Ilija Bogunovic

Learning-Based Compressive Subsampling

The problem of recovering a structured signal x ∈ C^p from a set of dimensionality-reduced linear measurements b = Ax arises in a variety of applications, such as medical imaging, spectroscopy, Fourier optics, and computerized tomography. Due to computational and storage complexity or physical constraints imposed by the problem, the measurement matrix A ∈ C^n×p is often of the form A = P_ΩΨ for some orthonormal basis matrix Ψ ∈ C^P×P and subsampling operator P_Ω : C^p → Cⁿ that selects the rows indexed by Ω. This raises the fundamental question of how best to choose the index set Ω in order to optimize the recovery performance. Previous approaches to addressing this question rely on nonuniform random subsampling using application-specific knowledge of the structure of x. In this paper, we instead take a principled learning-based approach in which affixed index set is chosen based on a set of training signals x₁, . . . , x_m. We formulate combinatorial optimization problems seeking to maximize the energy captured in these signals in an average-case or worst-case sense, and we show that these can be efficiently solved either exactly or approximately via the identification of modularity and submodularity structures. We provide both deterministic and statistical theoretical guarantees showing how the resulting measurement matrices perform on signals differing from the training signals, and we provide numerical examples showing our approach to be effective on a variety of data sets.

Active learning of self-concordant like multi-index functions

We study the problem of actively learning a multi-index function of the form f(x) = g₀(A₀x) from its point evaluations, where A₀ ∈ ℝ^k×d with k ≫ d. We build on the assumptions and techniques of an existing approach based on low-rank matrix recovery (Tyagi and Cevher, 2012). Specifically, by introducing an additional self- concordant like assumption on g0 and adapting the sampling scheme and its analysis accordingly, we provide a bound on the sampling complexity with a weaker dependence on d in the presence of additive Gaussian sampling noise. For example, under natural assumptions on certain other parameters, the dependence decreases from O(d^3/2) to O(d^¾).