Tensor decomposition for learning Gaussian mixtures from moments

Rima Khouja (1) , Pierre-Alexandre Mattei (2) , Bernard Mourrain (1)

1 AROMATH - AlgebRe, geOmetrie, Modelisation et AlgoriTHmes
2 MAASAI - Modèles et algorithmes pour l’intelligence artificielle

In data processing and machine learning, an important challenge is to recover and exploit models that can represent accurately the data. We consider the problem of recovering Gaussian mixture models from datasets. We investigate symmetric tensor decomposition methods for tackling this problem, where the tensor is built from empirical moments of the data distribution. We consider identifiable tensors, which have a unique decomposition, showing that moment tensors built from spherical Gaussian mixtures have this property. We prove that symmetric tensors with interpolation degree strictly less than half their order are identifiable and we present an algorithm, based on simple linear algebra operations, to compute their decomposition. Illustrative experimentations show the impact of the tensor decomposition method for recovering Gaussian mixtures, in comparison with other state-of-the-art approaches.

Mots clés: Tensor decomposition; Method of moments; Gaussian mixture; Symmetric tensor; Clustering; Simultaneous diagonalisation

Domaines: Géométrie algébrique [math.AG]; Machine Learning [stat.ML]

Rima Khouja, Pierre-Alexandre Mattei, Bernard Mourrain. Tensor decomposition for learning Gaussian mixtures from moments. Journal of Symbolic Computation, 2022, 113, pp.193-210. ⟨10.1016/j.jsc.2022.04.002⟩. ⟨hal-03244448v2⟩