In this paper, we introduce a novel theoretical framework for multi-task regression, applying random matrix theory to provide precise performance estimations, under high-dimensional, non-Gaussian data distributions. We formulate a multi-task optimization problem as a regularization technique to enable single-task models to leverage multi-task learning information. We derive a closed-form solution for multi-task optimization in the context of linear models. Our analysis provides valuable insights by linking the multi-task learning performance to various model statistics such as raw data covariances, signal-generating hyperplanes, noise levels, as well as the size and number of datasets. We finally propose a consistent estimation of training and testing errors, thereby offering a robust foundation for hyperparameter optimization in multi-task regression scenarios. Experimental validations on both synthetic and real-world datasets in regression and multivariate time series forecasting demonstrate improvements on univariate models, incorporating our method into the training loss and thus leveraging multivariate information.

This paper provides a large dimensional analysis of the Softmax classifier. We discover and prove that, when the classifier is trained on data satisfying loose statistical modeling assumptions, its weights become deterministic and solely depend on the data statistical means and covariances. As a striking consequence, despite the implicit and non-linear nature of the underlying optimization problem, the performance of the Softmax classifier is the same as if performed on a mere Gaussian mixture model, thereby disrupting the intuition that non-linearities inherently extract advanced statistical features from the data. Our findings are theoretically as well as numerically sustained on CNN representations of images produced by GANs.

Inspired by human learning, which transfers knowledge from learned tasks to solve new tasks, multitask learning aims at simultaneously solving multiple tasks by a smart exploitation of their similarities. How to relate the tasks so to optimize their performances is however a largely open problem.Based on a random matrix approach, this article proposes an asymptotic analysis of a support vector machine-inspired multitask learning scheme. The asymptotic performance of the algorithm, validated on both synthetic and real data, sets forth the relation between the statistics of the data in each task and the hyperparameters relating the tasks together. The article, as such, provides first insights on an offline control of multitask learning, which finds natural connections to the currently popular transfer learning paradigm.

This paper shows that deep learning (DL) representations of data produced by generative adversarial nets (GANs) are random vectors which fall within the class of so-called \emph{concentrated} random vectors. Further exploiting the fact that Gram matrices, of the type $G = X^\intercal X$ with $X=[x_1,\ldots,x_n]\in \mathbb{R}^{p\times n}$ and $x_i$ independent concentrated random vectors from a mixture model, behave asymptotically (as $n,p\to \infty$) as if the $x_i$ were drawn from a Gaussian mixture, suggests that DL representations of GAN-data can be fully described by their first two statistical moments for a wide range of standard classifiers. Our theoretical findings are validated by generating images with the BigGAN model and across different popular deep representation networks.

This article provides a theoretical analysis of the asymptotic performance of a regression or classification task performed by a simple random neural network. This result is obtained by leveraging a new framework at the crossroads between random matrix theory and the concentration of measure theory. This approach is of utmost interest for neural network analysis at large in that it naturally dismisses the difficulty induced by the non-linear activation functions, so long that these are Lipschitz functions. As an application, we provide formulas for the limiting law of the random neural network output and compare them conclusively to those obtained practically on handwritten digits databases.

This article proposes an original approach to the performance understanding of large dimensional neural networks. In this preliminary study, we study a single hidden layer feed-forward network with random input connections (also called extreme learning machine) which performs a simple regression task. By means of a new random matrix result, we prove that, as the size and cardinality of the input data and the number of neurons grow large, the network performance is asymptotically deterministic. This entails a better comprehension of the effects of the hyper-parameters (activation function, number of neurons, etc.) under this simple setting, thereby paving the path to the harnessing of more involved structures.