Tensor models play an increasingly prominent role in many fields, notably in machine learning. In several applications of such models, such as community detection, topic modeling and Gaussian mixture learning, one must estimate a low-rank signal from a noisy tensor. Hence, understanding the fundamental limits and the attainable performance of estimators of that signal inevitably calls for the study of random tensors. Substantial progress has been achieved on this subject thanks to recent efforts, under the assumption that the tensor dimensions grow large. Yet, some of the most significant among these results--in particular, a precise characterization of the abrupt phase transition (in terms of signal-to-noise ratio) that governs the performance of the maximum likelihood (ML) estimator of a symmetric rank-one model with Gaussian noise--were derived on the basis of statistical physics ideas, which are not easily accessible to non-experts. In this work, we develop a sharply distinct approach, relying instead on standard but powerful tools brought by years of advances in random matrix theory. The key idea is to study the spectra of random matrices arising from contractions of a given random tensor. We show how this gives access to spectral properties of the random tensor itself. In the specific case of a symmetric rank-one model with Gaussian noise, our technique yields a hitherto unknown characterization of the local maximum of the ML problem that is global above the phase transition threshold. This characterization is in terms of a fixed-point equation satisfied by a formula that had only been previously obtained via statistical physics methods. Moreover, our analysis sheds light on certain properties of the landscape of the ML problem in the large-dimensional setting. Our approach is versatile and can be extended to other models, such as asymmetric, non-Gaussian and higher-order ones.

In this paper, we study a polynomial decomposition model that arises in problems of system identification, signal processing and machine learning. We show that this decomposition is a special case of the X-rank decomposition --- a powerful novel concept in algebraic geometry that generalizes the tensor CP decomposition. We prove new results on generic/maximal rank and on identifiability of a particular polynomial decomposition model. In the paper, we try to make results and basic tools accessible for general audience (assuming no knowledge of algebraic geometry or its prerequisites).

This paper proposes an original approach to cluster multi-component data sets, including an estimation of the number of clusters. From the construction of a minimal spanning tree with Prim's algorithm, and the assumption that the vertices are approximately distributed according to a Poisson distribution, the number of clusters is estimated by thresholding the Prim's trajectory. The corresponding cluster centroids are then computed in order to initialize the generalized Lloyd's algorithm, also known as $K$-means, which allows to circumvent initialization problems. Some results are derived for evaluating the false positive rate of our cluster detection algorithm, with the help of approximations relevant in Euclidean spaces. Metrics used for measuring similarity between multi-dimensional data points are based on symmetrical divergences. The use of these informational divergences together with the proposed method leads to better results, compared to other clustering methods for the problem of astrophysical data processing. Some applications of this method in the multi/hyper-spectral imagery domain to a satellite view of Paris and to an image of the Mars planet are also presented. In order to demonstrate the usefulness of divergences in our problem, the method with informational divergence as similarity measure is compared with the same method using classical metrics. In the astrophysics application, we also compare the method with the spectral clustering algorithms.