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Recent works have shown that imposing tensor structures on the coefficient tensor in regression problems can lead to more reliable parameter estimation and lower sample complexity compared to vector-based methods. This work investigates a new low-rank tensor model, called Low Separation Rank (LSR), in Generalized Linear Model (GLM) problems. The LSR model – which generalizes the well-known Tucker and CANDECOMP/PARAFAC (CP) models, and is a special case of the Block Tensor Decomposition (BTD) model – is imposed onto the coefficient tensor in the GLM model. This work proposes a block coordinate descent algorithm for parameter estimation in LSR-structured tensor GLMs. Most importantly, it derives a minimax lower bound on the error threshold on estimating the coefficient tensor in LSR tensor GLM problems. The minimax bound is proportional to the intrinsic degrees of freedom in the LSR tensor GLM problem, suggesting that its sample complexity may be significantly lower than that of vectorized GLMs. This result can also be specialised to lower bound the estimation error in CP and Tucker-structured GLMs. The derived bounds are comparable to tight bounds in the literature for Tucker linear regression, and the tightness of the minimax lower bound is further assessed numerically. Finally, numerical experiments on synthetic datasets demonstrate the efficacy of the proposed LSR tensor model for three regression types (linear, logistic and Poisson). Experiments on a collection of medical imaging datasets demonstrate the usefulness of the LSR model over other tensor models (Tucker and CP) on real, imbalanced data with limited available samples. License: Creative Commons Attribution 4.0 International (CC BY 4.0) 

The accuracy of many downstream machine learning algorithms is tied to the training data having uncorrelated features. With the modern-day data often being streaming in nature, geographically distributed, and having large dimensions, it is paramount to apply both uncorrelated feature learning and dimensionality reduction techniques in this scenario. Principal Component Analysis (PCA) is a state-of-the-art tool that simultaneously yields uncorrelated features and reduces data dimensions by projecting data onto the eigenvectors of the population covariance matrix. This paper introduces a novel algorithm called Consensus-DIstributEd Generalized Oja (C-DIEGO), which is based on Oja's method, to estimate the dominant eigenvector of a population covariance matrix in a distributed, streaming setting. The algorithm considers a distributed network of arbitrarily connected nodes without a central coordinator and assumes data samples continuously arrive at the individual nodes in a streaming manner. It is established in the paper that C-DIEGO can achieve an order-optimal convergence rate if nodes in the network are allowed to have enough consensus rounds per algorithmic iteration. Numerical results are also reported in the paper that showcase the efficacy of the proposed algorithm. 

This paper considers the problem of understanding the exit time for trajectories of gradient-related first-order methods from saddle neighborhoods under some initial boundary conditions. Given the ‘flat’ geometry around saddle points, first-order methods can struggle to escape these regions in a fast manner due to the small magnitudes of gradients encountered. In particular, while it is known that gradient-related first-order methods escape strict-saddle neighborhoods, existing analytic techniques do not explicitly leverage the local geometry around saddle points in order to control behavior of gradient trajectories. It is in this context that this paper puts forth a rigorous geometric analysis of the gradient-descent method around strict-saddle neighborhoods using matrix perturbation theory. In doing so, it provides a key result that can be used to generate an approximate gradient trajectory for any given initial conditions. In addition, the analysis leads to a linear exit-time solution for gradient-descent method under certain necessary initial conditions, which explicitly bring out the dependence on problem dimension, conditioning of the saddle neighborhood, and more, for a class of strict-saddle functions.