[Paper] 희소 복구에서 plug-and-play priors까지, 일반화된 투영 경사 하강법을 이용한 안정적인 복구를 위한 trade‑offs 이해

Source: arXiv - 2512.07397v1

Overview

노이즈가 섞인 측정값이 훨씬 적은 상황에서 고차원 신호를 복원하는 문제는 신호 처리, 컴퓨터 비전, 그리고 다양한 AI 기반 응용 분야의 핵심 과제이다. 이 논문은 Generalized Projected Gradient Descent (GPGD) 를 연구한다—전통적인 희소 복원 기법과 딥 뉴럴 네트워크 기반의 최신 “plug‑and‑play” 사전(prior)을 연결하는 유연한 알고리즘 프레임워크이다. 모델 불일치와 불완전한 투영을 고려하도록 수렴 보장을 확장함으로써, 저자들은 identifiability(진짜 신호를 얼마나 정확히 찾아낼 수 있는가)와 stability(복원이 노이즈와 모델 오류에 얼마나 강인한가) 사이의 트레이드‑오프를 보다 명확히 제시한다.

Key Contributions

• Unified analysis of GPGD that covers both traditional convex sparsity projections and learned deep projectors.
• Robust convergence proofs that tolerate both measurement noise and errors in the projection operator (e.g., imperfect neural network priors).
• Introduction of generalized back‑projection schemes to handle structured noise such as sparse outliers.
• Proposal of a normalized idempotent regularization technique that stabilizes the learning of deep projective priors.
• Comprehensive empirical evaluation on synthetic sparse recovery and real‑world image inverse problems, illustrating practical trade‑offs.

Methodology

1. Problem setup – The goal is to estimate a low‑dimensional vector (x^*) from measurements
   [ y = A x^* + \eta, ]
   where (A) is an underdetermined linear operator (more unknowns than equations) and (\eta) is noise.

2. Generalized Projected Gradient Descent (GPGD) – Starting from an initial guess (x_0), GPGD iterates:
   [ x_{k+1} = \mathcal{P}\bigl(x_k - \mu_k A^\top (A x_k - y)\bigr), ]
   where (\mathcal{P}) is a projector onto a set that encodes prior knowledge (e.g., sparsity, a deep denoiser).

3. Extending the theory – The authors prove that, even when (\mathcal{P}) is only an approximate projector (as is the case for learned networks), the iterates converge to a point whose error can be bounded by:

   • The measurement noise level,
   • The model error (how far the true signal lies outside the assumed prior set), and
   • The projection error (how well (\mathcal{P}) approximates an ideal projection).

4. Generalized back‑projection – Instead of the standard gradient step (A^\top (A x_k - y)), they replace it with a structured back‑projection that can suppress specific noise patterns (e.g., sparse outliers).

5. Normalized idempotent regularization – When training a deep network to act as (\mathcal{P}), they enforce a regularizer that encourages the network to behave like an idempotent operator (i.e., (\mathcal{P}(\mathcal{P}(z)) \approx \mathcal{P}(z))) while keeping its output norm consistent. This improves stability without sacrificing expressive power.

6. Experiments – Two families of tests:

   • Synthetic sparse vectors with varying sparsity levels and noise types.
   • Image inverse problems (deblurring, compressive sensing MRI) using a learned denoiser as the projector.

Results & Findings

• Stability gains: The normalized idempotent regularization reduced the sensitivity of the learned projector to small perturbations by ~30 % (measured via Lipschitz‑type constants).
• Robustness to outliers: The generalized back‑projection dramatically lowered reconstruction error when up to 5 % of measurements were corrupted by sparse spikes.
• Trade‑off curves: By varying the projection error (e.g., using a less‑trained network), the authors plotted identifiability vs. stability, confirming the theoretical prediction that improving one often worsens the other unless the regularization is applied.

Practical Implications

• Plug‑and‑play pipelines: Developers can replace hand‑crafted priors (like wavelet sparsity) with a pre‑trained denoiser and still retain provable convergence guarantees, provided the network respects the idempotent regularization.
• Robust sensing hardware: In applications such as LiDAR or compressed‑sensing cameras where occasional sensor glitches appear, the generalized back‑projection can be integrated into existing reconstruction code with minimal overhead (just a different residual computation).
• Fast prototyping: Because GPGD is a simple iterative scheme, it can be embedded in real‑time systems (e.g., video streaming) where each iteration is a cheap matrix‑vector multiply plus a forward pass through a neural net.
• Model‑error budgeting: The paper’s error bounds give engineers a quantitative way to allocate resources—e.g., decide whether to invest in better measurement matrices, cleaner hardware, or more expressive priors.

Limitations & Future Work

• Projection quality dependence: The theoretical guarantees degrade gracefully but still rely on the projector being “close enough” to an exact projection; extremely over‑parameterized networks may violate this.
• Computational cost of back‑projection: Structured back‑projection matrices (e.g., designed to null out outliers) can be expensive to compute for very large‑scale problems.
• Scope of experiments: The empirical validation focuses on relatively low‑dimensional synthetic data and a handful of imaging tasks; broader domains such as natural language or graph signals remain unexplored.
• Future directions suggested by the authors include:
  • Extending the framework to non‑linear measurement operators (e.g., phase retrieval).
  • Learning projectors that are adaptive across iterations rather than fixed.
  • Investigating tighter, data‑dependent bounds that could further shrink the identifiability‑stability gap.

Authors

• Ali Joundi
• Yann Traonmilin
• Jean‑François Aujol

Paper Information

• arXiv ID: 2512.07397v1
• Categories: eess.IV, cs.NE, math.OC
• Published: December 8, 2025
• PDF: Download PDF