Simple, Efficient, and Neural Algorithms for Sparse Coding (1503.00778v1)

Abstract: Sparse coding is a basic task in many fields including signal processing, neuroscience and machine learning where the goal is to learn a basis that enables a sparse representation of a given set of data, if one exists. Its standard formulation is as a non-convex optimization problem which is solved in practice by heuristics based on alternating minimization. Re- cent work has resulted in several algorithms for sparse coding with provable guarantees, but somewhat surprisingly these are outperformed by the simple alternating minimization heuristics. Here we give a general framework for understanding alternating minimization which we leverage to analyze existing heuristics and to design new ones also with provable guarantees. Some of these algorithms seem implementable on simple neural architectures, which was the original motivation of Olshausen and Field (1997a) in introducing sparse coding. We also give the first efficient algorithm for sparse coding that works almost up to the information theoretic limit for sparse recovery on incoherent dictionaries. All previous algorithms that approached or surpassed this limit run in time exponential in some natural parameter. Finally, our algorithms improve upon the sample complexity of existing approaches. We believe that our analysis framework will have applications in other settings where simple iterative algorithms are used.

• The paper presents a novel framework that reinterprets alternating minimization as approximating gradients of an unknown convex function to achieve efficient sparse coding.
• The paper introduces neural algorithms with provable recovery guarantees that operate near the information-theoretic limit while lowering computational complexity.
• The paper improves sample complexity and initialization accuracy through pairwise reweighting, enhancing practical applicability in neuroscience and machine learning.

Analysis and Algorithms for Sparse Coding in Neural Architectures

The paper "Simple, Efficient, and Neural Algorithms for Sparse Coding" by Arora et al. presents novel approaches to solving the sparse coding problem, which is widely encountered in fields such as signal processing, neuroscience, and machine learning. Sparse coding aims to find a basis that allows a sparse representation of data, often formulated as a non-convex optimization problem traditionally solved by heuristics like alternating minimization.

In this work, the authors offer a new theoretical framework for understanding and analyzing these heuristic methods. They focus on the alternating minimization heuristic widely regarded in practice for sparse coding, providing formal guarantees of convergence. Notably, they propose new algorithms implementable on neural architectures, addressing the initial motivations from neuroscience where sparse coding models neural activity patterns efficiently.

Key Contributions

1. Framework for Alternating Minimization: The authors present an analysis that reinterprets alternating minimization not as minimization of a known non-convex function, but as attempting to minimize an unknown convex function given an approximation to its gradient. This shift in focus allows leveraging techniques from convex optimization and applies uniformly across variants of sparse coding.
2. Neural Algorithms with Provable Guarantees: The paper introduces an efficient algorithm for sparse coding that operates close to the information-theoretic limit for sparse recovery on incoherent dictionaries, without the exponential time complexity of prior methods. Remarkably, this algorithm incorporates mechanisms plausible for neural computation.
3. Improved Sample Complexity: A significant advance of the algorithms introduced is their improved sample complexity over the existing approaches, making them more practical for real-world applications.
4. Initialization via Pairwise Reweighting: The authors propose a novel initialization technique using pairwise reweighting of samples. This method shows high probability of correct initialization, surpassing typical heuristic methods previously lacking theoretical justification.

Theoretical and Practical Implications

The theoretical implications of this work are profound. It demonstrates that the empirically successful yet theoretically opaque heuristic algorithms for sparse coding can be understood rigorously. The analysis provided extends to emphasize the importance of initializing alternating minimization approximately close to the optimal solution. Moreover, the framework and algorithms can apply to other scenarios where iterative and heuristic methods are routinely applied, offering a pathway to formal analysis in those contexts.

Practically, the ability to implement these algorithms in neural architectures directly speaks to their potential adaptation in biologically inspired systems and real-time processing environments. This work bridges neuroscientific modeling goals with algorithmic efficiency, contributing to both computational neuroscience and machine learning.

Future Directions

The authors highlight several future directions, such as exploring the neural plausibility and real-world implementation of these algorithms further, addressing computational efficiency in practice, and extending this framework to other non-convex optimization problems. Continued exploration of simple neural models with provable properties could enhance understanding of brain functionality and inspire novel machine learning architectures.

In conclusion, this paper provides not only robust algorithms for a challenging computational problem but also opens pathways for deeper integration of theoretical and practical research in neural computation and sparse coding. The anticipation of further developments and applications of these findings remains high.