Nonlinear LoRA

Personalizing text-to-image diffusion models from a few reference images requires preserving subject identity while following prompts that describe new contexts. Full-model fine-tuning is parameter-intensive, whereas low-rank adaptation (LoRA) reduces the number of trainable parameters but leaves open how adaptation capacity should be distributed across layers. We introduce Diffu-LoRA, a parameter-efficient method that learns this allocation through gated low-rank adaptation. Diffu-LoRA inserts trainable low-rank components into the linear layers of Transformer blocks and assigns a learnable gate to each component. Bilevel optimization updates the adaptation weights and gate parameters on separate data splits, while progressive pruning removes components with the lowest gate values to meet a prescribed rank budget. This procedure allocates adaptation capacity nonuniformly across layers while keeping the pretrained backbone frozen. Experiments with Stable Diffusion on subjects from DreamBooth and additional collected datasets show improved overall subject fidelity and prompt alignment relative to the evaluated fine-tuning baselines. Ablation studies examine the contributions of bilevel optimization, progressive pruning, and adapter placement. These results support learned rank allocation as a practical approach to parameter-efficient diffusion model personalization.

Low-rank adaptation (LoRA) is commonly viewed as an update-space approximation to full fine-tuning, yet this view is incomplete for self-gated Transformer feed-forward networks. In gated FFNs, a low-rank residual can change not only projected features but also the nonlinear selection weights that determine which channels contribute to the output. We formalize this effect as selection misalignment and connect it to the local effective homogeneity of self-gated activations. This motivates a nonlinearity-aware principle for parameter-efficient fine-tuning: low-rank updates should allocate capacity to gate channels whose nonlinear states remain responsive and should shape the temporal evolution of selection. We propose NA-LoRA, a training-only method with two lightweight mechanisms: a derivative-based temporal-importance mask for gate-related LoRA updates and an activation-specific step-scaling rule when a meaningful coarse effective-homogeneity partition is available. NA-LoRA adds no auxiliary loss and incurs no inference-time overhead. Experiments on language-model fine-tuning and vision-language transfer benchmarks show that NA-LoRA consistently improves over vanilla LoRA and is competitive with or better than strong PEFT variants.

Medical multiple-choice question answering requires parameter-efficient adaptation across heterogeneous knowledge domains and reasoning operations. A medication question, a diagnostic decision, a public-health item, and a nursing-action item may require different low-rank updates, while some recall items should preserve the base model's representation with only mild adapter intervention. We propose BiRG-LoRA, a single-adapter rank-gated LoRA method for medical question answering. BiRG-LoRA keeps one LoRA module per target layer but makes its rank dimension input-conditioned: for each question, a biaxial gate combines hidden semantic evidence with specialty/profession priors, clinical-operation priors, and their interaction to select a sparse top-$k$ subset of rank atoms. A scalar injection coefficient further controls the strength of the selected adapter update. Under a matched Qwen3-8B CMB-source protocol, BiRG-LoRA achieves the highest four-benchmark macro-average accuracy among trainable PEFT baselines and matched routing controls: 69.31% averaged over CMB, CMExam, MedQA, and MedMCQA. It improves over MoELoRA by 0.89 percentage points while using 28.1% fewer trainable parameters; a paired, benchmark-stratified bootstrap over final predictions gives a 95% confidence interval of [0.42, 1.37] for this macro-average gain. Basic controls show that BiRG-LoRA also improves over vanilla LoRA r16 and active-rank-matched LoRA r4 by 0.83 macro points, and an evaluation-time weak-axis perturbation check suggests that performance is not brittle to moderate tag noise. The results support a bounded claim: clinically structured rank allocation improves cross-benchmark medical QA under a matched single-seed protocol, while training-seed variance remains future work.

Low-Rank Adaptation (LoRA) is a popular parameter-efficient fine-tuning (PEFT) method that restricts weight updates to low-rank adapters, introducing a fixed low-rank inductive bias by optimizing in a low-dimensional subspace. In this work, we question whether a fixed-rank constraint is the most effective inductive bias for parameter-efficient fine-tuning. We introduce Learnable Rank LoRA (LR-LoRA), a PEFT method in which the adapter rank is learned during the training process. Instead of prescribing a uniform rank for all adapter layers, LR-LoRA allows the optimizer to determine the appropriate rank for each layer. Using this approach, we find substantial layer-wise variation in the learned ranks, with the attention and MLP layers in the transformer models exhibiting systematically different rank preferences. Across a range of language understanding and commonsense reasoning benchmarks, LR-LoRA achieves state-of-the-art performance in most settings and consistently outperforms strong PEFT baselines, demonstrating that a learnable rank provides a more flexible and effective inductive bias than fixed-rank adaptations.

With the rise in scale for deep learning models to billions of parameters, the computational cost of fine-tuning remains a significant barrier to deployment. While Low-Rank Adaptation (LoRA) has become the standard for parameter-efficient fine-tuning, the need to set a predefined, static rank $r$ requires exhaustive grid searches to balance efficiency and performance. Existing rank-adaptive solutions such as DyLoRA mitigate this by sampling ranks during the training from a predefined distribution. However, they often yield sub-optimal results at higher ranks due to lack of consistent gradient signals across the full hierarchy of ranks, thus making these methods data-inefficient. In this paper, we propose MatryoshkaLoRA, a general, Matryoshka-inspired training framework for LoRA that learns accurate hierarchical low-rank representations by inserting a fixed, carefully crafted diagonal matrix $P$ between the existing LoRA adapters to scale their sub-ranks accordingly. By introducing this simple modification, our general framework recovers LoRA and DyLoRA only by changing $P$ and ensures all sub-ranks embed the available gradient information efficiently. Our MatryoshkaLoRA supports dynamic rank selection with minimal degradation in accuracy. We further propose Area Under the Rank Accuracy Curve (AURAC), a metric that consistently evaluates the performance of hierarchical low-rank adapters. Our results demonstrate that MatryoshkaLoRA learns more accurate hierarchical low-rank representations than prior rank-adaptive approaches and achieves superior accuracy-performance trade-offs across ranks on the evaluated datasets. Our code is available at https://github.com/IST-DASLab/MatryoshkaLoRA.

Low-rank adaptation (LoRA) is a widely used strategy for efficient fine-tuning of large language models (LLMs), but its strictly linear structure fundamentally limits expressive capacity. The bilinear formulation of weight updates captures only first-order dependencies between low-rank factors, restricting the modeling of nonlinear and higher-order parameter interactions. In this paper, we propose Polynomial Expansion Rank Adaptation (PERA), a novel method that introduces structured polynomial expansion directly into the low-rank factor space. By expanding each low-rank factor to synthesize high-order interaction terms before composition, PERA transforms the adaptation space into a polynomial manifold capable of modeling richer nonlinear coupling without increasing rank or inference cost. We provide theoretical analysis demonstrating that PERA offers enhanced expressive capacity and more effective feature utilization compare to existing linear adaptation approaches. Empirically, PERA consistently outperforms state-of-the-art methods across diverse benchmarks. Notably, our experiments show that incorporating high-order nonlinear components particularly square terms is crucial for enhancing expressive capacity and maintaining strong and robust performance under various rank settings. Our code is available at https://github.com/zhangwenhao6/PERA

Adapting models pre-trained on large-scale datasets is a proven way to reach strong performance quickly for down-stream tasks. However, the growth of state-of-the-art mod-els makes traditional full fine-tuning unsuitable and difficult, especially for multi-task learning (MTL) where cost scales with the number of tasks. As a result, recent studies investigate parameter-efficient fine-tuning (PEFT) using low-rank adaptation to significantly reduce the number of trainable parameters. However, these existing methods use a single, fixed rank, which may not be optimal for differ-ent tasks or positions in the MTL architecture. Moreover, these methods fail to learn spatial information that cap-tures inter-task relationships and helps to improve diverse task predictions. This paper introduces Frequency-Aware and Automatic Rank (FAAR) for efficient MTL fine-tuning. Our method introduces Performance-Driven Rank Shrink-ing (PDRS) to allocate the optimal rank per adapter location and per task. Moreover, by analyzing the image frequency spectrum, FAAR proposes a Task-Spectral Pyramidal Decoder (TS-PD) that injects input-specific context into spatial bias learning to better reflect cross-task relationships. Experiments performed on dense visual task benchmarks show the superiority of our method in terms of both accuracy and efficiency compared to other PEFT methods in MTL. FAAR reduces the number of parameters by up to 9 times compared to traditional MTL fine-tuning whilst improving overall performance. Our code is available.

Vision language action models (VLAs) are increasingly used for Physical AI, but deploying a pre-trained VLA model to unseen environments, embodiments, or tasks still requires adaptation. Parameter-efficient fine-tuning (PEFT), especially LoRA, is common for VLA policies, yet the exposed capacity knob, the rank, does not transfer uniformly: robotics transfer exhibits a higher and task-varying intrinsic rank than language fine-tuning. Small ranks suffice for LLMs (e.g., $r \in \{4, 8\}$), while spectral analyses indicate VLAs may require much larger ranks (e.g., $r \approx 128$) or near-full rank, a mismatch that worsens in multi-task settings. We present LoRA-SP (Select-Prune), a rank-adaptive fine-tuning method that replaces fixed-rank updates with input- and layer-wise capacity. LoRA-SP uses an SVD-style parameterization with a small router whose nonnegative scores act as singular values over a shared vector bank. The active set is chosen by an energy target on the cumulative squared scores $E(k) \ge η$, providing a direct link to approximation error via our spectral analysis. During training, $η$ concentrates energy on a few directions and teaches the router to rely on fewer vectors while preserving accuracy. This yields compact adapters that reduce cross-task interference and improve generalization. On four real-robot manipulation tasks collected on an unseen AgileX PiPER arm, across two VLA backbones ($π_0$ and SmolVLA), LoRA-SP matches or exceeds full fine-tuning with far fewer trainable parameters, and improves multi-task success by up to 31.6% over standard LoRA while remaining robust to rank choice.

Low-Rank Adaptation (LoRA) dominates parameter-efficient fine-tuning (PEFT). However, it faces a ``linear ceiling'': increasing the rank yields diminishing returns in expressive capacity due to linear constraints. We introduce CeRA (Capacity-enhanced Rank Adaptation), a weight-level parallel adapter that injects SiLU gating and dropout to induce non-linearity during inference, thereby placing it in a different function class from adapters whose non-linearity exists during training and collapses to an affine map at inference time. On both the basic arithmetic (GSM8K) and the complex MATH benchmark, CeRA is markedly more parameter-efficient. Across a full rank $\times$ learning rate sweep, CeRA at rank 64 achieves the highest MATH pass@1 of any configuration in the grid (23.6%), matching or exceeding both a rank-512 LoRA (22.4%) and DoRA (19.8%) while using only 1/8 of the parameter budget. With the rank and learning rate fixed, CeRA equals or outperforms LoRA in 10 of 12 matched settings. Spectrally, CeRA's learned updates utilize the singular-value spectrum more broadly than linear adapters, which exhibit rank collapse at high rank, although a scale-matched control shows that this difference stems mostly from output scale and partially from non-linearity. Additionally, dropout appears to contribute to regularization rather than rank expansion. We release the code for reproducibility: https://github.com/hhchen1105/cera.

Low-rank adaptation (LoRA) approximates the update of a pretrained weight matrix using the product of two low-rank matrices. However, standard LoRA follows an explicit-rank paradigm, where increasing model capacity requires adding more rows or columns (i.e., basis vectors) to the low-rank matrices, leading to substantial parameter growth. In this paper, we find that these basis vectors exhibit significant parameter redundancy and can be compactly represented by lightweight nonlinear functions. Therefore, we propose Generative Low-Rank Adapter (GenLoRA), which replaces explicit basis vector storage with nonlinear basis vector generation. Specifically, GenLoRA maintains a latent vector for each low-rank matrix and employs a set of lightweight radial basis functions (RBFs) to synthesize the basis vectors. Each RBF requires far fewer parameters than an explicit basis vector, enabling higher parameter efficiency in GenLoRA. Extensive experiments across multiple datasets and architectures show that GenLoRA attains higher effective LoRA ranks under smaller parameter budgets, resulting in superior fine-tuning performance. The code is available at https://anonymous.4open.science/r/GenLoRA.