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arxiv: 2510.01025 · v2 · submitted 2025-10-01 · 💻 cs.AI · cs.CL

Hypothesis-Driven Feature Manifold Analysis in LLMs via Supervised Multi-Dimensional Scaling

Federico Tiblias , Irina Bigoulaeva , Jingcheng Niu , Simone Balloccu , Iryna Gurevych This is my paper

Pith reviewed 2026-05-18 10:36 UTC · model grok-4.3

classification 💻 cs.AI cs.CL
keywords feature manifoldslinear representation hypothesissupervised multi-dimensional scalingtemporal reasoninglarge language modelslatent spacegeometric structuresentity-based reasoning
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The pith

Language models encode temporal concepts as distinct geometric structures like circles, lines, and clusters in their latent space.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper proposes Supervised Multi-Dimensional Scaling to test the linear representation hypothesis by examining how language models organize features into manifolds. Using temporal reasoning as an example, it demonstrates that these manifolds take on specific shapes that match the meaning of the concepts, hold steady no matter the model size or type, help the model reason, and change when the context shifts. Understanding this would matter to a reader because it suggests language models build and manipulate structured internal representations for entities during reasoning rather than relying solely on surface patterns.

Core claim

Supervised Multi-Dimensional Scaling reveals that different features in language models instantiate distinct geometric structures, including circles, lines, and clusters. These structures reflect the semantic properties of the concepts they represent, remain stable across model families and sizes, actively support reasoning, and dynamically reshape in response to contextual changes, thereby supporting a model of entity-based reasoning in which LMs encode and transform structured representations.

What carries the argument

Supervised Multi-Dimensional Scaling (SMDS), a model-agnostic method that evaluates competing hypotheses about the geometry of feature manifolds by projecting representations while respecting supervised information.

If this is right

  • Feature manifolds reflect the semantic properties of the concepts they represent.
  • These structures remain stable across model families and sizes.
  • They actively support reasoning in the model.
  • They dynamically reshape in response to contextual changes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar manifold structures might appear in other reasoning tasks, allowing SMDS to map how models handle spatial or causal concepts.
  • The finding of dynamic reshaping suggests that prompting strategies could be designed to control or stabilize these internal geometries for more reliable outputs.
  • Entity-based reasoning may imply that models treat concepts as objects with properties that can be transformed, opening ways to debug specific failures in multi-step inference.

Load-bearing premise

The assumption that Supervised Multi-Dimensional Scaling reliably recovers the true underlying geometric structures of feature manifolds without artifacts introduced by the supervision signals, distance metrics, or the specific choice of temporal reasoning as the case study domain.

What would settle it

A calculation showing that disrupting the identified geometric structures through targeted interventions does not affect the model's accuracy on temporal reasoning tasks would falsify the claim that these structures actively support reasoning.

Figures

Figures reproduced from arXiv: 2510.01025 by Federico Tiblias, Irina Bigoulaeva, Iryna Gurevych, Jingcheng Niu, Simone Balloccu.

Figure 1
Figure 1. Figure 1: Our contributions. Supervised Multi-Dimensional Scaling is a novel dimensionality reduction technique [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Feature Manifold Discovery and the Limita [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Feature manifolds retrieved from the LP site. We can observe that models represent features in a similar way, and the resulting manifolds are interpretable and match an intuitive progression (linear, circular or categorical) of the underlying features. The scatter plots on the left show the first two components of SMDS dimensionality reduction; the bar plots on the right depict scoring of different manifol… view at source ↗
Figure 6
Figure 6. Figure 6: Manifold quality at different layer depths and [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 5
Figure 5. Figure 5: Feature manifolds of Llama-3.2-3B-Instruct [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: Downstream accuracy of Llama-3.2-3B￾Instruct at increasing levels of noise. Intervening on a low-dimensional manifold subspace is just as effective at degrading model performance as perturbing much larger activation spaces. Layer 1 Layer 14 Layer 27 1st Jan 31st Dec Manifolds before and after intervention [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Llama-3.2-3B-Instruct on date task. Latent space of the LP token before and after applying noise on the TE token (top and bottom respectively). Interven￾tions on early tokens cause disruptions to the manifolds of later ones. the disruption of structures located at subsequent tokens and layers ( [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Accuracy on temporal tasks. Accuracy is low across the board, with only Llama models achieving [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: More manifolds for different tasks and models. Continued from Figure [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Feature manifolds for models at different sizes. There is a preferential manifold also across scales. [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Feature manifolds for base models. Geometries are consistent with the instruction-tuned counterparts in [PITH_FULL_IMAGE:figures/full_fig_p021_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Circular manifolds on the time_of_day task. SMDS cannot find any structure on LP and A despite one being present at the TE site. 600 400 200 0 200 400 Component 0 200 150 100 50 0 50 100 150 Component 1 2D feature manifold [PITH_FULL_IMAGE:figures/full_fig_p022_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Multidimensional manifold constructed by [PITH_FULL_IMAGE:figures/full_fig_p022_14.png] view at source ↗
Figure 16
Figure 16. Figure 16: SMDS of gemma-2-2b-it on the cities task. The recovered projection shows the relative position of continents. The chord distance is again computed as the Eu￾clidean distance in 3D. For the geodesic manifold, we compute the great-circle distance between two cities—i.e., the shortest path along the surface of a sphere. We first convert latitude and longitude to radians and compute the differences: ϕi = radi… view at source ↗
Figure 15
Figure 15. Figure 15: Additional accuracy plots on intervention [PITH_FULL_IMAGE:figures/full_fig_p023_15.png] view at source ↗
read the original abstract

The linear representation hypothesis states that language models (LMs) encode concepts as directions in their latent space, forming organized, multidimensional manifolds. Prior work has largely focused on identifying specific geometries for individual features, limiting its ability to generalize. We introduce Supervised Multi-Dimensional Scaling (SMDS), a model-agnostic method for evaluating and comparing competing feature manifold hypotheses. We apply SMDS to temporal reasoning as a case study and find that different features instantiate distinct geometric structures, including circles, lines, and clusters. SMDS reveals several consistent characteristics of these structures: they reflect the semantic properties of the concepts they represent, remain stable across model families and sizes, actively support reasoning, and dynamically reshape in response to contextual changes. Together, our findings shed light on the functional role of feature manifolds, supporting a model of entity-based reasoning in which LMs encode and transform structured representations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces Supervised Multi-Dimensional Scaling (SMDS) as a model-agnostic method to evaluate competing hypotheses about feature manifolds in LLMs. Applied to temporal reasoning as a case study, it claims that distinct features instantiate different geometric structures (circles, lines, clusters) that reflect semantic properties of the concepts, remain stable across model families and sizes, actively support reasoning, and dynamically reshape under contextual changes, thereby supporting an entity-based model of reasoning in which LLMs encode and transform structured representations.

Significance. If SMDS can be shown to recover intrinsic manifold geometries rather than artifacts of its supervision, the work would provide a useful general-purpose tool for hypothesis-driven analysis of internal representations. The reported cross-model stability and context-dependent reshaping, if quantitatively substantiated with controls, would strengthen evidence for structured representations that functionally contribute to reasoning.

major comments (2)
  1. [Method (SMDS formulation and loss)] The central assumption that SMDS recovers intrinsic geometries is load-bearing for all claims about semantic reflection, stability, and reasoning support. Because SMDS is supervised on temporal-reasoning labels and employs a chosen distance metric, the recovered circles, lines, and clusters could be imposed by the objective. No ablations are presented that vary supervision strength, remove labels entirely, or compare against unsupervised MDS/PCA on identical activations to rule out method-induced artifacts.
  2. [Experiments and Results] The results section reports qualitative observations of distinct geometries and their properties but supplies no quantitative metrics, error bars, statistical tests, or baseline comparisons. This absence makes it impossible to evaluate the strength of the stability-across-models claim or the dynamic-reshaping claim.
minor comments (2)
  1. [Abstract] The abstract would be strengthened by a single sentence summarizing the key quantitative controls or metrics that support the qualitative findings.
  2. [Preliminaries] Notation for the supervised scaling objective and the distance function could be introduced with a short table of symbols for readers outside the immediate subfield.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the detailed review and constructive feedback on our manuscript. We have carefully considered the major comments and will revise the paper to address the concerns regarding potential method-induced artifacts and the lack of quantitative evaluations. Below we provide point-by-point responses.

read point-by-point responses

  1. Referee: [Method (SMDS formulation and loss)] The central assumption that SMDS recovers intrinsic geometries is load-bearing for all claims about semantic reflection, stability, and reasoning support. Because SMDS is supervised on temporal-reasoning labels and employs a chosen distance metric, the recovered circles, lines, and clusters could be imposed by the objective. No ablations are presented that vary supervision strength, remove labels entirely, or compare against unsupervised MDS/PCA on identical activations to rule out method-induced artifacts.

    Authors: We thank the referee for highlighting this important methodological concern. While SMDS is intentionally supervised to evaluate specific hypotheses about feature manifolds, we agree that it is essential to demonstrate that the recovered structures are not solely artifacts of the supervision or chosen metric. In the revised manuscript, we will add a series of ablations, including: (1) comparisons of SMDS results with those from unsupervised MDS and PCA applied to the same activation data, (2) experiments varying the strength of supervision (e.g., using partial or noisy labels), and (3) a fully unsupervised baseline. These additions will provide evidence that the observed geometric structures reflect intrinsic properties of the representations rather than being imposed by the method. revision: yes

  2. Referee: [Experiments and Results] The results section reports qualitative observations of distinct geometries and their properties but supplies no quantitative metrics, error bars, statistical tests, or baseline comparisons. This absence makes it impossible to evaluate the strength of the stability-across-models claim or the dynamic-reshaping claim.

    Authors: We acknowledge that the current presentation relies primarily on qualitative visualizations, which limits the ability to rigorously assess the claims. In the revision, we will introduce quantitative metrics to evaluate manifold properties, such as measures of geometric fidelity (e.g., stress or reconstruction error for the scaling), cross-model stability quantified via Procrustes analysis or correlation of distances with error bars from multiple seeds, and statistical tests (e.g., permutation tests) for the significance of observed differences in reshaping under context changes. We will also include baseline comparisons to quantify the improvements or differences from SMDS. revision: yes

Circularity Check

0 steps flagged

No circularity: SMDS applied as independent analysis tool to observed activations

full rationale

The paper introduces Supervised Multi-Dimensional Scaling (SMDS) as a model-agnostic method to evaluate competing feature manifold hypotheses on LLM activations for temporal reasoning. The central findings (distinct geometries like circles/lines/clusters that reflect semantics, remain stable, support reasoning, and reshape with context) are presented as outcomes of applying this method to empirical data rather than being defined into existence or recovered by construction from fitted parameters. No self-definitional loops, fitted-input predictions, or load-bearing self-citations that reduce the derivation to its own inputs are evident in the abstract or described approach. The method is positioned as an evaluation tool against external model behavior, making the derivation self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the linear representation hypothesis as a background domain assumption in LLM interpretability research. No free parameters, additional axioms, or invented entities are described in the abstract.

axioms (1)
  • domain assumption Language models encode concepts as directions in latent space that form organized multidimensional manifolds (linear representation hypothesis).
    Explicitly stated as the foundational premise at the start of the abstract.

pith-pipeline@v0.9.0 · 5694 in / 1375 out tokens · 49438 ms · 2026-05-18T10:36:17.742433+00:00 · methodology

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Forward citations

Cited by 2 Pith papers

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  1. Latent Trajectory Dynamics in Large Language Models: A Manifold Evolution Framework with Empirical Validation

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    DMET models LLM generation as controlled dynamical trajectories on a semantic manifold, with three proxy metrics that predict output quality and support adaptive decoding to lower perplexity.

  2. H-Probes: Extracting Hierarchical Structures From Latent Representations of Language Models

    cs.CL 2026-04 unverdicted novelty 5.0

    H-probes locate low-dimensional subspaces encoding hierarchy in LLM activations for synthetic tree tasks, show causal importance and generalization, and detect weaker signals in mathematical reasoning traces.

Reference graph

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