Submodular meets Spectral: Greedy Algorithms for Subset Selection, Sparse Approximation and Dictionary Selection (1102.3975v2)

Abstract: 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.

• The paper introduces the submodularity ratio to explain the surprising effectiveness of greedy algorithms in subset selection.
• It establishes strong theoretical approximation guarantees based on the submodularity ratio and the smallest k-sparse eigenvalue of the covariance matrix.
• Empirical results on real and synthetic data confirm that these greedy methods robustly manage high feature correlation in sparse approximation tasks.

Overview of Greedy Algorithms for Subset Selection and Dictionary Selection

The paper "Submodular meets Spectral: Greedy Algorithms for Subset Selection, Sparse Approximation and Dictionary Selection" by Abhimanyu Das and David Kempe addresses the critical problem of selecting a subset of variables for optimal linear prediction, situated within both feature selection and sparse approximation domains.

The authors focus on the efficacy of greedy algorithms, such as Forward Regression and Orthogonal Matching Pursuit (OMP), in tackling this problem, leveraging concepts from submodular function maximization and spectral analysis. A key contribution is the introduction of the submodularity ratio, a novel metric that explains the surprising effectiveness of these algorithms, even under high correlation scenarios among variables.

Main Contributions

1. Submodularity Ratio: The paper introduces the submodularity ratio as a measure to predict the performance of greedy algorithms. It provides insight into cases where the $R^2$ objective may approximate submodular behavior, explaining why greedy methods outperform expectations especially in near-singular matrix conditions.
2. Theoretical Guarantees: The authors derive the strongest known approximation guarantees for greedy algorithm performance in subset selection, based on both the submodularity ratio and the smallest $k$-sparse eigenvalue of the covariance matrix. These guarantees are shown to be superior to those derived from traditional spectral bounds.
3. Extended Framework: The techniques are extended to analyze dictionary selection problems, achieving improved theoretical bounds compared to previous results, particularly enhancing understanding of greedy algorithms in this context through the submodularity perspective.
4. Empirical Validation: Experiments on both real-world and synthetic datasets corroborate the theoretical insights, demonstrating the robustness of greedy algorithms despite high feature correlation, with the submodularity ratio emerging as a stronger performance predictor than traditional spectral parameters.

Implications and Speculations

The implications of this research are profound in various domains like machine learning and signal processing, where efficient feature selection is crucial. The proposed framework could steer future algorithmic designs towards leveraging approximate submodularity, potentially leading to more refined approaches in sparse approximation and beyond.

Furthermore, understanding the submodularity ratio could influence the development of more adaptive greedy strategies, tailored for datasets exhibiting specific submodular properties. These insights may extend to other combinatorial optimization challenges where greedy methodologies are often employed.

Future Directions

Future investigations may involve deeper exploration of the submodularity ratio in other algorithmic contexts or its potential in hybrid algorithm frameworks, combining greedy methods with other optimization techniques. Additionally, the scalability of these approaches on larger datasets and their integration with advanced machine learning pipelines warrant further paper.

Overall, this paper compellingly integrates submodular and spectral concepts, advancing the theoretical foundation and practical effectiveness of greedy algorithms in subset selection and related tasks.