Optimal Memory Encoding Through Fluctuation-Response Structure

Lianxiang Cui; Kohei Nakajima; Kazuyuki Aihara

cs.NEMar 23, 2026

Optimal Memory Encoding Through Fluctuation-Response Structure

Authors: Lianxiang Cui, Kohei Nakajima, Kazuyuki Aihara

Organizations: International Research Center for Neurointelligence, University of Tokyo Institutes for Advanced Study, University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan · Graduate School of Information Science and Technology, University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan

Abstract

Physical reservoir computing exploits the intrinsic dynamics of physical systems for information processing, while keeping the internal dynamics fixed and training only linear readouts; yet the role of input encoding remains poorly understood. We show that optimal input encoding is a geometric problem governed by the system's fluctuation-response structure. By measuring steady-state fluctuations and linear response, we derive an analytical criterion for the input direction that maximizes task-specific linear memory under a fixed power constraint, termed Response-based Optimal Memory Encoding (ROME). Backpropagation-based encoder optimization is shown to be equivalent to ROME, revealing a trade-off between task-dependent feature mixing and intrinsic noise. We apply ROME to various reservoir platforms, including spin-wave waveguides and spiking neural networks, demonstrating effective encoder design across physical and neuromorphic reservoirs, even in non-differentiable systems.

Figures & tables

Figure 1: Numerical demonstrations of ROME. (a) Polar plot of the MF for a linear reservoir, evaluated in the plane spanned by the task-only direction (blue) and the ROME-optimal direction (red); gray curve: MF landscape; black line: noise-minimizing direction projected onto this plane. (b) MF versus delay $k$ for a linear reservoir with different linear memory tasks. Blue curve and shaded band: mean $\pm$ s.d. over random encoders; colored curves: short-term ( $k=2,3,4$ ; orange) and long-term ( $k=20,22,24$ ; green) ROME encoders, MC-optimal encoder (black, $\mathrm{MC}=\sum_{k}\mathrm{MF}(k)$ ), and single-delay–optimal encoder (red). (c) NARMA10 performance $R^{2}$ versus input power $P$ for a nonlinear ESN. Blue circles and shaded band: mean $\pm$ s.d. over random encoders; Red squares: ROME encoder. (d) Training evolution for the BP encoder in a linear reservoir (single-delay task). Blue circles: alignment $|\cos(G_{\rm BP},G^{*})|$ between learned input direction and ROME direction; red squares: BP performance $R^{2}$ ; dashed line: $R^{2}$ of the ROME direction $G^{*}$ .

Figure 2: PRC demonstrations of ROME. (a) Schematic illustration of the spin-wave physical reservoir and the performance $R^{2}$ versus delay $k$ on a single-delay memory task. Blue curve and shaded band: mean $\pm$ s.d. over random encoders; red and black: ROME encoders optimized for target delays $k_{0}$ . (b) Performance of the heterogeneous E/I SNN neuromorphic PRC systems on a single-delay memory task. Red dashed line: ROME; blue line: mean over random encoders; gray dotted lines: standard deviation of random encoders.

Reservoir Computing

Reservoir computing uses recurrent dynamical systems to transform temporal inputs into features that support efficient learning with simple readouts.

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Organization of computation in reservoir computing

Mohab Abdalla, Damien Rontani

CentraleSupélec, Université de Lorraine, LMOPS EA-4423 Laboratory, 57070 Metz, France

Reservoir Computing

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Reservoir computing exploits nonlinear dynamical systems to encode temporal inputs into high-dimensional state space representations. Although reservoir performance is often characterized through memory, nonlinearity, and their tradeoff, such aggregate measures do not reveal how task-relevant information is organized within the state space. Here, we introduce an eigen-spectral decomposition framework linking the degree-wise information processing capacity to the corresponding state space modes. As a result, we are able to quantify the degree-wise representation energy, and show that in some cases, substantial amounts of information processing capacity may reside in low-energy modes that are vulnerable to experimental noise. These results suggest that useful reservoir computation depends not only on dimensionality expansion, but also on the geometric organization of task-relevant information, with direct implications on physical reservoir computers.

Abstract

Reservoir Computing

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