Continual Learning as Shared-Manifold Continuation Under Compatible Shift

Catastrophic forgetting is often viewed as the destruction of previously learned knowledge during sequential learning. Building on the Accessibility Collapse framework, we investigate the geometric structure of recoverability in continual learning. Using Split CIFAR-100 and a sequentially trained ResNet-18, we analyze recoverability, representational drift, and recovery complexity across ten tasks. We introduce Recovery Subspace Dimensionality (k_t), a measure of the minimum number of singular directions required to preserve 90 percent of full probe performance. Contrary to our Recoverability Diffusion hypothesis, recovery dimensionality remains stable throughout training (mean k_t = 8.0) despite substantial representational drift. Principal-angle drift strongly predicts recoverability (r = -0.862), and a simple geometric model explains 82.2 percent of recoverability variance. These findings support the Stable Recovery Manifold hypothesis, suggesting that forgotten knowledge remains compactly decodable despite representational reorganization. The results indicate that catastrophic forgetting is primarily an accessibility and manifold-alignment problem rather than information destruction.

Rehearsal-free class-incremental learning (CIL) with LoRA adapters remains challenging because the low-rank subspaces updated across tasks evolve without geometric control, causing unstable shared representations and repetitive collapse of task-specific updates into previously occupied directions. We introduce Geo-LoRA, a geometry-aware framework that explicitly regulates how low-rank subspaces, both shared and task-specific, evolve during continual learning. For the shared branch, Subspace Projection Preservation (SPP) constrains consecutive updates to follow smooth trajectories on the Grassmann manifold, and Adaptive Core-Slack Alignment (ACSA) decomposes transitions into principal and residual components, aligning the former while modulating the latter to balance stability and plasticity. For the task-specific branch, Median-Calibrated Block Overlap (MCBO) imposes a statistical constraint via normalized projection overlap, penalizing excessive reuse to mitigate subspace crowding. These constraints jointly regulate the evolution of all LoRA subspaces across layers and tasks without introducing additional adapter types beyond standard LoRA. Geo-LoRA provides a principled geometric formulation for continual low-rank adaptation and consistently achieves state-of-the-art performance across multiple benchmark datasets and different task lengths.

Catastrophic forgetting remains a fundamental obstacle to continual learning, where neural networks lose previously acquired knowledge while learning new tasks. Existing methods primarily mitigate forgetting through parameter regularization or experience replay, while the representation-space dynamics associated with forgetting remain less understood. We investigate latent representation evolution during sequential learning and introduce representation flux, a geometric measure of sample-level representation displacement across training. We show that representation flux is strongly associated with catastrophic forgetting across multiple benchmarks, with temporal analyses indicating that elevated flux can precede subsequent performance degradation. Representation displacement is also associated with confidence degradation, while complementary geometric properties provide additional information about sample-level forgetting. Motivated by these observations, we propose FlowLess-R, a representation-space regularization method that constrains replay representations relative to stored references while allowing continued learning. FlowLess-R is architecture-agnostic and integrates into replay-based methods through a representation-matching term. Experiments on SplitMNIST, SplitFashionMNIST, SplitCIFAR10, and SplitTinyImageNet show improved final average accuracy and reduced forgetting with ER, DER++, and ER-ACE. Our results identify representation flux as an informative geometric marker of forgetting and show that stabilizing latent representations provides a simple strategy for mitigating catastrophic forgetting.