Xiaoyi Mai

Mohammed Racim Moussa Boudjemaa, Alper Kalle, Xiaoyi Mai et al.

Whitening is a classical technique in unsupervised learning that can facilitate estimation tasks by standardizing data. An important application is the estimation of latent variable models via the decomposition of tensors built from high-order moments. In particular, whitening orthogonalizes the means of a spherical Gaussian mixture model (GMM), thereby making the corresponding moment tensor orthogonally decomposable, hence easier to decompose. However, in the large-dimensional regime (LDR) where data are high-dimensional and scarce, the standard whitening matrix built from the sample covariance becomes ineffective because the latter is spectrally distorted. Consequently, whitened means of a spherical GMM are no longer orthogonal. Using random matrix theory, we derive exact limits for their dot products, which are generally nonzero in the LDR. As our main contribution, we then construct a corrected whitening matrix that restores asymptotic orthogonality, allowing for performance gains in spherical GMM estimation.

Xiaoyi Mai, Romain Couillet

Semi-supervised Laplacian regularization, a standard graph-based approach for learning from both labelled and unlabelled data, was recently demonstrated to have an insignificant high dimensional learning efficiency with respect to unlabelled data (Mai and Couillet 2018), causing it to be outperformed by its unsupervised counterpart, spectral clustering, given sufficient unlabelled data. Following a detailed discussion on the origin of this inconsistency problem, a novel regularization approach involving centering operation is proposed as solution, supported by both theoretical analysis and empirical results.

Xiaoyi Mai, Zhenyu Liao

This article provides, through theoretical analysis, an in-depth understanding of the classification performance of the empirical risk minimization framework, in both ridge-regularized and unregularized cases, when high dimensional data are considered. Focusing on the fundamental problem of separating a two-class Gaussian mixture, the proposed analysis allows for a precise prediction of the classification error for a set of numerous data vectors $\mathbf{x} \in \mathbb R^p$ of sufficiently large dimension $p$. This precise error depends on the loss function, the number of training samples, and the statistics of the mixture data model. It is shown to hold beyond Gaussian distribution under some additional non-sparsity condition of the data statistics. Building upon this quantitative error analysis, we identify the simple square loss as the optimal choice for high dimensional classification in both ridge-regularized and unregularized cases, regardless of the number of training samples.

Xiaoyi Mai, Romain Couillet

This article provides an original understanding of the behavior of a class of graph-oriented semi-supervised learning algorithms in the limit of large and numerous data. It is demonstrated that the intuition at the root of these methods collapses in this limit and that, as a result, most of them become inconsistent. Corrective measures and a new data-driven parametrization scheme are proposed along with a theoretical analysis of the asymptotic performances of the resulting approach. A surprisingly close behavior between theoretical performances on Gaussian mixture models and on real datasets is also illustrated throughout the article, thereby suggesting the importance of the proposed analysis for dealing with practical data. As a result, significant performance gains are observed on practical data classification using the proposed parametrization.