Physical-Support Confidence Sets for Highly Coherent Dictionaries

What the math gives to ML

The paper's transferable asset is not dictionary learning itself, but its separation between coordinate-level sparsity and physically justified support, together with an exact uncertainty correspondence across calibration-compatible models and deployment-compatible representations. The resolution law \(\delta_{\mathrm{opt}}(N,s)\asymp\min\{s,(\sqrt{N}s^2)^{-1}\}\) gives a principled criterion for when additional calibration data can resolve highly coherent alternatives. A neural analogue is uncertainty-aware sparse routing: retain all router or expert assignments compatible with calibration and the current input, report only the coarsest expert group justified by that set, and use active endpoint bracketing to avoid evaluating candidates that cannot change the result. This is most promising for mixture-of-experts routing, retrieval, and structured prediction where a confident but wrong discrete choice causes large downstream errors.