Related papers: Hotelling Deflation on Large Symmetric Spiked Tens…

This work introduces an asymptotic study of Hotelling-type tensor deflation in the presence of noise, in the regime of large tensor dimensions. Specifically, we consider a low-rank asymmetric tensor model of the form $\sum_{i=1}^r…

Leveraging on recent advances in random tensor theory, we consider in this paper a rank-$r$ asymmetric spiked tensor model of the form $\sum_{i=1}^r \beta_i A_i + W$ where $\beta_i\geq 0$ and the $A_i$'s are rank-one tensors such that…

This paper tackles the problem of recovering a low-rank signal tensor with possibly correlated components from a random noisy tensor, or so-called spiked tensor model. When the underlying components are orthogonal, they can be recovered…

We consider rank-one symmetric tensor estimation when the tensor is corrupted by Gaussian noise and the spike forming the tensor is a structured signal coming from a generalized linear model. The latter is a mathematically tractable model…

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…

Large loss spikes in stochastic gradient descent are studied through a rigorous large-deviations analysis for a shallow, fully connected network in the NTK scaling. In contrast to full-batch gradient descent, the catapult phase is shown to…

In this paper, we study the estimation of a rank-one spiked tensor in the presence of heavy tailed noise. Our results highlight some of the fundamental similarities and differences in the tradeoff between statistical and computational…

Tensor decompositions are promising tools for big data analytics as they bring multiple modes and aspects of data to a unified framework, which allows us to discover complex internal structures and correlations of data. Unfortunately most…

Deflation techniques are typically used to shift isolated clusters of small eigenvalues in order to obtain a tighter distribution and a smaller condition number. Such changes induce a positive effect in the convergence behavior of Krylov…

Relying on random matrix theory (RMT), this paper studies asymmetric order-$d$ spiked tensor models with Gaussian noise. Using the variational definition of the singular vectors and values of (Lim, 2005), we show that the analysis of the…

Because of the attractiveness of the canonical polyadic (CP) tensor decomposition in various applications, several algorithms have been designed to compute it, but efficient ones are still lacking. Iterative deflation algorithms based on…

Across many disciplines from neuroscience and genomics to machine learning, atmospheric science and finance, the problems of denoising large data matrices to recover signals obscured by noise, and of estimating the structure of these…

Consider a spiked random tensor obtained as a mixture of two components: noise in the form of a symmetric Gaussian $p$-tensor for $p\geq 3$ and signal in the form of a symmetric low-rank random tensor. The latter is defined as a linear…

This work considers a computationally and statistically efficient parameter estimation method for a wide class of latent variable models---including Gaussian mixture models, hidden Markov models, and latent Dirichlet allocation---which…

We study the statistical limits of both detecting and estimating a rank-one deformation of a symmetric random Gaussian tensor. We establish upper and lower bounds on the critical signal-to-noise ratio, under a variety of priors for the…

This chapter studies the problem of decomposing a tensor into a sum of constituent rank one tensors. While tensor decompositions are very useful in designing learning algorithms and data analysis, they are NP-hard in the worst-case. We will…

Principal Component Analysis (PCA) aims to find subspaces spanned by the so-called principal components that best represent the variance in the dataset. The deflation method is a popular meta-algorithm that sequentially finds individual…

We consider the problem of estimating a rank-one nonsymmetric matrix under additive white Gaussian noise. The matrix to estimate can be written as the outer product of two vectors and we look at the special case in which both vectors are…

Tensor train (TT) decomposition provides a space-efficient representation for higher-order tensors. Despite its advantage, we face two crucial limitations when we apply the TT decomposition to machine learning problems: the lack of…

We study asymmetric rank-one spiked tensor models in the high-dimensional regime, where the noise entries are independent and identically distributed with zero mean, unit variance, and finite fourth moment. This extends the classical…