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 = PΩΨ for some orthonormal basis matrix Ψ ∈ C p×p and subsampling operator PΩ : C p → C n 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 non-uniform random subsampling using application-specific knowledge of the structure of x. In this paper, we instead take a principled learning-based approach in which a fixed index set is chosen based on a set of training signals x1, . . . , xm. 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.
In this paper, we consider the problem of maximizing a monotone submodular function subject to a cardinality constraint, with two added twists: The computation is distributed across a number of machines, and we require the solution to be robust against adversarial removals. We provide two versions of a partitioned robust algorithm for this problem, with the difference amounting to whether or not the centralized machine is informed (only in the final stage of the algorithm) which elements will be removed. In both of these cases, we provide a novel constant-factor approximation guarantee with respect to the optimal algorithm. Finally, we validate our algorithms via numerical experiments on real-world data sets in influence maximization and data summarization.
We consider a stochastic linear bandit problem in which the rewards are not only subject to random noise, but also adversarial attacks subject to a suitable budget C (i.e., an upper bound on the sum of corruption magnitudes across the time horizon). We provide two variants of a Robust Phased Elimination algorithm, one that knows C and one that does not. Both variants are shown to attain near-optimal regret in the non-corrupted case C = 0, while incurring additional additive terms respectively having a linear and quadratic dependency on C in general. We present algorithmindependent lower bounds showing that these additive terms are near-optimal. In addition, in a contextual setting, we revisit a setup of diverse contexts, and show that a simple greedy algorithm is provably robust with a near-optimal additive regret term, despite performing no explicit exploration and not knowing C.
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