Exact Recovery Under Deterministic Partial Views: Confusability Graphs, Strong Powers, and Capacity
arXiv:2602.23520v4 Announce Type: replace-cross Abstract: We study exact recovery from deterministic partial views of a finite latent tuple. A family of admissible views induces a confusability graph on latent states, and this graph is the structural object governing zero-error recovery. In the exact coordinate-view model on the full labeled tuple space, we characterize the realizable confusability relations exactly: they are precisely those determined by upward-closed families of coordinate-agreement sets. We show that exact recovery with a $T$-ary auxiliary tag is equivalent to $T$-colorability of the induced graph, while exact recovery on a designated success set is equivalent to colorability of the corresponding induced subgraph. Under repeated composition, the block confusability grap
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Abstract:We study exact recovery from deterministic partial views of a finite latent tuple. A family of admissible views induces a confusability graph on latent states, and this graph is the structural object governing zero-error recovery. In the exact coordinate-view model on the full labeled tuple space, we characterize the realizable confusability relations exactly: they are precisely those determined by upward-closed families of coordinate-agreement sets. We show that exact recovery with a $T$-ary auxiliary tag is equivalent to $T$-colorability of the induced graph, while exact recovery on a designated success set is equivalent to colorability of the corresponding induced subgraph. Under repeated composition, the block confusability graph is the strong power of the one-shot graph, so the normalized zero-error rates converge to the Shannon capacity of the induced graph and inherit the standard Lovász-$\vartheta$ upper theory. We also identify a structural equality route: when confusability is transitive, the induced graph collapses to a cluster graph, yielding capacity--$\vartheta$ equality, with meet-witnessing and fiber coherence as sufficient conditions. Finally, under an affine restriction on the realized state family, the coordinate side carries a representable matroid whose rank gives tractable upper bounds on confusability and capacity. A classification of representative channel families shows that the majority of widely deployed deterministic partial-view architectures operate above the zero-incoherence boundary, rendering the graph-capacity limits operationally unavoidable.
Comments: 17 pages, 1 figure, 2 tables. Lean 4 artifact and supplementary available at this https URL
Subjects:
Information Theory (cs.IT); Programming Languages (cs.PL)
MSC classes: 94A17, 94A15, 94A24, 68Q25, 68Q30, 03B35
ACM classes: E.4; F.2.2; G.2.2; D.2.4; F.4.1
Cite as: arXiv:2602.23520 [cs.IT]
(or arXiv:2602.23520v4 [cs.IT] for this version)
https://doi.org/10.48550/arXiv.2602.23520
arXiv-issued DOI via DataCite
Submission history
From: Tristan Simas [view email] [v1] Thu, 26 Feb 2026 21:47:11 UTC (351 KB) [v2] Mon, 2 Mar 2026 06:07:23 UTC (547 KB) [v3] Mon, 16 Mar 2026 23:08:03 UTC (1,114 KB) [v4] Tue, 31 Mar 2026 15:22:22 UTC (751 KB)
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