Discrete history operators: Exact state-space realization for sequence memory
Abstract
A prescribed finite-dimensional analysis of an evolving discrete history need not admit an exact recurrence driven only by its current state and the entering input. We study exact realizations of causal linear discrete history operators (DHOs). Operator factorization and kernel inclusion characterize exact closure, while the cross-projection defect characterizes closure failure for orthonormal analyses. For finite-dimensional history spaces over fixed finite horizons, we distinguish two minimality problems. Reachable task quotients determine the minimum state dimension for preserving prescribed future tasks on reachable histories. Backward invariant envelopes characterize the minimum dimension of exact linear realizations recovering prescribed analyses on full history spaces within the admissible class. The minimum number of additional independent state directions is given by the rank of the propagated closure defect operator restricted to the kernel of the current analysis. For geometrically weighted countable histories, matrix mode transport yields structured exact recurrences and a uniform exponential decay estimate for state transitions. Additional conditions provide uniform Gram bounds and refined decay estimates. A stationary Meixner specialization yields explicit recurrence matrices, an O(N) arithmetic state update, and minimum augmentation dimensions over finite horizons for selected coefficients. Numerical experiments examine recurrence accuracy, state transition bounds, Gram conditions, and minimum realization dimensions.
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BibTeX
@misc{zhou2026discrete,
title = {{Discrete history operators: Exact state-space realization for sequence memory}},
author = {Bin Zhou and Rong Gao and Zhao-Yan Li},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07396},
eprint = {2611.07396},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07396},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Discrete history operators: Exact state-space realization for sequence memory
AU - Bin Zhou
AU - Rong Gao
AU - Zhao-Yan Li
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07396
UR - https://arxiv.org/abs/2611.07396
AB - A prescribed finite-dimensional analysis of an evolving discrete history need not admit an exact recurrence driven only by its current state and the entering input. We study exact realizations of causal linear discrete history operators (DHOs). Operator factorization and kernel inclusion characterize exact closure, while the cross-projection defect characterizes closure failure for orthonormal analyses. For finite-dimensional history spaces over fixed finite horizons, we distinguish two minimality problems. Reachable task quotients determine the minimum state dimension for preserving prescribed future tasks on reachable histories. Backward invariant envelopes characterize the minimum dimension of exact linear realizations recovering prescribed analyses on full history spaces within the admissible class. The minimum number of additional independent state directions is given by the rank of the propagated closure defect operator restricted to the kernel of the current analysis. For geometrically weighted countable histories, matrix mode transport yields structured exact recurrences and a uniform exponential decay estimate for state transitions. Additional conditions provide uniform Gram bounds and refined decay estimates. A stationary Meixner specialization yields explicit recurrence matrices, an O(N) arithmetic state update, and minimum augmentation dimensions over finite horizons for selected coefficients. Numerical experiments examine recurrence accuracy, state transition bounds, Gram conditions, and minimum realization dimensions.
ER -