Year: 2026
Type: preprint
Abstract: A state may need to be compressed before its exact downstream task is known. For a finite family of linear task operators $\{T_u\}_{u\in\U}$, we characterize exactly how much state must survive when only one of $K$ coarse task messages is available at compression time: $$p^*(K)= \min_{\substack{\Pcal\text{ partition of }\U\\|\Pcal|\le K}} \max_{C\in\Pcal}\rank(T_C),$$ where $T_C$ stacks the tasks that remain unresolved within cell $C$. Thus task information reduces retained state precisely by separating tasks whose joint task-visible subspace is expensive. We derive the dual bit frontier, an irreducible common-core floor, and a sharp singular-value characterization for nonzero tolerance. For overlapping task families, a cumulative overlap deficit places the exact frontier within an additive $Δ$ of weighted load balancing. Optimal partitioning is strongly NP-hard, yet direct-sum families admit classical approximation guarantees and a PTAS. We stress-test the theory on learned task operators from CIFAR-10, Fashion-MNIST, SVHN, and Burgers observables. For Burgers with $K=2$ and $\eps=0.35$, a constructive partition attains $p=4$, while balanced random partitions require median $p=6$, with held-out worst standardized RMSE $0.0368$. The results distinguish uniform task sufficiency from distributional and decision-level performance.
Source: arXiv (Cornell University)
Author: Katende, Ronald
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213939074
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