提出可解释性增强的稀疏表示推断方法,解决高相干字典下的物理意义不确定性问题。
Physical-Support Confidence Sets for Highly Coherent Dictionaries

- 联合考虑字典学习与信号表示的不确定性,构建物理支持置信集
- 理论证明最小分辨率随校准样本数呈1/√N衰减,方向信息影响相对精度
- 引入自适应筛选机制AEB,减少无效计算,避免过度精确的误判
字典学习后的稀疏追踪即使在缺乏校准数据支持的情况下仍可能给出精确的原子支撑,尤其在高度相干字典中,不同校准兼容字典会对同一支撑赋予不同物理含义。本文提出感知分辨率的物理支撑推断方法,同时考虑学习字典与部署信号表示的不确定性。通过跨字典置信对应保留校准兼容字典与部署兼容稀疏表示,并将剩余解释投影至物理支撑空间。对于分离尺度为s的局部相干原子类,一旦部署数据确定了相干块解释及其原子支撑,基于N个校准信号的最小极大物理分辨率为δ_opt(N,s) ≍ min{s, 1/(√N s²)},相对分辨率由方向信息尺度Ns⁶决定。仅当方向变化无法通过调整激活系数吸收时,重复部署才有助于提升物理定位。计算方面,提出主动端点括号法(AEB),一种自适应有限候选集方法,仅评估可能影响物理报告的候选项,其余则安全粗化或放弃。有限候选集实验(包括四区域合成应用)表明,点估计插值选择器可能产生过度精确的物理推断,而AEB在更少候选评估下避免了无依据的细化。
原文摘要 · Abstract (English)
Sparse pursuit after dictionary learning can yield a precise atom support even when its physical interpretation is not justified by the calibration data, especially for highly coherent dictionaries where alternative calibration-compatible dictionaries may assign different physical meanings to the same selected support. We develop resolution-aware physical-support inference that jointly accounts for uncertainty in the learned dictionary and in the representation of a deployment signal. Our cross-dictionary confidence correspondence retains calibration-compatible dictionaries and deployment-compatible sparse representations, then projects the surviving explanations onto physical-support space. For local coherent-atom classes with separation scale s, once the deployment data resolve the coherent-block explanation and its atom support, the minimax physical resolution from N calibration signals satisfies $δ_{\mathrm{opt}}(N,s)\asymp\min\{s,\frac{1}{\sqrt{N}s^2}\}$, with relative resolution governed by the orientation-information scale $Ns^6$. Deployment replication improves physical localization only when orientation changes cannot be absorbed by adjusting the active coefficients. For computation, we introduce active endpoint bracketing (AEB), an adaptive finite-bank procedure that evaluates only candidates that can still affect the physical report and otherwise safely coarsens or abstains. Finite-bank experiments, including a four-region synthetic application, show that a point-valued plug-in selector can be physically overprecise, whereas AEB avoids unsupported refinement with fewer candidate evaluations.
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