Low-rank compression reduces the cost of pretrained language models by replacing linear transformations with low-rank factorizations. However, conventional methods use a fixed rank allocation during inference, assigning the same amount of compute regardless of the input token. We introduce Low-Rank Conditional Computation (LRCC), which adds token-dependent computation to pretrained models by training one lightweight router per Transformer block to select among a small set of nested low-rank paths. During training, the low-rank factors remain frozen, and only the routers are optimized. We evaluate LRCC on Llama and Qwen models for language modeling and zero-shot downstream tasks. Within the same average active-parameter budget, LRCC improves the predictive performance over static low-rank compression, including a 7.6 percentage-point gain in average downstream accuracy on Llama-2-7B over static methods. At matched batch-size-1 decoding latency, LRCC improves both perplexity and downstream accuracy on Llama-3.2-1B and remains competitive on Llama-2-7B, without specialized kernels. Finally, we assess the usefulness of assigning a token-wise path by analyzing the routers' path choices.
Low-rank decomposition is a promising compression paradigm for large language models (LLMs), yet its effectiveness hinges on rank budget allocation across weight matrices: uniform or hand-crafted rules ignore module-wise importance, while learning-based allocation incurs substantial training overhead. We formulate rank allocation as a global sorting-and-truncation pipeline that scores every singular component by combining local singular energy with global functional importance, estimated via layer-wise input--output cosine similarity on a tiny calibration set. We show, both geometrically and empirically, that high input--output cosine similarity implies low effective rank. We further propose rank-preserving fine-tuning (RPFT), which adapts only a small subset of retained singular components so that the allocated rank stays bounded without re-decomposition. Experimental results show that UniRank cuts zero-shot perplexity by up to 50%, improves average reasoning accuracy by 3.0% over LoRAP at 25% sparsity, and boosts four SVD-based decomposition methods as a plug-and-play module.
Chao Han, Yongjie Du, Junjie Tan +1
Ningbo Institute of Digital Twin, Eastern Institute of Technology, Ningbo
Matrix-level low-rank compression is a promising way to reduce the cost of large language models, but running compression and evaluating the resulting models on language tasks can be prohibitively expensive. Can compression-induced degradation be predicted before committing to this compute? We systematically analyze the Qwen3 and Gemma3 model families across four representative low-rank compression methods: vanilla SVD, two ASVD variants, and SVD-LLM. We find that stable rank and information density, measured in bits per parameter, dominate performance degradation. The interaction term γ⋅ρˉs, defined as compression ratio times stable rank, is a robust predictor of accuracy degradation, achieving leave-one-out cross-validation Pearson correlations of 0.890 for attention layers and 0.839 for MLP layers. We provide theoretical intuition for why this predictor succeeds by connecting it to standard SVD truncation bounds and error composition mechanisms in transformer layers. These findings enable a predict-then-compress workflow: compute γ⋅ρˉs from weights, estimate degradation, and invest compute only in desirable configurations.
Mingxue Xu
Department of Electrical and Electronic Engineering, Imperial College London, London, United Kingdom
Large language models have driven recent progress in language and multimodal AI, yet pre-training them at scale is prohibitively expensive. Low-rank pre-training, which factorizes each weight matrix into a rank-r product to reduce both parameters and FLOPs, is a promising response but typically lags full-rank training in quality. We propose Duplicated Latent Residual (DLR), a training-only, parameter-free, foldable plug-in for low-rank pre-training. DLR augments the standard low-rank output Bz with a fixed structured residual alpha/sqrt(K) * Expand_K(z) that replicates each latent coordinate K = ceil(d_out/r) times across the output. With alpha fixed, DLR adds zero learnable parameters per layer; after training, it is absorbed into the up-projection in closed form, B* = B + alpha/sqrt(K) R^T, so deployment parameter count, FLOPs and memory match the underlying low-rank backbone exactly. Across LLaMA models from 60M to 7B parameters, DLR strengthens low-rank pre-training on C4 validation perplexity in most settings, with the clearest gains at 130M and above; folded checkpoints transfer cleanly to supervised fine-tuning on standard benchmarks.
Dong Wang, Wenwu Tang, Yun Cheng +1
Graz University of Technology, Austria · Swiss Data Science Center, Switzerland