Rethinking the Role of Tensor Decompositions in Post-Training LLM Compression

Large language models (LLMs) are built from structured high-dimensional objects such as token representations, weights, adaptation updates, caches, and activations, whose multilinear structure is underexploited by the conventional matrix-centric view. Tensor decompositions and tensor networks provide a principled algebraic language for this structure, yet the literature often treats them as isolated compression mechanisms. This survey organizes tensor methods for LLMs through two complementary views: a seven-stage lifecycle taxonomy covering tokenization, embeddings, pre-training, adaptation, compression, inference, and interpretability, and a component view covering embeddings, attention, and feed-forward networks. We provide unified notation and theoretical foundations, analyze tensorization strategies for individual Transformer components, and compare methods at each lifecycle stage while making differences in evaluation protocols and model scales explicit. We further connect tensor methods to neighboring efficiency techniques and probabilistic tensor networks. Finally, we synthesize open challenges and introduce $ρ_{\rm gap}$, a metric for the compression-realization gap between theoretical memory reduction and measured system-level speedup. By treating tensorization as a common structural principle, the survey provides a structured entry point to tensorized language models and clarifies when parameter savings can plausibly translate into memory efficiency, computational efficiency, or interpretability. The GitHub page dedicated to this paper is accessible at \href{https://github.com/ma-tt-a/awesome-tensor-methods-for-llms}{this https URL}.

Large language models (LLMs) are dominated by dense linear transformations, whose storage, memory and computational overheads hinder efficient adaptation and deployment while masking the functional impacts of structural simplification. Here we present Tensor Mixture (MixT), a general tensor-structured compression scheme that replaces targeted dense linear layers with natively executable mixtures of tensor operators. Operating directly on generic linear projections instead of model-specific components, MixT is potentially applicable across Transformer-based LLMs and other dense neural mappings. We evaluate MixT on Qwen3-8B and LLaMA2-7B under a unified recovery protocol, identifying a broad compressible regime in which MMLU accuracy is largely preserved before an abrupt transition at model-specific boundaries. This transition coincides with coordinated shifts in output entropy, prediction entropy and inter-layer geometry. At the LLaMA2-7B transition boundary, MixT reduces full-model parameters by 47.5%, inference FLOPs by 37.1%, training FLOPs by 52.1% and peak inference memory by 60.4%, demonstrating its practical potential for lower-cost LLM compression.

Tensor networks provide efficient representations for compressing large neural networks. By carefully designing shapes and topologies, they can significantly reduce memory and computational costs. However, identifying implicit low-rank structures in large foundation models remains challenging due to their enormous scale and un-structured weight distributions. We propose an adaptive tensorization method that discovers inherent low-rank structure in a target tensor by index ordering. Experiments on weight and KV-cache compression demonstrate improved reconstruction quality compared to baselines.