The Metric Slingshot: Navigational Reuse as Width-Optimal Structural Decoupling in Continual Learning
Authors: Xin Li
Organizations: Department of Computer Science University at Albany, SUNY Albany, NY 12222, USA
Abstract
The mammalian brain, most extensively studied in rodents and bats, solves an enormous variety of non-spatial cognitive tasks using neural circuitry, including grid cells, place cells, and hippocampal indexing, that originally evolved for physical navigation. We formalize the above observation within the local Urysohn width (LUW) framework for continual learning. The central construct is the \emph{metric slingshot}: a learned embedding φ:X→Z that maps an arbitrary learning problem into a navigational latent space Z where pre-evolved contraction maps (grid cells) already provide the metric machinery, so that only the topological indexing subproblem must be solved de novo. We prove three results. First, the optimal spacing of multi-scale grid cell modules is a geometric series whose ratio is determined by the Ω(wlogw) sample complexity bound of the LUW framework; for ecologically plausible parameters, optimality yields r∗≈1.4--1.7, matching electrophysiological measurements in rodent medial entorhinal cortex. Second, the slingshot preserves the width hierarchy with a Lipschitz-controlled transfer bound: a contractive embedding into a fine-resolution navigational space reduces the effective number of contexts the learner must discover. Third, the anatomical separation of the ventral (what'') and dorsal (where'') visual streams achieves the structural decoupling required by Metric-Topology Factorization (MTF) \emph{by architecture}, without gradient-routing mechanisms. We demonstrate that such metric slingshot is applicable to both perception cognition and motor control. Together, these results provide a unified, complexity-theoretic account of navigation in non-spatial domain, grid cell multi-modularity, and hippocampal-neocortical complementary learning as consequences of a single exaptation principle: metric slingshot.
Continual learning systems face a fundamental geometric obstacle: as experience accumulates on a fixed-capacity manifold, covering numbers grow linearly with time, eventually forcing representational overlap and catastrophic interference. Prevailing approaches attack this problem by \emph{expansion} - projecting into higher-dimensional spaces via kernels, overparameterization, or replay. We argue the solution is the opposite: \emph{contraction}. We formalize abstraction as the \textbf{Urysohn Ladder}, a hierarchy of quotient maps that recursively collapse validated metric neighborhoods into compact tokens, converting unbounded ambient-space search into bounded navigation on a low-dimensional intrinsic scaffold. Geometrically, each collapsed token acts as a shortcut - a region of extreme metric contraction that bridges distant experiences, much like a wormhole in the representational manifold. We establish four results that collectively guarantee \emph{separability} (metric contraction renders nonlinearly entangled structure linearly separable at each quotient level, and this separability propagates faithfully through the entire hierarchy), \emph{bounded capacity} (covering numbers remain O(1) per quotient level, independent of stream length), \emph{stability} (parity-partitioned flow/scaffold subspaces enable unbounded plasticity without catastrophic interference), and \emph{scalability} (inference cost scales with quotient distance, not ambient distance). We validate each claim empirically with pretrained models and real-world datasets. Moreover, we demonstrate the potential of Urysohn Ladder for scalable continual learning via scaffold amortization.
We introduce Neural Subspace Reallocation (NSR), which reframes continual learning as memory management over parameter subspaces. Instead of treating Low-Rank Adaptation (LoRA) modules as disposable per-task adapters, NSR manages them as compressible, retrievable memory units on a frozen backbone through a recurring cycle: (1) compress learned LoRAs via SVD, (2) reserve them in a TaskKnowledgeBank, (3) recall related past LoRAs by embedding similarity to warm-start new or returning tasks, and (4) reallocate the active subspace accordingly, with distillation protecting prior tasks. We prove that in cyclic environments any memoryless allocation policy incurs cumulative regret Omega(T(M-1)Delta_switch) relative to a history-aware policy backed by the Bank (Theorem 1). Empirically, on Split-CIFAR-100 the Bank reduces cyclic recovery time by 10x, exactly as predicted, and on the heterogeneous 5-Datasets benchmark NSR achieves the highest accuracy and the least forgetting, about 9x closer to zero backward transfer than the memoryless heuristics. Crucially, we run a controlled study that isolates which component matters: holding the Bank fixed and varying only the allocation rule, we find that a simple similarity-based retrieval rule matches or beats a learned reinforcement-learning controller (recovering recurring tasks in 0 vs 1.8 steps and reaching equal accuracy). Our central, honest finding is therefore that the memory mechanism -- compression and similarity retrieval -- rather than a learned allocation policy, drives continual-learning performance under fixed capacity. A memory-budget analysis confirms the compressed Bank stays small -- 0.29 MB of parameter memory per task -- so a top-K retention cap bounds the total footprint while preserving fast recovery for retained tasks.
Recent progress in visual navigation has largely been driven by scale: end-to-end policies with hundreds of millions of parameters trained on billions of frames or large-scale simulated data. We ask how much of this scale a single task family actually requires, and what structure can substitute for it. We propose a decomposed navigation model in which operations with known closed-form structure, such as projective geometry, occupancy, and coordinate transforms, are computed analytically and serve as interfaces between three small learned modules: an egress predictor that grounds the episode goal as a local subgoal in the current view, a navigation predictor that estimates a goal-conditioned posterior over where trajectories travel, and an endpoint-pinned residual diffusion generator that samples trajectory shapes from this posterior. The system trains only 0.58M out of a total of 22.7M parameters, on 44k frames in under one GPU-hour, yet approaches the performance of state-of-the-art models on navigation tasks across 6060 point-goal episodes and 60 environments, while having 233x fewer trainable parameters, the lowest collision rate among all evaluated methods, and 50 Hz inference speed. The decomposition further transfers to no-goal exploration by retraining only the 123k-parameter egress head, and its failure modes under sensor corruption are transparent and analytically correctable.