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Sequences of Logits Reveal the Low Rank Structure of Language Models

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 7.3Download Report PDF
Large language modelslow-rank structure
Abstract:

A major problem in the study of large language models is to understand their inherent low-dimensional structure. We introduce an approach to study the low-dimensional structure of language models at a model-agnostic level: as sequential probabilistic models. We first empirically demonstrate that a wide range of modern language models exhibit low-rank structure: in particular, matrices built from the model's logits for varying sets of prompts and responses have low approximate rank. We then show that this low-rank structure can be leveraged for generation --- in particular, we can generate a response to a target prompt using a linear combination of the model's outputs on unrelated, or even nonsensical prompts.

On the theoretical front, we observe that studying the approximate rank of language models in the sense discussed above yields a simple universal abstraction whose theoretical predictions parallel our experiments. We then analyze the representation power of the abstraction and give provable learning guarantees.

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This report is AI-GENERATED using Large Language Models and WisPaper (A scholar search engine). It analyzes academic papers' tasks and contributions against retrieved prior work. While this system identifies POTENTIAL overlaps and novel directions, ITS COVERAGE IS NOT EXHAUSTIVE AND JUDGMENTS ARE APPROXIMATE. These results are intended to assist human reviewers and SHOULD NOT be relied upon as a definitive verdict on novelty.
NOTE that some papers exist in multiple, slightly different versions (e.g., with different titles or URLs). The system may retrieve several versions of the same underlying work. The current automated pipeline does not reliably align or distinguish these cases, so human reviewers will need to disambiguate them manually.
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Overview

Overall Novelty Assessment

The paper introduces a framework for studying language models as sequential probabilistic systems by analyzing the rank structure of logit matrices constructed from varying prompts and responses. It resides in the 'Intrinsic Dimensionality and Rank Analysis' leaf alongside two sibling papers examining effective dimensionality and layer-wise dimensional evolution. This leaf sits within the broader 'Theoretical Foundations and Empirical Analysis' branch, which contains only three leaves and roughly ten papers total. The positioning suggests a relatively sparse research direction focused on fundamental structural properties rather than applied compression or adaptation techniques.

The taxonomy reveals that most related work clusters in adjacent branches: low-rank adaptation methods (LoRA and variants, comprising roughly 20 papers across seven leaves) and compression via factorization (spanning four leaves with methods like SVD-based and tensor decomposition approaches). The paper's theoretical emphasis distinguishes it from these application-oriented neighbors. Within its own branch, the 'Geometric and Algebraic Frameworks' leaf explores connections between next-token prediction and nuclear norm regularization, while 'Representation Analysis' examines how models encode linguistic constructs through latent dimensions. The paper bridges these by linking logit-level rank structure to generation capabilities.

Among eight candidates examined across three contributions, none clearly refuted the proposed ideas. The extended logit matrix framework examined two candidates with no overlaps identified. The linear generation procedure reviewed four candidates without finding substantial prior work on generation via linear combinations of unrelated prompt outputs. The theoretical characterization through time-varying Input Switched Affine Networks examined two candidates, again without clear precedent. This limited search scope—eight papers rather than an exhaustive review—suggests the analysis captures nearby semantic matches but may not reflect the full landscape of rank-based language model theory.

Given the sparse population of the theoretical analysis branch and the absence of refuting work among examined candidates, the contributions appear to occupy relatively unexplored territory within the taxonomy. However, the small search scale and the paper's position in a less-crowded branch mean this assessment reflects local novelty rather than comprehensive field coverage. The dynamic, generation-focused perspective on logit rank structure distinguishes it from static dimensionality measurements in sibling papers, though the limited candidate pool prevents definitive claims about broader originality.

Taxonomy

Core-task Taxonomy Papers
50
3
Claimed Contributions
8
Contribution Candidate Papers Compared
0
Refutable Paper

Research Landscape Overview

Core task: Understanding low-dimensional structure of language models through logit matrices. The field has organized itself around several complementary perspectives on how neural language models exhibit and exploit low-rank structure. At the highest level, one branch focuses on theoretical foundations and empirical analysis—examining intrinsic dimensionality, rank properties, and the geometric organization of representations (e.g., Intrinsic Dimensionality[1], Dimensional Chasm[6]). A second major branch centers on low-rank adaptation methods for parameter-efficient fine-tuning, exemplified by LoRA[10] and its many descendants (SoLA[11], X-LoRA[19], RoseLoRA[18]), which leverage low-rank updates to adapt large models with minimal overhead. Other branches address model compression via matrix and tensor factorization (Tensorized Transformer[12], SVD-LLM[8]), quantization-aware low-rank techniques (QA-LoRA[35], LQ-LoRA[36]), dimensionality reduction for embeddings (Embedtextnet[37]), and specialized applications ranging from federated learning (Federated LoRA[13]) to fairness (Fairness LoRA[26]) and continual unlearning (Continual Unlearning[14]). Within this landscape, a particularly active line of work explores the intrinsic dimensionality and rank properties of model internals, asking how many degrees of freedom are truly necessary to capture linguistic structure and how this varies across layers, tasks, and architectures. Sequences of Logits[0] sits squarely in this theoretical and empirical analysis branch, specifically within the cluster examining intrinsic dimensionality and rank. It shares close kinship with Intrinsic Dimensionality[1], which investigates the effective dimensionality of learned representations, and Dimensional Chasm[6], which probes discrepancies between nominal and effective dimensions. Compared to these neighbors, Sequences of Logits[0] emphasizes the temporal evolution of logit matrices across generation steps, offering a dynamic lens on low-rank structure rather than a static snapshot. This contrasts with compression-focused work like Low-Rank Prune Factorize[3] or adaptation methods like LoRA[10], which exploit low-rank structure for practical efficiency rather than analyzing its fundamental origins.

Claimed Contributions

Extended logit matrix framework for studying low-dimensional structure of language models

The authors propose studying language models through extended logit matrices, which are constructed from model logits over varying sets of prompts (histories) and responses (futures). This framework is architecture-agnostic and treats language models as sequential probabilistic mappings, enabling analysis of their low-dimensional structure without requiring architecture-specific details.

2 retrieved papers
Linear generation procedure exploiting low-rank structure

The authors demonstrate that the low-rank structure of extended logit matrices can be leveraged for generation through a procedure called LINGEN. This method generates continuations to a target prompt by only querying the model on unrelated or nonsensical prompts, using linear combinations of their outputs.

4 retrieved papers
Theoretical characterization via time-varying Input Switched Affine Networks

The authors establish theoretical foundations by proving that low logit rank is equivalent to expressibility as a time-varying ISAN (Input Switched Affine Network). They analyze the representation power of this model and provide efficient learning algorithms with logit query access, demonstrating polynomial-time learnability under this query model.

2 retrieved papers

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Extended logit matrix framework for studying low-dimensional structure of language models

The authors propose studying language models through extended logit matrices, which are constructed from model logits over varying sets of prompts (histories) and responses (futures). This framework is architecture-agnostic and treats language models as sequential probabilistic mappings, enabling analysis of their low-dimensional structure without requiring architecture-specific details.

Contribution

Linear generation procedure exploiting low-rank structure

The authors demonstrate that the low-rank structure of extended logit matrices can be leveraged for generation through a procedure called LINGEN. This method generates continuations to a target prompt by only querying the model on unrelated or nonsensical prompts, using linear combinations of their outputs.

Contribution

Theoretical characterization via time-varying Input Switched Affine Networks

The authors establish theoretical foundations by proving that low logit rank is equivalent to expressibility as a time-varying ISAN (Input Switched Affine Network). They analyze the representation power of this model and provide efficient learning algorithms with logit query access, demonstrating polynomial-time learnability under this query model.

Table of Contents

Sequences of Logits Reveal the Low Rank Structure of Language Models | Novelty Validation