An arXiv preprint proposes a way to stop sparse-model analyses from claiming more physical precision than their evidence supports when their dictionaries contain highly coherent atoms. The document is version 1, dated 20 August 2026.
Its cross-dictionary confidence correspondence keeps dictionaries compatible with calibration data and sparse representations compatible with deployment data. It then projects the surviving explanations into physical-support space—the physical location or orientation represented by the signal—and reports the finest conclusion they share.
Precision has a physical limit
The continuous model contains one supplied coherent block of q atoms and a separated atom, with q ≥ 4 and n = q + 1. It uses N independent latent sparse-mixture calibration observations and T independent deployment measurements; the calibration coefficients follow a Bernoulli–Gaussian model that combines activation with Gaussian variation.
Under the stated model, the procedure retains the true explanation and physical-support target on the joint calibration and deployment event. If the two error sources are independent, their combined error is α = 1 − (1 − αD)(1 − αT), giving coverage of at least 1 − α.
Once the local explanation and atom support have been resolved, the paper gives the best worst-case physical resolution, up to constant factors, as δopt(N, s) ≍ min{s, 1/(√N s²)}. In practical terms, the remaining resolution follows the smaller of the coherence scale s and the calibration term 1/(√N s²), while relative resolution is governed by Ns⁶. This is a result for the supplied local class, with s treated as given.
Deployment replication does not automatically add orientation information. After the coefficients have been profiled out—allowed to adjust to the data—an orientation change with χtan = 0 is invisible, while χtan > 0 is informative. The matching rate requires the stated restricted-orbit regularity conditions.
Synthetic tests favored caution
A theory-guided fixed-grid diagnostic used a balanced q = 4, r = 2 Bernoulli–Gaussian submodel and an exact 32-component calibration mixture. Equal coefficients served as the deployment-invariance control, while unequal coefficients represented the informative condition. The calculation was a mechanism check, not an independent coverage study.
The diagnostic produced a Jeffreys-divergence scaling slope of 5.935, with R² = 0.99999 and a stability range of 5.925 to 5.947, supporting the proposed sixth-order calibration sensitivity. Its collapse spread was 0.0167 against a prespecified tolerance of 0.015, so that check narrowly missed its target.
A broader four-region synthetic study compared active endpoint bracketing, or AEB, with exhaustive evaluation of the same 216-explanation bank and with a point-valued plug-in selector across 15 fresh datasets. One dataset had an empty exhaustive finite-bank profile.
At the primary cap of 162/216 = 0.75, AEB matched exhaustive results in 10 of 11 A-fine profiles, 5 of 5 B group or sector profiles, 5 of 5 C-ambiguity profiles, and 10 of 10 D-absence profiles. It produced no false D-absence conclusion in 3 controls.
The plug-in selector returned unsupported fine B localization and definitive C-absence conclusions in all 5 of 5 eligible weak-C profiles. AEB abstained on the remaining A-fine profile.
A finite-bank result, not a universal guarantee
The larger global finite-bank study comprised 18 independent cases with 3 predeclared reporting profiles per case, producing 54 reporting traces across low-, intermediate-, and high-information regimes.
At a displayed budget of 0.50, AEB returned substantive reports in 41 of 54 traces, recovered 33 of 34 ambiguity conclusions and 0 of 7 fine conclusions, and produced 0 of 54 unsafe finer-than-reference reports. At 0.75, the corresponding counts were 54 of 54, 34 of 34, 5 of 7, and 0 of 54; the median queried fraction was 0.209 at both budgets.
The numerical evidence is synthetic and finite-bank. AEB certification is deterministic relative to exhaustive evaluation of the same specified bank and statistically valid only for data-generating candidates in that bank; the paper makes no off-bank coverage or continuous-space outer-cover claim. The continuous analysis is likewise for a supplied fixed-dimensional local model.
Within those limits, AEB is designed to certify fine, coarser, or ambiguous same-bank conclusions and to return abstain when a requested report cannot be certified. The method's central message is that physical detail should survive across calibration-compatible dictionaries and deployment-compatible explanations before it is reported as a precise result.
Paper data and sources
Original title: Physical-Support Confidence Sets for Highly Coherent Dictionaries
Authors: Guan-Ju Peng
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text