# Finite-Horizon Fisher Memory in Two-Sided Power-Bounded Recurrent Systems

arXiv: https://arxiv.org/abs/2609.39800 · HTML: https://mlsanity.com/paper/2609.39800/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 52 references that are in the map.
- First posted on arXiv 30 Sep 2026.

## Builds on (most cited first, 17 of 17)

- [Titans: Learning to Memorize at Test Time](https://mlsanity.com/paper/2501.00663/) (2024)
- [xLSTM: Extended Long Short-Term Memory](https://mlsanity.com/paper/2405.04517/) (2024)
- [Using Fast Weights to Attend to the Recent Past](https://mlsanity.com/paper/1610.06258/) (2016)
- [Recurrent Kalman Networks: Factorized Inference in High-Dimensional Deep Feature Spaces](https://mlsanity.com/paper/1905.07357/) (2019)
- [UnICORNN: A recurrent model for learning very long time dependencies](https://mlsanity.com/paper/2103.05487/) (2021)
- [Non-normal Recurrent Neural Network (nnRNN): learning long time dependencies while improving expressivity with transient dynamics](https://mlsanity.com/paper/1905.12080/) (2019)
- [On the Curse of Memory in Recurrent Neural Networks: Approximation and Optimization Analysis](https://mlsanity.com/paper/2009.07799/) (2020)
- [Reversible Recurrent Neural Networks](https://mlsanity.com/paper/1810.10999/) (2018)
- [Improved memory in recurrent neural networks with sequential non-normal dynamics](https://mlsanity.com/paper/1905.13715/) (2019)
- [Covariance inflation in the ensemble Kalman filter: a residual nudging perspective and some implications](https://mlsanity.com/paper/1305.4496/) (2013)
- [Robust information propagation through noisy neural circuits](https://mlsanity.com/paper/1608.05706/) (2016)
- [How noise affects memory in linear recurrent networks](https://mlsanity.com/paper/2409.03187/) (2024)
- [Efficient Communication over Complex Dynamical Networks: The Role of Matrix Non-Normality](https://mlsanity.com/paper/1904.02447/) (2019)
- [Optimal Memory Encoding Through Fluctuation-Response Structure](https://mlsanity.com/paper/2603.21666/) (2026)
- [Memory Uncertainty Relation and Harmonic Memory in Random Recurrent Networks](https://mlsanity.com/paper/2605.24628/) (2026)
- [Lindblad-Inspired Multi-Timescale Reservoir Computing with Separable Rotation and Dissipation](https://mlsanity.com/paper/2608.04028/) (2026)
- [Universal Estimation of the Fisher Information for Processes with Memory](https://mlsanity.com/paper/2609.14582/) (2026)

## Abstract (arXiv)

We analyse allocation, admission and post-write retention in finite-horizon linear-Gaussian noisy recurrent memories. At every horizon, the directional Fisher memory Mn satisfies operatornametrMn=N: non-normality redistributes information but cannot raise its spherical average, while normal carriers satisfy Mn=I. For bi-power-bounded carriers, we derive uniform 1/n lag bounds, identify the limit of Mn with the inverse of the classical Ces`aro asymptotic limit of W^top, and give finite-horizon error bounds. A time-varying coupling defines an end-to-end store operator. The writer-optimal direction need not be store-optimal. After writing ends, an invertible hold preserves the full stored Fisher matrix. Additive contamination bounded by α times the closure covariance retains at least 1/(1+α) of that matrix; a covariance-aware decoder attains the corresponding accuracy. With recurrent carriers held fixed, training input masks and linear readouts approached the task-specific optimum in 160 runs, with median normalized Rayleigh efficiency above 0.998. Binary accuracy matched the Gaussian prediction to mean absolute error below 0.002 over more than four orders of magnitude in J. In a separate pre-specified study of 320 runs, trained masks followed the designated input-time objective in both carrier types, in 16 of 16 draws. These studies used development-seen carriers and are pre-specified validations, not blind holdouts. The same fixed design reproduced the objective-specific result in 16 of 16 draws on carriers unused before run commitment. Exact isolation preserved information, while a decoder fixed at its training horizon fell to chance; inverse-adjoint transport restored its sampled decisions to numerical precision.