PersonaManifold: Revealing and Exploiting Curved Geometry in LLM Persona Representations

1 Introduction

• Linear-isotropic methods report coupled trait dimensions, asymmetric steering effects, and deviations in multi-trait composition, motivating a curved, anisotropic account.Related reports identify non-orthogonal traits [Feng et al., 2026, Liu et al., 2026], asymmetric effects [Feng et al., 2026, Chen et al., 2026], and composition deviation [Sun et al., 2025, Xu et al., 2025].Source: §1 · p. 2
• PersonaManifold models activations as points on a curved, low-dimensional Riemannian submanifold and estimates local metrics, geodesic distances, and Ollivier–Ricci curvature.Source: §1 · p. 2
• Across multiple open-source LLMs, geodesic distance predicts behavioral similarity better than Euclidean distance, and geodesic steering yields more coherent intermediate personas, especially in high-deviation regions.The reported advantage holds on BST and external evaluations.Source: §1 · p. 2
• Density-controlled analysis finds heterogeneous curvature in persona representations and shows it is intrinsic rather than a sampling artifact.Source: §1 · p. 2
• Geodesic steering interpolates along manifold geodesics, producing more coherent intermediate personas than linear interpolation on internal and external benchmarks.Source: §1 · p. 2
• BST defines persona similarity through situational behavioral responses grounded in established psychological constructs, rather than self-report questionnaires.Source: §1 · p. 2

2 Related Work

• PersonaManifold models persona representations as points on a curved Riemannian submanifold and reports improved distance prediction and steering.Source: §2 · p. 2
• Prior persona-control methods manipulate activation- or weight-space vectors algebraically under a linear-isotropic assumption, whereas analyses report heterogeneous trait circuits and sparse opposing-trait subnetworks.The cited work also finds personality structures more complex than five dimensions [Liu et al., 2026].Source: §2 · p. 2
• The overview figure depicts activation extraction, geometry estimation, geodesic steering, and BST evaluation as the framework's four stages.Source: Figure 1 · p. 3
• Classical manifold learning recovers intrinsic structure, but the cited work does not characterize local curvature or metric anisotropy; persona geometry had not been characterized or exploited for activation steering.Related analyses discussed in this section include graph shortest paths, diffusion, and discrete Ollivier–Ricci curvature.Source: §2 · p. 3

3.1 Persona Activation Extraction

• Persona activations are neutral-baseline-subtracted mean response activations, with a TCR-selected layer and PCA retaining 99% of variance.The selected layer falls in the 12–16 range across all three models, and PCA reduces dimensionality from about 4096 to 200–500.Source: §3.1 · p. 4

3.2 Manifold Geometry Estimation

• The geometry analysis combines intrinsic-dimensionality estimates, local metric tensors, graph geodesics, and Ollivier–Ricci curvature, with density matching to test whether curvature differences persist.Geodesic distance is the Dijkstra shortest-path length, accounting for anisotropic costs and paths through data-dense regions rather than empty space.Source: §3.2 · p. 4

3.3 Geodesic Activation Steering

• Geodesic steering follows a smoothed manifold path between personas and samples intermediate points at equal arc-length intervals, unlike straight-line interpolation.The reported advantage is largest when the geodesic deviates substantially from the straight line; locally flat regions make the methods equivalent.Source: §3.3 · p. 5

3.4 Behavioral Similarity Triplet (BST)

• BST defines persona similarity through behavioral responses to situational questions grounded in six psychological constructs, rather than self-report questionnaires.It combines forced-choice agreement and open-ended response embeddings, then forms 10K triplets with at least 85% human agreement; TCR-choice is reported to check against embedding-metric circularity.Source: §3.4 · p. 5

4.1 Experimental Setup

• Experiments span three open-source LLMs and approximately 2K personas, using BST for behavioral similarity and internal, external, and human evaluations for steering.Steering metrics include Interpolation Coherence, Transition Smoothness, and Manifold Adherence.Source: §4 · p. 5, §4.1 · p. 6

4.2 Persona Activations Form a Curved Manifold (RQ1)

• Persona activations form a robust 15–23-dimensional manifold versus ambient dimension D′≈300; professional personas trend positive in curvature, while fictional and rare combinations trend negative.Curvature differences remain significant within density-matched subsets (p < 0.001), and intrinsic dimensionality substantially exceeds the five dimensions assumed by Big Five models.Source: §4.2 · pp. 6–7, Table 1 · p. 6, Figure 2 · p. 7

4.3 Geodesic Distance Predicts Behavioral Similarity (RQ2)

• Geodesic distance achieves the highest TCR across all three splits and models, improving TCR-combined over Euclidean by 5.1–6.1 percentage points.Anisotropy and graph topology each contribute significantly; improvement reaches 11–12% in the top deviation quartile versus 1–1.5% in the bottom quartile.Source: Table 2 · p. 7, §4.3 · p. 7, Figure 3 · p. 8

4.4 Geodesic Steering Produces More Coherent Intermediates (RQ3)

• On high-deviation pairs, geodesic steering leads IC and TS across three models and MA on two; external and human evaluations also favor its smoother, more natural transitions.Human ratings average 3.82 for geodesic steering versus 3.41 for linear steering; the retrieval extension improves MRR on rare personas but offers little advantage on common ones.Source: Table 3 · p. 8, §4.4 · p. 8
• Geodesic steering benefits most when the path departs substantially from a straight line; across architectures, it is most beneficial when dg/dE > 1.2, covering approximately 60% of random persona pairs.On low-deviation pairs, methods perform comparably, consistent with geodesics approximating straight lines in locally flat regions.Source: §4.4 · p. 8

5.1 Ablation Studies

• Local PCA achieves the best TCR, isotropic Euclidean degrades substantially, and steering refinements improve high-deviation performance.For Llama, norm-aware α improves IC and TS by 3–4 points; spline smoothing improves TS without sacrificing IC or MA. TCR varies by at most 1.3% across tested k values.Source: §5.1 · p. 9

5.2 Geometry–Performance Interaction across Persona Types

• Anisotropy contributes most for dense professional personas, whereas graph topology contributes more for fictional characters and rare combinations in sparse regions.The full geodesic combines both factors and consistently outperforms either alone, with the largest advantage in high-deviation regions.Source: §5.2 · p. 9

6 Conclusion

• Across the BST benchmark and multiple external evaluations, geodesic distance better predicts behavioral similarity and geodesic steering yields more coherent intermediates, especially in regions far from flat.The findings support intrinsic geometry as a practically useful inductive bias for persona control.Source: §6 · p. 9

Limitations and Broader Impact

• Experiments cover 7–8B models; each persona is a single mean activation, BST covers six constructs, and full geometry estimation requires O(N^2) pairwise computations.The authors identify larger-scale analysis, conversational variation, and additional psychological dimensions as directions for extending the work.Source: Limitations and Broader Impact · p. 12
• More coherent persona steering may lower barriers to impersonation or social engineering, so the authors recommend safety filters and restricting contexts where impersonation poses direct harm.They characterize this risk as incremental because the method uses existing open-source LLMs and adds no fundamentally new generative capabilities.Source: Limitations and Broader Impact · p. 12

A Implementation Details

• For approximately 2K personas, geometry estimation adds about 15 minutes after 45-minute activation extraction, while inference-time path queries take less than 1 ms.The reported 33% geometry-estimation overhead is paid once per model and amortized across evaluations.Source: A Implementation Details · pp. 12–13

B Probe Questions

• The 40 initial probe questions adapt BFI-44, HEXACO-60, and TIPI items into situational behavioral prompts, with all six HEXACO domains represented.Pilot testing found non-degenerate response distributions across the questions.Source: B Probe Questions · p. 13

C BST Question Generation Prompts

• Filtering 180 candidate BST questions on 500 personas retained 150, removing items with low response entropy or low embedding variance.The forced-choice pool lost 20% of questions and the open-ended pool lost 15%; socially desirable or ambiguous items were among those removed.Source: C BST Question Generation Prompts · pp. 14–15

D Human Agreement Evaluation

• 86.1% of evaluated triplets aligned with the automated BST label, and inter-annotator agreement was κ = 0.81; evaluators judged behavior without seeing persona descriptions.The 200 triplets were assessed by three annotators each.Source: D Human Agreement Evaluation · pp. 15–16
• Geodesic steering received a 3.82 mean naturalness score versus 3.41 for linear steering, with the largest gap on high-deviation dialogues.On low-deviation dialogues, the scores were 3.72 versus 3.65.Source: D Human Agreement Evaluation · p. 16

E Density–Curvature Decoupling

• Curvature differences across persona types persist within density bins (p < 0.001), while density and mean curvature have a non-significant Spearman correlation of ρ = 0.08.The pattern is qualitatively replicated on Qwen and Mistral, with non-significant correlations after Bonferroni correction.Source: E Density–Curvature Decoupling · pp. 17–18, Figure E.1 · p. 18

F Persona Retrieval

• Geodesic distance improves MRR over Euclidean distance by +0.12–0.16 on rare personas and +0.04–0.06 on common personas across all three models.The advantage is largest in sparse, high-curvature regions, where topology adds information beyond local anisotropy.Source: F Persona Retrieval · p. 18, Table F.1 · p. 19

G Manifold Visualization

• Layer-16 projections show dense, partially overlapping professional clusters, scattered fictional-character neighborhoods, and broadly distributed demographic personas across Llama and Mistral.These qualitative structures are consistent with the described positive curvature of professional personas and negative curvature of fictional characters.Source: G Manifold Visualization · pp. 18–19, Figure G.1 · p. 19

H Geodesic Deviation Ratio Distribution

• Approximately 60% of persona pairs have dg/dE > 1.2, the threshold associated with >3% improvement on IC and TS; the top quartile exceeds 1.5 and reaches +9–12% TCR.Llama has slightly heavier deviation tails than Mistral, alongside higher intrinsic dimensionality (d∗= 22–23 vs. 16–17).Source: H Geodesic Deviation Ratio Distribution · p. 19

I Layer Selection Analysis

• Layers 12–16 yield the strongest TCR-combined across all three models, while shallow and deep layers underperform.The selected range captures the peak for all three 32-layer architectures; shallow layers encode more syntactic and positional information, while deep-layer underperformance may reflect entanglement with output planning.Source: I Layer Selection Analysis · pp. 19–20, Figure I.1 · p. 20

J Steering Trajectory Visualization

• For high-deviation pairs, linear paths cross low-density regions while geodesic paths follow data-supported regions; for low-deviation pairs, the paths nearly overlap.The authors describe geodesic deviation as an indicator of when curvature-aware steering will help.Source: J Steering Trajectory Visualization · pp. 20–21, Figure J.1 · p. 21

K Full Per-Model Results

• Geodesic distance shows its largest TCR-combined advantage on fictional and rare-trait personas and in the top dg/dE quartile across all three models.Table K.1 presents the breakdown by persona type and deviation quartile, comparing Euclidean and Geodesic results.Source: K Full Per-Model Results · p. 21, Table K.1 · p. 21
• The reported TCR-combined breakdown covers persona type and dg/dE quartile for all three models.The table caption identifies Euclidean and Geodesic as the compared methods and defines Δ as absolute improvement.Source: Table K.1 · p. 21
• On low-deviation pairs, steering-quality differences between methods are small, as expected on near-flat manifold regions.Table K.2 reports results for the bottom 25% of pairs by dg/dE.Source: Table K.2 · p. 21, K Full Per-Model Results · p. 21

• The advantage is largest on PersonaGym PersonaScore and BFI-44 Monotonicity, where curved paths avoid off-manifold regions that produce incoherent trait profiles.Source: L External Benchmark Results · p. 22
• Across high-deviation pairs, geodesic steering consistently outperforms linear steering across all external-benchmark metrics and models.Source: Table L.1 · p. 22, L External Benchmark Results · p. 22
• The external results cover PersonaGym PersonaScore (1–5), BFI-44 Monotonicity (%), and Trait Expression Score (0–100).Source: Table L.1 · p. 22

M Synthetic Manifold Validation

• On the Swiss Roll, estimated-versus-true geodesic-distance correlation is ρ = 0.94 at σ = 0.1 and ρ = 0.89 at σ = 0.3, falling to ρ = 0.86 in R10.Source: M Synthetic Manifold Validation · p. 22
• On the sphere, estimated geodesic distances correlate with great-circle distances at ρ = 0.92, with antipodal pairs the main error source.Source: M Synthetic Manifold Validation · p. 22
• Under 10% subsampling repeated 100 times, mean Kendall τ is 0.82 ± 0.04 on the Swiss Roll and 0.85 ± 0.03 on the sphere, indicating moderate ranking stability.Source: M Synthetic Manifold Validation · p. 22
• The distance pipeline’s real-manifold accuracy is likely lower than its synthetic results because k-NN graph approximation is biased in sparse regions and at manifold boundaries.The authors note that the real persona manifold has d*≈20 and D′≈300, while downstream results suggest its distance rankings remain sufficient for the evaluated tasks.Source: M Synthetic Manifold Validation · p. 22

N.1 Case 1: Pediatrician →Trial Lawyer (High-Deviation)

• The pediatrician-to-trial-lawyer case shows higher geodesic scores for IC (0.71 vs. 0.54), TS (0.74 vs. 0.51), and MA (0.65 vs. 0.40); linear IC falls to 0.38 at t = 0.5.Source: §N.1 · p. 23

N.2 Case 2: Cautious Accountant →Impulsive Artist (Medium-Deviation)

• For the accountant-to-artist case, geodesic scores exceed linear scores for IC (0.71 vs. 0.62), TS (0.69 vs. 0.59), and MA (0.63 vs. 0.53), with a smaller gap than Case 1.Source: §N.2 · p. 24

N.3 Data-Level Analysis of Steering Coherence

• For high-deviation pairs, linear steering’s IC drops by an average of 0.20 at t ∈[0.3, 0.7], while geodesic IC stays within 0.06 of endpoint values; neither method shows a dip for low-deviation pairs.Source: §N.3 · p. 24, Table N.1 · p. 24