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From Generality to Specificity: Prior-Driven Optimal Sparse Transformation in Compressed Sensing

Zhihan Zhu, Yanhao Zhang, Yong Xia

TL;DR

The paper presents the POST framework to fuse generalizable classical transforms with task-specific reference knowledge, yielding the HOT transform that applies to real and complex signals. The key contributions include a formal POST optimization with rank-based solutions leading to a Householder-based HOT, theoretical guarantees on generalization and specificity, and an extension to multiple references. The authors demonstrate substantial performance gains across audio sensing, 5G channel estimation, and image compression while maintaining low computational overhead. This approach offers a practical, theory-backed route to improved sparse representations in compressed sensing and related transform-dependent tasks.

Abstract

This paper introduces a new paradigm for sparse transformation: the Prior-to-Posterior Sparse Transform (POST) framework, designed to overcome long-standing limitation on generalization and specificity in classical sparse transforms for compressed sensing. POST systematically unifies the generalization capacity of any existing transform domains with the specificity of reference knowledge, enabling flexible adaptation to diverse signal characteristics. Within this framework, we derive an explicit sparse transform domain termed HOT, which adaptively handles both real and complex-valued signals. We theoretically establish HOT's sparse representation properties under single and multiple reference settings, demonstrating its ability to preserve generalization while enhancing specificity even under weak reference information. Extensive experiments confirm that HOT delivers substantial meta-gains across audio sensing, 5G channel estimation, and image compression tasks, consistently boosting multiple compressed sensing algorithms under diverse multimodal settings with negligible computational overhead.

From Generality to Specificity: Prior-Driven Optimal Sparse Transformation in Compressed Sensing

TL;DR

The paper presents the POST framework to fuse generalizable classical transforms with task-specific reference knowledge, yielding the HOT transform that applies to real and complex signals. The key contributions include a formal POST optimization with rank-based solutions leading to a Householder-based HOT, theoretical guarantees on generalization and specificity, and an extension to multiple references. The authors demonstrate substantial performance gains across audio sensing, 5G channel estimation, and image compression while maintaining low computational overhead. This approach offers a practical, theory-backed route to improved sparse representations in compressed sensing and related transform-dependent tasks.

Abstract

This paper introduces a new paradigm for sparse transformation: the Prior-to-Posterior Sparse Transform (POST) framework, designed to overcome long-standing limitation on generalization and specificity in classical sparse transforms for compressed sensing. POST systematically unifies the generalization capacity of any existing transform domains with the specificity of reference knowledge, enabling flexible adaptation to diverse signal characteristics. Within this framework, we derive an explicit sparse transform domain termed HOT, which adaptively handles both real and complex-valued signals. We theoretically establish HOT's sparse representation properties under single and multiple reference settings, demonstrating its ability to preserve generalization while enhancing specificity even under weak reference information. Extensive experiments confirm that HOT delivers substantial meta-gains across audio sensing, 5G channel estimation, and image compression tasks, consistently boosting multiple compressed sensing algorithms under diverse multimodal settings with negligible computational overhead.

Paper Structure

This paper contains 10 sections, 8 theorems, 90 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Prior-driven Optimal Sparse Transformation
  3. Theoretical Properties of HOT
  4. Choice of $j$ in HOT
  5. HOT with Multiple Reference Knowledge
  6. Experiment
  7. Audio Sensing with HOT
  8. 5G Channel Estimation with HOT
  9. Image Compression with HOT
  10. Conclusion

Key Result

Theorem 1

Choosing $\mathcal{L}\left(\mathbf{D}_{post},\mathbf{D}_{prior}\right)$ as $\mathop{\mathrm{rank}}\nolimits\left(\mathbf{D}_{post}-\mathbf{D}_{prior}\right)$, then for $\forall \mathbf{r}\in\mathbb{C}^{N\times1}$, the global optimal solution of POST is formulated as (a) $\mathbf{r} = \alpha\mathbf{D (b) $\mathbf{r} \neq \alpha\mathbf{D}_{prior, j}$: where $\mathbf{w} =\mathbf{D}_{prior}^H \mathbf

Figures (10)

  • Figure 1: Wireless channel, images and audio signal exhibit sparsity on classic transform domains, yet the transform sparsity may still not be sufficient.
  • Figure 2: Open the door for classic transform and idea of POST.
  • Figure 3: Specificity of HOT.
  • Figure 4: Generalizability of HOT.
  • Figure 5: HOT with two reference knowledge (each is average of columns in 4 images).
  • ...and 5 more figures

Theorems & Definitions (31)

  • Theorem 1
  • proof
  • Corollary 1
  • Remark 1
  • Theorem 2
  • proof
  • Remark 2
  • Remark 3
  • Theorem 3
  • proof
  • ...and 21 more