Physical-Support Confidence Sets for Highly Coherent Dictionaries

Guan-Ju Peng

cs.LGAug 20, 2026

Physical-Support Confidence Sets for Highly Coherent Dictionaries

Authors: Guan-Ju Peng

Organizations: Institute of Data Science and Information Computing, National Chung Hsing University, Taichung, Taiwan

Abstract

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.

Figures & tables

Panel of Figure 1 — Figure 1: Resolution-aware physical-support inference after dictionary learning. (a) illustrates the retain–project–coarsen principle for joint dictionary–support–coefficient explanations. (b) specializes the framework to one supplied coherent block and shows how unresolved deployment and calibration uncertainty limit the attainable physical resolution.

Panel of Figure 2 — Figure 2: Theory-guided check of the information mechanism on the stored controlled calculation grid. This is a mechanism illustration, not a numerical coverage study for the continuous confidence correspondence.

Panel of Figure 3 — Figure 3: Four-region physical reports relative to the represented 216-explanation bank. The representative weak-C report shows that AEB preserves the exhaustive same-bank group/ambiguity resolution while the point-valued plug-in gives unsupported fine B localization and definite C absence; the aggregate panel shows both discrepancies in all five eligible completed weak-C profiles. The study uses 15 fresh datasets, one of which has an empty exhaustive finite-bank profile.

Figure 4: Global AEB status relative to exhaustive same-bank evaluation over 18 independent cases and three predeclared profiles per case (54 reporting traces). At each of the six stored query budgets, exact-reference match, safe coarsening, abstention, and unsafe finer-than-reference counts sum to 54; the unsafe count is zero throughout.

Diagnostic	Result	Interpretation
Jeffreys $s$ -exponent	5.9355; central 95% paired-batch MC range [5.9249, 5.9468]	sixth-order scaling; numerical stability
Jeffreys log–log $R^{2}$	0.99999	stable scaling
Order-14 quadrature exponent	5.9342	independent check
Maximum cross- $s$ affinity spread	0.0243	approximate collapse
Median transition-band spread	0.0167	finite- $s$ drift
Prespecified spread tolerance	0.015	not met

Table S1: Theory-guided numerical diagnostics. Monte Carlo intervals use the 128 paired batches; quadrature and deployment checks are deterministic.

Figure S1: Across-scale orientation-information collapse diagnostic underlying Figure 2 . At each stored

I_{D}=Ns^{6}

, the plotted vertical spread is the maximum minus the minimum product affinity across the saved separation scales. Over the prespecified transition budgets

I_{D}\in\{4,8,16,32,64\}

, the median spread is

0.0167

, which narrowly exceeds the prespecified tolerance

0.015

. The criterion is therefore recorded as missed and is not used as evidence of numerical coverage for the continuous confidence correspondence.

Figure S2: Coefficient-profiled deployment residual at

s=0.1

. Equal coefficients are invariant to numerical precision, whereas nonzero coefficient contrasts follow the closed form in Equation ( 54 ).

e\in\operatornamewithlimits{arg\,min}_{\begin{subarray}{c}e^{\prime}\in U\\ e^{\prime}\text{ unqueried}\end{subarray}}\pi_{\mathcal{G}}(e^{\prime};L,U)

Algorithm S1 Active endpoint bracketing on a finite explanation bank

Condition	$N$	$s$	$\nu_{\rm truth}$	Bank size	Cases
Low information	4096	0.35	1.20	1025	6
Intermediate information	65536	0.50	0.90	1025	6
High information	131072	0.50	0.80	369	6

Table S2: Global finite-bank evaluation conditions.

Outcome relative to exhaustive same-bank reference	Budget $0.50$	Budget $0.75$
Non-abstaining (substantive) report	$41/54$	$54/54$
Ambiguity conclusion recovered	$33/34$	$34/34$
Fine conclusion recovered	$0/7$	$5/7$
Unsafe finer-than-reference conclusion	$0/54$	$0/54$
Median queried bank fraction	$0.209$	$0.209$

Table S3: Global AEB outcomes over 18 independent cases and 54 reporting traces, relative to exhaustive scoring of the same finite candidate bank.

Figure S3: Global AEB recovery at query budgets

0.50

and

0.75

. The 54 reporting traces arise from three predeclared profiles applied to 18 independent cases; the recovery denominators are 34 exhaustive-bank ambiguity traces and seven exhaustive-bank fine traces.

Panel of Figure 12 — Figure S4: Four-region finite-bank construction used for the application protocol. The library contains 12 atoms arranged into the four physical regions, and the coherence matrix shows the within- and between-region atom similarities that determine the available physical resolution.

Region / phenomenon	Exhaustive finite-bank reference	AEB outcome relative to reference	Point-valued plug-in selector
A: isolated persistent	Fine localization in 11 eligible main profiles	Fine: $10/11$	—
B: highly coherent persistent	Group/sector level in five eligible completed weak-C profiles	Group/sector: $5/5$	Unsupported fine localization: $5/5$
C: weak optional	Support ambiguity in the same five eligible completed weak-C profiles	Support ambiguity: $5/5$	False certainty: $5/5$
D: represented-scale absence	Absence at the represented scale in 10 eligible main profiles	Absence: $10/10$	—
D-present controls	D present in all three eligible controls	False absence: $0/3$	—

Table S4: Physical conclusions across 15 fresh datasets in the four-region synthetic application. Every reference conclusion is obtained by exhaustive evaluation of the same finite 216-explanation bank.

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Honest Physical-Support Inference after Latent Dictionary Learning: Collision Singularities and Minimax Resolution

Guan-Ju Peng

Institute of Data Science and Information Computing, National Chung Hsing University.

Sparse Recovery Dictionary Learning

Open paper AskPDF arXiv

Sparse-support uncertainty is usually quantified by treating the dictionary as known, an assumption that can produce overconfident, label-dependent conclusions when the dictionary is learned from latent sparse mixtures. Near collisions of coherent atoms, a test signal may identify the active physical group even though the training data cannot distinguish the physical rays within it. We develop inference for active physical rays, unit atoms modulo sign, after latent dictionary learning. In a fixed-dimensional Gaussian train-test experiment, we retain all dictionaries compatible with a robust training-moment region, profile the test representation over them, and project surviving configurations onto a permutation-invariant support space. The resulting confidence correspondence can report cross-sheet inconclusiveness, group resolution with child ambiguity, or fine-support resolution. We characterize both its statistical cost and decision-theoretic benefit. Residual block orientation first affects the latent training density at cubic order, yielding information of order

s^6

, where

s

is the within-block collision scale. The correspondence provides high-probability-over-training conditional test coverage, with resolution governed separately by parent detectability, test-time support separation, and learned-dictionary orientation. In the resolved fixed-shell regime, its projective Hausdorff diameter contracts at the minimax-optimal rate

s \wedge (\sqrt{N}s^2)^{-1}

, up to constants. A restricted-task theorem further determines when coefficient asymmetry allows test replication to supplement training information and when calibration uncertainty remains irreducible. The framework thus yields honest, resolution-adaptive support statements and guides the allocation of training versus test measurements.

Abstract

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Honest Physical-Support Inference after Latent Dictionary Learning: Collision Singularities and Minimax Resolution

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