Submodular meets Spectral: Greedy Algorithms for Subset Selection, Sparse Approximation and Dictionary Selection

We study the problem of selecting a subset of k random variables from a large set, in order to obtain the best linear prediction of another variable of interest. This problem can be viewed in the context of both feature selection and sparse approximation. We analyze the performance of widely used greedy heuristics, using insights from the maximization of submodular functions and spectral analysis. We introduce the submodularity ratio as a key quantity to help understand why greedy algorithms perform well even when the variables are highly correlated. Using our techniques, we obtain the strongest known approximation guarantees for this problem, both in terms of the submodularity ratio and the smallest k-sparse eigenvalue of the covariance matrix. We further demonstrate the wide applicability of our techniques by analyzing greedy algorithms for the dictionary selection problem, and significantly improve the previously known guarantees. Our theoretical analysis is complemented by experiments on real-world and synthetic data sets; the experiments show that the submodularity ratio is a stronger predictor of the performance of greedy algorithms than other spectral parameters

Improved Batch Reverse Osmosis Configuration for Better Energy Effiency

Recent progress in batch and semi-batch reverse osmosis processes such as CCRO have shown the promise to be the most efficient desalination systems. Despite their progress, it is critical to further increase their efficiencies, and reduce the downtime between cycles that worsens their cost performance. In this study, we model in new detail a further improved batch desalination system that uses a high pressure feed tank with a reciprocating piston. A high-pressure pump fills the inactive side with the following cycle’s feedwater, providing two main benefits. First, no tank emptying step is needed because feed is already present, thus reducing downtime. Second, the tank fully empties each cycle, thus avoiding the small energy losses from brine mixing with the new feed that past best designs had. The modeling methodology is the most thorough yet for batch processes, as it uses a discretized module that includes transient mass transport equations for salt boundary layers, membrane permeability effects, and minute salt permeation through the membrane. Comparing the new configuration to standard reverse osmosis with and without energy recovery, the new process vastly outperforms, with the potential to be below 2 kWh/m3 for seawater. The new process has less downtime too, around 2% of cycle time, compared with 10% for CCRO or 16% from past batch studies

Trimmed Maximum Likelihood Estimation for Robust Learning in Generalized Linear Models

We study the problem of learning generalized linear models under adversarial corruptions. We analyze a classical heuristic called the iterative trimmed maximum likelihood estimator which is known to be effective against label corruptions in practice. Under label corruptions, we prove that this simple estimator achieves minimax near-optimal risk on a wide range of generalized linear models, including Gaussian regression, Poisson regression and Binomial regression. Finally, we extend the estimator to the more challenging setting of label and covariate corruptions and demonstrate its robustness and optimality in that setting as well