arxiv: 2604.07382 · v2 · submitted 2026-04-08 · 💻 cs.LG · cs.AI

Recognition: 2 theorem links

· Lean Theorem

Latent Structure of Affective Representations in Large Language Models

Benjamin J. Choi , Melanie Weber

Authors on Pith no claims yet

Pith reviewed 2026-05-10 18:50 UTC · model grok-4.3

classification 💻 cs.LG cs.AI

keywords large language modelslatent representationsaffective emotionsvalence-arousalgeometric analysismodel interpretabilityemotion processing

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The pith

Large language models develop coherent latent representations of emotions that align with psychological valence-arousal dimensions.

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

The paper tests whether LLMs internally organize emotions along the same continuous dimensions that psychologists use. It applies geometric analysis to the embeddings of emotion stimuli and finds patterns that match valence and arousal axes. This alignment matters because it offers a concrete way to inspect how models handle affective content without needing external labels. The work also shows the structure is nonlinear yet can be captured well by linear methods, and that the space itself can measure uncertainty during emotion tasks. These results point toward using established psychological models as a reference for interpreting and auditing LLM behavior.

Core claim

LLMs learn coherent latent representations of affective emotions that align with widely used valence-arousal models from psychology. These representations exhibit nonlinear geometric structure that can nonetheless be well-approximated linearly, providing empirical support for the linear representation hypothesis. The learned latent representation space can be leveraged to quantify uncertainty in emotion processing tasks. The findings indicate that LLMs acquire affective representations with geometric structure paralleling established models of human emotion.

What carries the argument

Geometric data analysis tools applied to the latent embeddings produced by LLMs when processing emotion stimuli.

If this is right

Emotion-related outputs in LLMs can be interpreted by projecting them onto established valence-arousal coordinates.
Standard linear techniques for model transparency remain applicable even when the underlying geometry is mildly nonlinear.
Uncertainty estimates for affective tasks can be read directly from distances or spreads in the representation space.

Where Pith is reading between the lines

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

The same geometric approach could be applied to other continuous attributes such as sentiment strength or moral valence to test for similar structure.
If the representations prove stable across model scales, targeted edits in the valence-arousal plane might offer a route to controlled changes in emotional tone of generated text.

Load-bearing premise

The chosen geometric analysis methods and emotion stimuli correctly recover the true underlying geometry of affective representations inside the models.

What would settle it

Finding no consistent alignment between the model's latent positions of emotion words and independent valence-arousal ratings from psychology, or discovering that the structure resists linear approximation, would undermine the central claims.

Figures

Figures reproduced from arXiv: 2604.07382 by Benjamin J. Choi, Melanie Weber.

**Figure 1.** Figure 1: Parabolic valence–arousal model of affective space, based on Maleki et al. (2023). To study the organization of emotional representations in LLMs, we begin by defining a notion of similarity between embeddings corresponding to different emotions. For each emotion pair (i, j), we train a pairwise logistic regression classifier with L2 regularization on mean-pooled hidden state activations (see Section 3.… view at source ↗

**Figure 2.** Figure 2: 2D classical MDS embeddings of internal latent emotion representations in Gemma [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: Isomap (left) appears to unroll parabolic structure in the emotion data manifold, resulting in improvements in capturing rank-1 structure over classical MDS (right). An artificial y-axis jitter is introduced for label visibility, and we include kNN background shading by valence to illustrate improved semantic coherence (left) under Isomap. that separate from the bulk and correspond to affective structure. … view at source ↗

**Figure 4.** Figure 4: Upper panels depict activation distance from separating hyperplane, with misclassifi [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗

**Figure 5.** Figure 5: Corroboration of key results on LLaMA-3-70B-Instruct. Top-left: MDS emotion lay [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗

**Figure 6.** Figure 6: Representative examples of a classical MDS eigenspectrum ( [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗

**Figure 7.** Figure 7: Mean emotional separability across layers for Gemma-2-9B (left) and Mistral-7B [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗

**Figure 8.** Figure 8: MDS emotion layouts for final layers in Gemma-2-9B ( [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗

**Figure 9.** Figure 9: R2 and p-values for the classical MDS LLM latent representations vs. established valence-arousal scores. conclusions depend critically on this choice, we performed an additional ablation based on a purely geometric cosine-distance metric over the mean-pooled activations. Concretely, for emotion class across layers, we first computed the mean activation vector by averaging the mean-pooled hidden states over… view at source ↗

**Figure 10.** Figure 10: Layer-by-layer activation distances from output class vs. ground-truth input class. [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗

**Figure 11.** Figure 11: UMAP visualizations for Gemma-2-9B, Mistral-7B, and LLaMA-3-70B-Instruct. [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗

**Figure 12.** Figure 12: Classical MDS embeddings obtained from cosine-based dissimilarities between mean [PITH_FULL_IMAGE:figures/full_fig_p021_12.png] view at source ↗

**Figure 13.** Figure 13: Additional results from experiments on LLaMA-3-70B-Instruct. Top-left: activation [PITH_FULL_IMAGE:figures/full_fig_p022_13.png] view at source ↗

**Figure 14.** Figure 14: We find that a similar parabolic “V”-shaped emotion layout exists in human brain [PITH_FULL_IMAGE:figures/full_fig_p023_14.png] view at source ↗

**Figure 15.** Figure 15: Isomap embedding of the ANEW valence–arousal space. [PITH_FULL_IMAGE:figures/full_fig_p024_15.png] view at source ↗

**Figure 16.** Figure 16: (a) Consensus valence ratings (mean ± SEM across 3 raters) by steering condition. Positive-target conditions (blue) are separated from negative-target conditions (red) by over six points on the 10-point scale. (b–d) Pairwise rater agreement on valence, colored by condition; all pairs show r > 0.94. can be characterized as near-distributional collapse toward a profanity-dominated output mode. This incohere… view at source ↗

read the original abstract

The geometric structure of latent representations in large language models (LLMs) is an active area of research, driven in part by its implications for model transparency and AI safety. Existing literature has focused mainly on general geometric and topological properties of the learnt representations, but due to a lack of ground-truth latent geometry, validating the findings of such approaches is challenging. Emotion processing provides an intriguing testbed for probing representational geometry, as emotions exhibit both categorical organization and continuous affective dimensions, which are well-established in the psychology literature. Moreover, understanding such representations carries safety relevance. In this work, we investigate the latent structure of affective representations in LLMs using geometric data analysis tools. We present three main findings. First, we show that LLMs learn coherent latent representations of affective emotions that align with widely used valence--arousal models from psychology. Second, we find that these representations exhibit nonlinear geometric structure that can nonetheless be well-approximated linearly, providing empirical support for the linear representation hypothesis commonly assumed in model transparency methods. Third, we demonstrate that the learned latent representation space can be leveraged to quantify uncertainty in emotion processing tasks. Our findings suggest that LLMs acquire affective representations with geometric structure paralleling established models of human emotion, with practical implications for model interpretability and safety.

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.

Desk Editor's Note private letter to a colleague

The paper applies geometric tools to affective stimuli in LLMs and reports alignment with valence-arousal plus a workable linear approximation, but the alignment may not be specific to affect.

read the letter

The main thing to know is that the authors treat emotion categories as a testbed with known psychological structure and find that LLM activations line up with valence-arousal dimensions while still allowing a decent linear fit. They also sketch a use for uncertainty quantification in emotion tasks. This is new because prior geometry work has stayed more general and lacked any external anchor; emotion gives one that psychology already maps out in two dimensions. The setup is straightforward and connects directly to interpretability and safety questions, which is useful even if the numbers are not yet shown in detail. The soft spot is the missing specificity check. The alignment could come from broad semantic or lexical clustering that any coherent input set would produce, and the abstract gives no sign of baselines such as frequency-matched non-emotion words, shuffled labels, or unrelated categories. Without those, the claim that the geometry is specifically affective rests on the choice of stimuli alone. The lack of reported error bars, alignment scores, or validation steps in the summary also leaves the strength of the three findings hard to judge. This is for people already working on representation geometry or on LLM safety who want a concrete domain to try the same tools. A reader who cares about the linear representation hypothesis would get value from the empirical angle even if the affective part needs tightening. It deserves a serious referee because the testbed idea is reasonable and the linear-approximation result is worth checking with proper controls. I would send it to review and ask for the baselines and quantitative details up front.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates the latent geometric structure of affective emotion representations in large language models using geometric data analysis tools applied to emotion stimuli. It presents three main findings: (1) LLMs learn coherent latent representations of affective emotions that align with valence-arousal models from psychology, (2) these representations exhibit nonlinear geometric structure that can nonetheless be well-approximated linearly, providing empirical support for the linear representation hypothesis, and (3) the learned latent representation space can be leveraged to quantify uncertainty in emotion processing tasks. The work positions these results as relevant to model transparency and AI safety.

Significance. If the central claims hold after addressing specificity concerns, the paper would provide useful empirical evidence that LLMs encode affective information with geometric structure paralleling established psychological models of emotion. This could strengthen the case for using geometric analysis in interpretability work and offer practical tools for uncertainty quantification in emotion-related tasks. The support for linear approximations in an affective domain is a modest but concrete addition to the linear representation hypothesis literature.

major comments (2)

[Results section presenting the first finding] The first main finding (alignment of LLM latent representations with valence-arousal dimensions) is load-bearing for the central claim that LLMs acquire specifically affective geometry. The reported alignment metrics are not accompanied by controls such as frequency-matched non-affective word sets, shuffled emotion labels, or non-emotion semantic categories. Without these, it is not possible to rule out that the observed geometry reflects general semantic clustering rather than dedicated affective structure.
[Section on nonlinear geometric structure and linear approximation] The claim that nonlinear structure is 'well-approximated linearly' (second finding) requires quantitative detail on approximation quality. The manuscript should report the specific error metric, the fraction of variance captured by the linear model versus nonlinear baselines, and whether this holds after correcting for the number of dimensions used.

minor comments (2)

[Abstract] The abstract summarizes the three findings without any quantitative values, error bars, or brief description of validation procedures; adding one or two key numbers would improve readability.
[Methods] Notation for the geometric analysis tools (e.g., specific manifold learning or dimensionality reduction methods) should be introduced with a short equation or reference in the methods section to aid reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments on our manuscript investigating the latent structure of affective representations in LLMs. We address each of the major comments point by point below and outline the revisions we will make to strengthen the paper.

read point-by-point responses

Referee: [Results section presenting the first finding] The first main finding (alignment of LLM latent representations with valence-arousal dimensions) is load-bearing for the central claim that LLMs acquire specifically affective geometry. The reported alignment metrics are not accompanied by controls such as frequency-matched non-affective word sets, shuffled emotion labels, or non-emotion semantic categories. Without these, it is not possible to rule out that the observed geometry reflects general semantic clustering rather than dedicated affective structure.

Authors: We agree that demonstrating the specificity of the affective geometry is crucial. While our stimuli consist of emotion-related terms and the alignment is measured against established psychological models, we acknowledge that explicit controls are necessary to rule out general semantic effects. In the revised version, we will add control experiments including: (1) shuffled emotion labels to assess if the structure is label-dependent, (2) frequency-matched non-affective word sets, and (3) comparisons with other semantic categories. These additions will provide stronger evidence that the observed valence-arousal alignment reflects dedicated affective representations rather than broad semantic clustering. revision: yes
Referee: [Section on nonlinear geometric structure and linear approximation] The claim that nonlinear structure is 'well-approximated linearly' (second finding) requires quantitative detail on approximation quality. The manuscript should report the specific error metric, the fraction of variance captured by the linear model versus nonlinear baselines, and whether this holds after correcting for the number of dimensions used.

Authors: We appreciate this suggestion for greater rigor. The original manuscript described the linear approximation qualitatively. In the revision, we will include quantitative details such as the reconstruction error (e.g., mean squared error) of the linear model, the fraction of variance explained by the linear approximation compared to the full nonlinear structure, and comparisons with nonlinear baselines like kernel methods or neural network-based dimensionality reduction. We will also address dimensionality correction by reporting results normalized by the number of dimensions or using appropriate statistical controls. This will better support the claim regarding the linear representation hypothesis. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical geometric analysis on activations

full rationale

The paper applies standard geometric data analysis tools (e.g., dimensionality reduction, distance metrics) directly to LLM hidden states for emotion stimuli and compares the resulting structure to independently established valence-arousal dimensions from psychology. No equations, parameters, or predictions are defined in terms of the target alignments; the reported coherence is a measured outcome rather than a fitted or self-referential construct. The derivation chain is self-contained against external benchmarks and does not reduce to self-citation or ansatz smuggling.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The analysis rests on standard geometric data analysis assumptions and the validity of valence-arousal models from psychology; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)

standard math Geometric data analysis tools can reveal meaningful structure in high-dimensional embedding spaces.
Invoked to interpret latent representations of emotions.
domain assumption Valence-arousal models from psychology provide a valid ground-truth organization for affective states.
Used as the reference for alignment of LLM representations.

pith-pipeline@v0.9.0 · 5518 in / 1262 out tokens · 58627 ms · 2026-05-10T18:50:03.268477+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We employ two manifold learning methods... classical multidimensional scaling (MDS) and Isometric Feature Mapping (Isomap)... pairwise logistic regression classifier... dissimilarity matrix D_ij = acc_ij... Procrustes R^2... valence-arousal model
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

nonlinear geometric structure... parabolic 'V'-shaped... Isomap recovers the expected parabolic structure

What do these tags mean?

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

2 extracted references · 1 canonical work pages

[1]

Classify this text into exactly one emotion from this list: . . . Text:{text}Emotion:

Springer, 2001. 15 Chen Xiong, Zhiyuan He, Pin-Yu Chen, Ching-Yun Ko, and Tsung-Yi Ho. Steering externalities: Benign activation steering unintentionally increases jailbreak risk for large language models. arXiv preprint arXiv:2602.04896, 2026. Bo Zhao, Maya Okawa, Eric J Bigelow, Rose Yu, Tomer Ullman, Ekdeep Singh Lubana, and Hidenori Tanaka. Emergence ...

work page arXiv 2001
[2]

emotion vectors

The probe directions identified in our earlier analyses are indeed causally efficacious axes in representation space that reliably shift the emotional register of generated text. D.5.3 Secondary Analysis: Coherence and the Neutral-First Hypothesis Coherence degrades asymmetrically.Steering toward joy leads to mildly perturbed flu- ency: pooled positive co...

2026