Subspace Tracking with Dynamical Models on the Grassmannian

Tracking signals in dynamic environments presents difficulties in both analysis and implementation. In this work, we expand on a class of subspace tracking algorithms which utilize the Grassmann manifold -- the set of linear subspaces of a high-dimensional vector space. We design regularized least squares algorithms based on common manifold operations and intuitive dynamical models. We demonstrate the efficacy of the approach for a narrowband beamforming scenario, where the dynamics of multiple signals of interest are captured by motion on the Grassmannian.

Decentralized Feature-Distributed Optimization for Generalized Linear Models

We consider the "all-for-one" decentralized learning problem for generalized linear models. The features of each sample are partitioned among several collaborating agents in a connected network, but only one agent observes the response variables. To solve the regularized empirical risk minimization in this distributed setting, we apply the Chambolle--Pock primal--dual algorithm to an equivalent saddle-point formulation of the problem. The primal and dual iterations are either in closed-form or reduce to coordinate-wise minimization of scalar convex functions. We establish convergence rates for the empirical risk minimization under two different assumptions on the loss function (Lipschitz and square root Lipschitz), and show how they depend on the characteristics of the design matrix and the Laplacian of the network.