Geo-LoRA: Geometry-Aware Subspace Evolution for Low-Rank Adaptation in Continual Learning

Continual Learning (CL) requires models to sequentially adapt to new tasks without forgetting old knowledge. Recently, Low-Rank Adaptation (LoRA), a representative Parameter-Efficient Fine-Tuning (PEFT) method, has gained increasing attention in CL. Several LoRA-based CL methods reduce interference across tasks by separating their update spaces, typically building the new space from the estimated null space of past tasks. However, they (i) overlook task-shared directions, which suppresses knowledge transfer, and (ii) fail to capture truly effective task-specific directions since these ``null bases" of old tasks can remain nearly inactive for new task under correlated tasks. To address this, we study LoRA learning capability from a projection energy perspective, and propose Low-rank Decomposition and Adaptation (LoDA). It performs a task-driven decomposition to build general and truly task-specific LoRA subspaces by solving two energy-based objectives, decoupling directions for knowledge sharing and isolation. LoDA fixes LoRA down-projections on two subspaces and learns robust up-projections via a Gradient-Aligned Optimization (GAO) approach. After each task, before integrating the LoRA updates into the backbone, LoDA derives a closed-form recalibration for the general update, approximating a feature-level joint optimum along this task-shared direction. Experiments indicate that LoDA outperforms existing CL methods. Our code is available at https://github.com/HHHLF/LoDA_ICML2026.

Pre-trained visual models have become fundamental in computer vision, but they face challenges in continual learning scenarios where data and tasks evolve over time. Low-Rank Adaptation (LoRA) offers efficient fine-tuning capabilities but remains limited for such dynamic environments. Standard LoRA cannot distinguish important subspaces, causing critical knowledge to be overwritten in sequential training. Existing approaches address this by dynamically expanding the set of LoRA adapters, either maintaining a growing pool of task-specific modules or merging new adapters into prior ones, at the cost of unbounded parameter growth or increasing inference complexity. We propose Continual Low-Rank Adaptation (C-LoRA), a method that enables a single, shared LoRA adapter to handle sequential tasks without catastrophic forgetting, without requiring any module selection or fusion at inference. The core of C-LoRA is a learnable routing matrix R that explicitly controls how each rank-one subspace contributes to the weight update. This matrix is decomposed into a stability component (R_base), which preserves knowledge from prior tasks, and a plasticity component (R_delta), which drives adaptation to the current task, providing direct control over the stability-plasticity trade-off. We analyze how R governs gradient flow during sequential training, and demonstrate competitive performance across multiple benchmarks.

While orthogonal subspace methods try to mitigate task interference in Continual Learning (CL), they often suffer from energy diffusion across the basis, hindering knowledge compaction and exhausting capacity for future tasks. We observe that output feature drift induced by parameter updates is inherently low-rank, and theoretically prove that preserving parameters along the principal directions of this drift minimizes the output reconstruction error. Motivated by this, we propose Energy-Concentrated and Energy-Ordered Low-Rank Adaptation (E$^2$-LoRA). By explicitly ordering and concentrating knowledge into leading ranks, E$^2$-LoRA frees capacity for subsequent tasks. Furthermore, we design a dynamic rank allocation strategy to balance stability and plasticity by jointly optimizing energy retention and model plasticity. Extensive experiments across multiple benchmarks demonstrate that E$^2$-LoRA achieves state-of-the-art performance. Code is available at https://github.com/kiddo127/E2-LoRA.