Learning efficient sparse and low rank models

Abstract: Parsimony, including sparsity and low rank, has been shown to successfully model data in numerous machine learning and signal processing tasks. Traditionally, such modeling approaches rely on an iterative algorithm that minimizes an objective function with parsimony-promoting terms. The inherently sequential structure and data-dependent complexity and latency of iterative optimization constitute a major limitation in many applications requiring real-time performance or involving large-scale data. Another limitation encountered by these modeling techniques is the difficulty of their inclusion in discriminative learning scenarios. In this work, we propose to move the emphasis from the model to the pursuit algorithm, and develop a process-centric view of parsimonious modeling, in which a learned deterministic fixed-complexity pursuit process is used in lieu of iterative optimization. We show a principled way to construct learnable pursuit process architectures for structured sparse and robust low rank models, derived from the iteration of proximal descent algorithms. These architectures learn to approximate the exact parsimonious representation at a fraction of the complexity of the standard optimization methods. We also show that appropriate training regimes allow to naturally extend parsimonious models to discriminative settings. State-of-the-art results are demonstrated on several challenging problems in image and audio processing with several orders of magnitude speedup compared to the exact optimization algorithms.

• The paper introduces a novel process-centric approach using deterministic, fixed-complexity pursuit processes derived from proximal descent for efficient sparse and low-rank model learning.
• The research demonstrates state-of-the-art performance in image and audio processing tasks with significantly reduced computation compared to classical optimization methods.
• This process-centric framework enables robust solutions for real-time applications and paves the way for future extensions to other data representations like cosparse models.

Efficient Sparse and Low-Rank Models

This paper presents a novel perspective on parsimonious modeling that prioritizes the pursuit algorithm over the model itself, especially in scenarios demanding real-time performance or handling large-scale data. The authors introduce deterministic, fixed-complexity pursuit processes as alternatives to traditional iterative optimization techniques for finding sparse and low-rank representations. These processes are derived from proximal descent algorithms and architected to approximate sparse and low-rank models efficiently. Additionally, the paper proposes training protocols for extending these parsimonious models into discriminative settings, demonstrating significant speedups compared to conventional optimization methods.

The research emphasizes a shift from model-centric to process-centric approaches in parsimonious modeling. Traditionally, parsimonious representation relies on complex iterative algorithms to optimize an objective function composed of data fitting and regularization terms. This technique can be computationally prohibitive, especially when the optimization must run sequentially. Moreover, integrating sparse models into discriminative learning hasn't been widely successful due to the inherent complexity and non-differentiability of bilevel optimization problems. In contrast, this paper's approach allows the encoder structures to learn the optimal sparse representations directly, enabling efficient computation and potential applications in supervised learning scenarios.

Strong numerical results in the study include state-of-the-art performance in image and audio processing tasks achieved with at least one order of magnitude reduction in computation compared to classical optimization methods. The encoders designed within this framework can be trained offline or online, accommodating dynamically evolving or particularly extensive datasets. Furthermore, deterministic encoders developed through supervised learning exhibit improved discriminative performance, which is vital for complex tasks like speaker identification and image recognition.

The implications of these findings span both theoretical and practical domains. Theoretically, they elevate process-centric modeling as a feasible alternative to optimization-centric approaches, inviting further exploration into learning architectures akin to autoencoders but driven by proximal algorithms. Practically, the encoders offer robust solutions for real-time applications in multimedia processing where quick and accurate signal decomposition is paramount.

Future directions envisioned by the authors include extending this framework to analysis cosparse models, further enhancing the versatility and efficiency of parsimonious representations. As the authors suggest, leveraging augmented Lagrangian methods could be instrumental in extrapolating these benefits to domains where data is inherently cosparse. This potentially unlocks advancements in fields such as compressed sensing and robust signal recovery, broadening the scope of process-centric learning beyond synthesis models.

In conclusion, the presented process-centric parsimonious modeling framework marks a significant step toward efficient, scalable, and adaptive signal representations. This work is a substantive contribution to the ongoing discourse on efficient model learning processes and sets a precedent for subsequent research to refine and expand these methodologies across various applications.