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

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.

Graph-Regularized Low-Rank Matrix Completion by Variable Projection

We address the low-rank matrix completion problem by incorporating graph regularization into the existing Riemannian Trust-Region Matrix Completion (RTRMC) framework. The latter uses the geometry of the low-rank constraint to remodel the problem as an unconstrained optimization problem on a single Grassmann manifold. Our approach, named Graph-Regularized RTRMC (GR-RTRMC), exploits the inherent relationships between rows and columns of the matrix. By using these relationships, we aim to improve the accuracy and robustness of matrix completion, particularly in scenarios where the underlying data exhibits strong correlations between rows or columns.

Scale-Calibrated Median-of-Means for Robust Distributed Principal Component Analysis

Distributed principal component analysis (PCA) produces node-level estimates of both a mean vector and a principal subspace. Robustly aggregating these heterogeneous objects requires a relative scale between mean error and subspace error. We study a scale-calibrated median-of-means estimator for this problem using the product geometry of Euclidean space and the Grassmann manifold. A node-level PCA expansion shows that the mean component has the usual linear influence, whereas the subspace component is an eigengap-weighted covariance perturbation. We prove a local reduction showing that the proposed product-manifold median-of-means estimator is asymptotically equivalent to a scaled spatial median of node influence errors. This yields fixed-node non-Gaussian limits, growing-node Gaussian limits with finite-block bias, and an explicit scale-dependent covariance formula. We propose robust block-scale and inference-optimal calibration rules, establish high-probability median-of-means bounds, characterize factorwise bad-node influence, and prove node-bootstrap validity. Simulations and large-scale single-cell RNA-seq data show that scale calibration adapts to eigengap-driven subspace uncertainty and provides a robust distributed PCA summary.