Physical-Support Confidence Sets for Highly Coherent Dictionaries

Sparse coding and dictionary learning have become fundamental in signal processing, enabling efficient representations in diverse fields such as array localization, spectral unmixing, and neuroimaging. Early works like Olshausen and Field (1996) established the importance of sparse priors, while subsequent algorithms like Mairal et al. (2014) improved scalability. Despite progress, the challenge of uncertainty quantification, especially in high coherence dictionaries, remains unresolved. High coherence leads to multiple plausible physical interpretations for the same sparse support, complicating physical localization. Existing methods often assume known dictionaries or ignore calibration uncertainty, limiting their reliability in real-world scenarios where calibration data are finite and noisy. Addressing this gap is critical for trustworthy physical inference.

The core problem is that in high coherence dictionaries, sparse supports do not translate uniquely into physical elements. Multiple calibration-compatible dictionaries can produce the same sparse support but assign different physical meanings, causing ambiguity. Traditional support recovery methods lack explicit uncertainty quantification, leading to overconfidence and unsupported physical claims. The fundamental question is: how to quantify and control the physical support uncertainty, given limited calibration data and deployment signals? This is crucial for applications demanding high physical interpretability, such as array source localization and brain imaging, where mislocalization can have serious consequences.

The paper introduces several key innovations:

1) Cross-dictionary confidence correspondence: a formal framework that guarantees support validity across multiple plausible dictionaries, incorporating calibration and deployment uncertainties.

2) Minimax physical resolution analysis: deriving the fundamental limit \(\delta_{opt}(N,s)\), which characterizes the best achievable physical localization accuracy under finite calibration samples.

3) Active Endpoint Bracketing (AEB): an adaptive algorithm that selectively evaluates only the most impactful candidates, reducing computational load while maintaining statistical coverage.

4) Theoretical proof that the resolution bound is minimax optimal, explicitly linking calibration sample size, coherence scale, and physical localization limits.

• �� Establish a joint model for calibration data (Gaussian mixtures with Bernoulli-Gaussian coefficients) and deployment signals, incorporating uncertainties in dictionary geometry and coefficients. • Use robust estimators for second and fourth moments to define calibration-compatible regions with finite samples. • Formalize the retain–project–coarsen principle: retain all plausible explanations, project them onto physical space via the physical mapping, and coarsen to shared support. • Derive the minimax physical resolution \(\delta_{opt}(N,s)\) by analyzing the Fisher information about the orientation of coherent atoms, showing it scales as \(\min\{s, 1/\sqrt{N}s^2\}\). • Develop AEB: an adaptive, finite-bank procedure that evaluates only candidates that can influence the physical report, certifying support at various resolutions or abstaining when uncertain. • Validate through synthetic experiments, comparing traditional plug-in methods with AEB in terms of coverage, resolution, and evaluation efficiency.

Simulations involve synthetic multi-region signals with controlled coherence scales and calibration sample sizes. The experiments compare traditional point estimators against AEB in terms of support diameter, coverage probability, and candidate evaluations. Different noise levels and support complexities test robustness. Results show that point estimators tend to overrefine support, leading to unsupported claims, while AEB maintains correct coverage with fewer evaluations, closely matching theoretical bounds. The experiments validate the minimax optimality of the derived resolution \(\delta_{opt}(N,s)\) and demonstrate the efficiency of the adaptive screening process in complex high-coherence scenarios.

The main findings confirm that the physical resolution cannot surpass \(\delta_{opt}(N,s) \asymp \min\{s, 1/\sqrt{N}s^2\}\), which is minimax optimal. AEB achieves this limit efficiently, reducing candidate evaluations by up to 70% compared to exhaustive search. Traditional plug-in selectors often produce overrefined supports, risking unsupported physical claims. The framework reliably quantifies support uncertainty, providing confidence sets with guaranteed coverage even under high coherence and limited calibration data. These results establish a rigorous foundation for physically meaningful sparse inference.

Applicable to array source localization, hyperspectral unmixing, Raman spectroscopy, and EEG source imaging, especially where calibration data are scarce and dictionaries are highly coherent. The method enables practitioners to quantify the physical support uncertainty, improve localization accuracy, and avoid unsupported claims. Its adaptive nature makes it suitable for real-time applications, and its theoretical guarantees ensure reliability in critical scientific and industrial tasks. Future integration with deep learning could further enhance scalability and robustness across diverse modalities.

Assumes Gaussian noise and sparsity, which may not hold in real signals with heavy-tailed noise or dense supports. Performance under extreme coherence or very limited calibration samples needs further validation. Computational complexity, though improved, remains a challenge for very large-scale problems. Extending the framework to non-Gaussian noise models and multi-scale data fusion is an important future direction.