Adapting large pre-trained models to unseen tasks under tight data and compute budgets remains challenging. Meta-learning approaches explicitly learn good initializations, but they require an additional meta-training phase over many tasks, incur high training cost, and can be unstable. At the same time, the number of task-specific pre-trained models continues to grow, yet the question of how to transfer them to new tasks with minimal additional training remains relatively underexplored. We propose BOLT (Basis-Oriented Low-rank Transfer), a framework that reuses existing fine-tuned models not by merging weights, but instead by extracting an orthogonal, task-informed spectral basis and adapting within that subspace. In the offline phase, BOLT collects dominant singular directions from multiple task vectors and orthogonalizes them per layer to form reusable bases. In the online phase, we freeze these bases and train only a small set of diagonal coefficients per layer for the new task, yielding a rank-controlled update with very few trainable parameters. This design provides (i) a strong, training-free initialization for unseen tasks, obtained by pooling source-task coefficients, along with a lightweight rescaling step while leveraging the shared orthogonal bases, and (ii) a parameter-efficient fine-tuning (PEFT) path that, in our experiments, achieves robust performance compared to common PEFT baselines as well as a representative meta-learned initialization. Our results show that constraining adaptation to a task-informed orthogonal subspace provides an effective alternative for unseen-task transfer.

Transformer-based language models exhibit complex and distributed behavior, yet their internal computations remain poorly understood. Existing mechanistic interpretability methods typically treat attention heads and multilayer perceptron layers (MLPs) (the building blocks of a transformer architecture) as indivisible units, overlooking possibilities of functional substructure learned within them. In this work, we introduce a more fine-grained perspective that decomposes these components into orthogonal singular directions, revealing superposed and independent computations within a single head or MLP. We validate our perspective on widely used standard tasks like Indirect Object Identification (IOI), Gender Pronoun (GP), and Greater Than (GT), showing that previously identified canonical functional heads, such as the name mover, encode multiple overlapping subfunctions aligned with distinct singular directions. Nodes in a computational graph, that are previously identified as circuit elements show strong activation along specific low-rank directions, suggesting that meaningful computations reside in compact subspaces. While some directions remain challenging to interpret fully, our results highlight that transformer computations are more distributed, structured, and compositional than previously assumed. This perspective opens new avenues for fine-grained mechanistic interpretability and a deeper understanding of model internals.

Low-Rank Adaptation (LoRA) enables efficient fine-tuning of large language models but suffers from catastrophic forgetting when learned updates interfere with the dominant singular directions that encode essential pre-trained knowledge. We propose Orthogonal Projection LoRA (OPLoRA), a theoretically grounded approach that prevents this interference through double-sided orthogonal projections. By decomposing frozen weights via SVD, OPLoRA constrains LoRA updates to lie entirely within the orthogonal complement of the top-$k$ singular subspace using projections $P_L = I - U_k U_k^\top$ and $P_R = I - V_k V_k^\top$. We prove that this construction exactly preserves the top-$k$ singular triples, providing mathematical guarantees for knowledge retention. To quantify subspace interference, we introduce $ρ_k$, a metric measuring update alignment with dominant directions. Extensive experiments across commonsense reasoning, mathematics, and code generation demonstrate that OPLoRA significantly reduces forgetting while maintaining competitive task-specific performance on LLaMA-2 7B and Qwen2.5 7B, establishing orthogonal projection as an effective mechanism for knowledge preservation in parameter-efficient fine-tuning.

J-PARSE is an algorithm for smooth first-order inverse kinematic control of a serial manipulator near kinematic singularities. The commanded end-effector velocity is interpreted component-wise, according to the available mobility in each dimension of the task space. First, a substitute "Safety" Jacobian matrix is created, keeping the aspect ratio of the manipulability ellipsoid above a threshold value. The desired motion is then projected onto non-singular and singular directions, and the latter projection scaled down by a factor informed by the threshold value. A right-inverse of the non-singular Safety Jacobian is applied to the modified command. In the absence of joint limits and collisions, this ensures safe transition into and out of low-rank configurations, guaranteeing asymptotic stability for reaching target poses within the workspace, and stability for those outside. Velocity control with J-PARSE is benchmarked against approaches from the literature, and shows high accuracy in reaching and leaving singular target poses. By expanding the available workspace of manipulators, the algorithm finds applications in teleoperation, servoing, and learning. Videos and code are available at https://jparse-manip.github.io/.

Large-scale foundation models have demonstrated remarkable versatility across a wide range of downstream tasks. However, fully fine-tuning these models incurs prohibitive computational costs, motivating the development of Parameter-Efficient Fine-Tuning (PEFT) methods such as LoRA, which introduces low-rank updates to pre-trained weights. Despite their empirical success, the underlying mechanisms by which PEFT modifies model parameters remain underexplored. In this work, we present a systematic investigation into the structural changes of weight matrices during fully fine-tuning. Through singular value decomposition (SVD), we reveal that fine-tuning predominantly amplifies the top singular values while leaving the remainder largely intact, suggesting that task-specific knowledge is injected into a low-dimensional subspace. Furthermore, we find that the dominant singular vectors are reoriented in task-specific directions, whereas the non-dominant subspace remains stable. Building on these insights, we propose a novel method that leverages learnable rescaling of top singular directions, enabling precise modulation of the most influential components without disrupting the global structure. Our approach achieves consistent improvements over strong baselines across multiple tasks, highlighting the efficacy of structurally informed fine-tuning.

It can be difficult to tell whether a trained generative model has learned to generate novel examples or has simply memorized a specific set of outputs. In published work, it is common to attempt to address this visually, for example by displaying a generated example and its nearest neighbor(s) in the training set (in, for example, the L2 metric). As any generative model induces a probability density on its output domain, we propose studying this density directly. We first study the geometry of the latent representation and generator, relate this to the output density, and then develop techniques to compute and inspect the output density. As an application, we demonstrate that "memorization" tends to a density made of delta functions concentrated on the memorized examples. We note that without first understanding the geometry, the measurement would be essentially impossible to make.