at $L$ consecutive time lags between each pair of time series. Let $N$ denote the total number of consecutively observed windows that are used to estimate these correlation matrices. We analyze the asymptotic regime where $M,L,N \rightarrow +\infty$ while $ML/N \rightarrow c_\star$, $0<c_\star<\infty$. We study the behavior of linear statistics of the eigenvalues of this block correlation matrix under these asymptotic conditions and show that the empirical eigenvalue distribution converges to a Marcenko-Pastur distribution. Our results are potentially useful in order to address the problem of testing whether a large number of time series are uncorrelated or not.

math.PR↗

arXiv2019

On the behaviour of large empirical autocovariance matrices between the past and the future

The asymptotic behaviour of the distribution of the squared singular values of the sample autocovariance matrix between the past and the future of a high-dimensional complex Gaussian uncorrelated sequence is studied. Using Gaussian tools, it is established the distribution behaves as a deterministic probability measure whose support S is characterized. It is also established that the singular values to the square are almost surely located in a neighbourhood of S.

math.PR↗

arXiv2018

On the detection of low rank matrices in the high-dimensional regime

We address the detection of a low rank $n\times n$deterministic matrix $\mathbf{X}_{0}$ from the noisy observation ${\bf X}_{0}+{\bf Z}$ when $n\to\infty$, where ${\bf Z}$ is a complex Gaussian random matrix with independent identically distributed $\mathcal{N}_{c}(0,\frac{1}{n})$ entries. Thanks to large random matrix theory results, it is now well-known that if the largest singular value $λ_{1}$ of ${\bf X}_{0}$ verifies $λ_{1}>1$, then it is possible to exhibit consistent tests. In this contribution, we prove a contrario that under the condition $λ_{1}<1$, there are no consistent tests. Our proof is rather simple, inspired by previous works devoted to the case of rank 1 matrices ${\bf X}_{0}$.

eess.SP↗

arXiv2018

On the non-detectability of spiked large random tensors

This paper addresses the detection of a low rank high-dimensional tensor corrupted by an additive complex Gaussian noise. In the asymptotic regime where all the dimensions of the tensor converge towards $+\infty$ at the same rate, existing results devoted to rank 1 tensors are extended. It is proved that if a certain parameter depending on the low rank tensor is below a threshold, then the null hypothesis and the presence of the low rank tensor are undistinguishable hypotheses in the sense that no test performs better than a random choice.

eess.SP↗

arXiv2017

Spectral Convergence of Large Block-Hankel Gaussian Random Matrices

This paper studies the behaviour of the empirical eigenvalue distribution of large random matrices W_N W_N* where W_N is a ML x N matrix, whose M block lines of dimensions L x N are mutually independent Hankel matrices constructed from complex Gaussian correlated stationary random sequences. In the asymptotic regime where M \rightarrow \infty, N \rightarrow +\infty and ML/N \rightarrow c > 0, it is shown using the Stieltjes transform approach that the empirical eigenvalue distribution of W_N W_N* has a deterministic behaviour which is characterized.

math.PR↗

arXiv2015

Large system analysis of a GLRT for detection with large sensor arrays in temporally white noise

This paper addresses the behaviour of a classical multi-antenna GLRT test that allows to detect the presence of a known signal corrupted by a multi-path propagation channel and by an additive white Gaussian noise with unknown spatial covariance matrix. The paper is focused on the case where the number of sensors M is large, and of the same order of magnitude as the sample size N, a context which is modeled by the large system asymptotic regime M goes to infinity, N goes to infinity in such a way that M/N goes to c for c in (0, infinity). The purpose of this paper is to study the behaviour of a GLRT test statistics in this regime, and to show that the corresponding theoretical analysis allows to accurately predict the performance of the test when M and N are of the same order of magnitude.

cs.IT↗

arXiv2015

A CLT for an improved subspace estimator with observations of increasing dimensions

This paper deals with subspace estimation in the small sample size regime, where the number of samples is comparable in magnitude with the observation dimension. The traditional estimators, mostly based on the sample correlation matrix, are known to perform well as long as the number of available samples is much larger than the observation dimension. However, in the small sample size regime, the performance degrades. Recently, based on random matrix theory results, a new subspace estimator was introduced, which was shown to be consistent in the asymptotic regime where the number of samples and the observation dimension converge to infinity at the same rate. In practice, this estimator outperforms the traditional ones even for certain scenarios where the observation dimension is small and of the same order of magnitude as the number of samples. In this paper, we address a performance analysis of this recent estimator, by proving a central limit theorem in the above asymptotic regime. We propose an accurate approximation of the mean square error, which can be evaluated numerically.

stat.ME↗

arXiv2015

Performance analysis of an improved MUSIC DoA estimator

This paper adresses the statistical performance of subspace DoA estimation using a sensor array, in the asymptotic regime where the number of samples and sensors both converge to infinity at the same rate. Improved subspace DoA estimators were derived (termed as G-MUSIC) in previous works, and were shown to be consistent and asymptotically Gaussian distributed in the case where the number of sources and their DoA remain fixed. In this case, which models widely spaced DoA scenarios, it is proved in the present paper that the traditional MUSIC method also provides DoA consistent estimates having the same asymptotic variances as the G-MUSIC estimates. The case of DoA that are spaced of the order of a beamwidth, which models closely spaced sources, is also considered. It is shown that G-MUSIC estimates are still able to consistently separate the sources, while it is no longer the case for the MUSIC ones. The asymptotic variances of G-MUSIC estimates are also evaluated.

stat.AP↗

arXiv2015

On the almost sure location of the singular values of certain Gaussian block-Hankel large random matrices

This paper studies the almost sure location of the eigenvalues of matrices ${\bf W}_N {\bf W}_N^{*}$ where ${\bf W}_N = ({\bf W}_N^{(1)T}, ..., {\bf W}_N^{(M)T})^{T}$ is a $ML \times N$ block-line matrix whose block-lines $({\bf W}_N^{(m)})_{m=1, ..., M}$ are independent identically distributed $L \times N$ Hankel matrices built from i.i.d. standard complex Gaussian sequences. It is shown that if $M \rightarrow +\infty$ and $\frac{ML}{N} \rightarrow c_*$ ($c_* \in (0, \infty)$), then the empirical eigenvalue distribution of ${\bf W}_N {\bf W}_N^{*}$ converges almost surely towards the Marcenko-Pastur distribution. More importantly, it is established that if $L = \mathcal{O}(N^α)$ with $α< 2/3$, then, almost surely, for $N$ large enough, the eigenvalues of ${\bf W}_N {\bf W}_N^{*}$ are located in the neighbourhood of the Marcenko-Pastur distribution.

math.PR↗

arXiv2015

Performance analysis of spatial smoothing schemes in the context of large arrays

This paper adresses the statistical behaviour of spatial smoothing subspace DoA estimation schemes using a sensor array in the case where the number of observations $N$ is significantly smaller than the number of sensors $M$, and that the smoothing parameter $L$ is such that $M$ and $NL$ are of the same order of magnitude. This context is modelled by an asymptotic regime in which $NL$ and $M$ both converge towards $\infty$ at the same rate. As in recent works devoted to the study of (unsmoothed) subspace methods in the case where $M$ and $N$ are of the same order of magnitude, it is shown that it is still possible to derive improved DoA estimators termed as Generalized-MUSIC with spatial smoothing (G-MUSIC SS). The key ingredient of this work is a technical result showing that the largest singular values and corresponding singular vectors of low rank deterministic perturbation of certain Gaussian block-Hankel large random matrices behave as if the entries of the latter random matrices were independent identically distributed. This allows to conclude that when the number of sources and their DoA do not scale with $M,N,L,$ a situation modelling widely spaced DoA scenarios, then both traditional and Generalized spatial smoothing subspace methods provide consistent DoA estimators whose convergence speed is faster than $\frac{1}{M}$. The case of DoA that are spaced of the order of a beamwidth, which models closely spaced sources, is also considered. It is shown that the convergence speed of G-MUSIC SS estimates is unchanged, but that it is no longer the case for MUSIC SS ones.

stat.ME↗

arXiv2012

A Subspace Estimator for Fixed Rank Perturbations of Large Random Matrices

This paper deals with the problem of parameter estimation based on certain eigenspaces of the empirical covariance matrix of an observed multidimensional time series, in the case where the time series dimension and the observation window grow to infinity at the same pace. In the area of large random matrix theory, recent contributions studied the behavior of the extreme eigenvalues of a random matrix and their associated eigenspaces when this matrix is subject to a fixed-rank perturbation. The present work is concerned with the situation where the parameters to be estimated determine the eigenspace structure of a certain fixed-rank perturbation of the empirical covariance matrix. An estimation algorithm in the spirit of the well-known MUSIC algorithm for parameter estimation is developed. It relies on an approach recently developed by Benaych-Georges and Nadakuditi, relating the eigenspaces of extreme eigenvalues of the empirical covariance matrix with eigenspaces of the perturbation matrix. First and second order analyses of the new algorithm are performed.

math.PR↗

arXiv2011

Diversity of the MMSE receiver in flat fading and frequency selective MIMO channels at fixed rate

In this contribution, the evaluation of the diversity of the MIMO MMSE receiver is addressed for finite rates in both flat fading channels and frequency selective fading channels with cyclic prefix. It has been observed recently that in contrast with the other MIMO receivers, the MMSE receiver has a diversity depending on the aimed finite rate, and that for sufficiently low rates the MMSE receiver reaches the full diversity - that is, the diversity of the ML receiver. This behavior has so far only been partially explained. The purpose of this paper is to provide complete proofs for flat fading MIMO channels, and to improve the partial existing results in frequency selective MIMO channels with cyclic prefix.

cs.IT↗