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This research introduces a novel, data-adaptive transform that combines fixed canonical matrices with sparse components to improve signal processing tasks like compression and noise reduction. By leveraging inexact proximal methods and a new closed-form projection operator, the approach achieves state-of-the-art performance with lower computational costs and better convergence than dense variants. The work addresses limitations of standard analytical transforms by introducing controllable adaptivity while maintaining stability, offering a fresh direction for efficient feature extraction.
Learning Doubly Sparse Explicitly Conditioned Transforms
arXiv:2606.10975v1 Announce Type: new Abstract: Finding convenient spaces in which certain hypotheses regarding an assumed sparse structure of natural signals hold true has become a desirable result in recent research, its implications being reflected in areas such as data compression, noise reduction and feature extraction. While the extensively used analytical transforms, such as DFT or DCT, already provide efficient algorithms and robust sparse representations, they assume a fixed prior about the data, failing to accurately capture the specific structure of more restrictive classes of signals. To address this, the concept of a data-adaptive, learnt transform has been introduced in the literature, allowing for the reduction of a residual term in the transform domain. More recent studies have shown that the condition number serves as a good metric in this context, where the desired outcome alternates between a generalizing tendency and one that achieves minimal approximation error. Motivated by these considerations, we introduce the learning of a structured, explicitly conditioned transform formulated as the product of a fixed canonical matrix and a refining data-adaptive sparse component. This approach seeks to preserve the advantages of fast and stable analytical transforms, while introducing controllable adaptivity to the data. No references that concern this specific formulation have been identified so far, indicating its novelty. The proposed algorithm is motivated within the framework of inexact proximal methods, leveraging a newly derived closed-form projection operator. Empirical observations demonstrate state-of-the-art results on the doubly sparse transform learning problem and comparable performance with its dense variant at significantly lower computational costs and sometimes faster convergence and better avoidance of bad local minima.
Abstract:Finding convenient spaces in which certain hypotheses regarding an assumed sparse structure of natural signals hold true has become a desirable result in recent research, its implications being reflected in areas such as data compression, noise reduction and feature extraction. While the extensively used analytical transforms, such as DFT or DCT, already provide efficient algorithms and robust sparse representations, they assume a fixed prior about the data, failing to accurately capture the specific structure of more restrictive classes of signals. To address this, the concept of a data-adaptive, learnt transform has been introduced in the literature, allowing for the reduction of a residual term in the transform domain. More recent studies have shown that the condition number serves as a good metric in this context, where the desired outcome alternates between a generalizing tendency and one that achieves minimal approximation error. Motivated by these considerations, we introduce the learning of a structured, explicitly conditioned transform formulated as the product of a fixed canonical matrix and a refining data-adaptive sparse component. This approach seeks to preserve the advantages of fast and stable analytical transforms, while introducing controllable adaptivity to the data. No references that concern this specific formulation have been identified so far, indicating its novelty. The proposed algorithm is motivated within the framework of inexact proximal methods, leveraging a newly derived closed-form projection operator. Empirical observations demonstrate state-of-the-art results on the doubly sparse transform learning problem and comparable performance with its dense variant at significantly lower computational costs and sometimes faster convergence and better avoidance of bad local minima.
10 pages, 1 figure, 1 table. Accepted for publication in Procedia Computer Science (30th International Conference on Knowledge-Based and Intelligent Information & Engineering Systems - KES 2026; Invited Session: Global and Constrained Optimization: Algorithms and Applications)
Machine Learning (cs.LG); Signal Processing (eess.SP); Optimization and Control (math.OC)
G.1.6; I.4.2; I.4.4; I.2.6
arXiv:2606.10975 [cs.LG]
(or arXiv:2606.10975v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2606.10975 arXiv-issued DOI via DataCite (pending registration)
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