PersonaManifold: Revealing and Exploiting Curved Geometry in LLM Persona Representations

Rui Xu; Yinghui Xu; Libo Wu

cs.AISep 28, 2026

PersonaManifold: Revealing and Exploiting Curved Geometry in LLM Persona Representations

Authors: Rui Xu, Yinghui Xu, Libo Wu

Organizations: Fudan University · Shanghai Innovation Institute

Abstract

Controlling persona in large language models (LLMs) at inference time is important for role-playing, personalized dialogue, and social simulation. Recent methods extract persona vectors from the model's activation space and apply Euclidean operations---addition, scaling, and linear interpolation---under the linear representation hypothesis. However, these methods themselves report systematic failures: non-orthogonal trait dimensions, asymmetric ceiling and resistance effects, and significant deviations in multi-trait composition, suggesting that the linear isotropic assumption does not hold. We propose PersonaManifold, a framework that models persona representations as points on a curved, low-dimensional Riemannian submanifold in activation space. We estimate the manifold's intrinsic geometry---local metric tensors, geodesic distances, and Ollivier-Ricci curvature---and introduce geodesic steering, which interpolates between personas along manifold geodesics rather than Euclidean straight lines. We also propose the Behavioral Similarity Triplet (BST) benchmark, which automatically generates situational questions grounded in six established psychological constructs and defines persona similarity through behavioral responses rather than self-report questionnaires. Experiments on three open-source LLMs show that persona activations form a manifold with heterogeneous curvature, geodesic distance predicts behavioral similarity more accurately than Euclidean alternatives with independent contributions from anisotropy and curvature, and geodesic steering produces more coherent intermediate personas on both our BST benchmark and external evaluations, with the advantage concentrated in high-deviation regions where the manifold deviates most from flatness.

Figures & tables

Figure 1 : Overview of PersonaManifold . (1) Extract persona activations from an LLM’s residual stream. (2) Estimate manifold geometry: local metric tensors, geodesic distances, and Ollivier–Ricci curvature. (3) Geodesic steering interpolates along the manifold surface. (4) BST benchmark evaluates behavioral similarity via triplets.

Model	Method	Layer 12	Layer 16	Layer 20
Llama-3.1-8B	MP	22	23	21
Llama-3.1-8B	LB	21	22	20
Qwen2.5-7B	MP	19	20	18
Qwen2.5-7B	LB	18	19	17
Mistral-7B	MP	16	17	15
Mistral-7B	LB	15	16	15

Table 1 : Intrinsic dimensionality estimates

d^{*}

across models and layers. MP: Marchenko–Pastur; LB: Levina–Bickel MLE. Both methods agree within

\pm 2

dimensions.

	Llama-3.1-8B			Qwen2.5-7B			Mistral-7B
Metric	Choice	Emb	Comb	Choice	Emb	Comb	Choice	Emb	Comb
Euclidean	60.2	62.8	62.1	59.5	61.3	61.5	58.8	60.7	60.8
Cosine	59.8	62.1	61.5	59.1	60.9	61.0	58.3	60.2	60.1
Mahalanobis	62.5	65.1	64.3	61.8	63.7	63.9	61.0	63.1	63.0
Isomap-Euclid.	63.8	66.2	65.8	63.0	64.9	65.1	62.5	64.5	64.3
Diffusion Map	63.1	65.5	65.0	62.4	64.7	64.5	61.8	63.8	63.7

Table 2 : Triplet Consistency Rate (%) on BST. Best in bold .

\Delta

: improvement over Euclidean. All significant at

p<0.01

(bootstrap).

	Llama-3.1-8B			Qwen2.5-7B			Mistral-7B
Method	IC	TS	MA	IC	TS	MA	IC	TS	MA
Euclidean-Linear	0.58	0.52	0.41	0.55	0.49	0.38	0.53	0.47	0.36
SLERP	0.60	0.55	0.43	0.58	0.53	0.41	0.56	0.51	0.39
Isomap-Euclid.	0.65	0.61	0.55	0.62	0.57	0.52	0.60	0.56	0.49
Diffusion Map	0.63	0.59	0.52	0.60	0.55	0.49	0.58	0.54	0.47
UMAP	0.61	0.56	0.48	0.58	0.53	0.45	0.57	0.51	0.43

Table 3 : Steering quality on high-deviation pairs (top-25% by

d_{g}/d_{E}

). IC: Interpolation Coherence, TS: Transition Smoothness, MA: Manifold Adherence. Best in bold .

		TCR-combined (%)			Steering (Llama)
Component	Variant	Llama	Qwen	Mistral	IC	TS
Metric est.	Local PCA †	68.2	67.0	65.9	0.72	0.69
	Diffusion kernel	67.5	66.8	65.5	0.72	0.68
	Isotropic Euclidean	63.8	63.1	62.0	0.65	0.61
Graph $k$	$k=3d^{*}$	66.9	65.7	64.5	0.70	0.67
	$k=5d^{*}$ †	68.2	67.0	65.9	0.72	0.69

Table 4 : Ablation studies. TCR: TCR-combined (%). IC/TS: Llama high-deviation pairs. Default marked with

\dagger

Appendix figures & tables14 assets

Supplementary material from the paper’s appendix.

Appendix

Component	Parameter	Value
Extraction	Probe questions per persona	40
	Target layers	12–16
	PCA variance threshold	99%
	Ambient $\to$ reduced dim ( $D\to D^{\prime}$ )	4096 $\to$ 200–500
Geometry	$k$ -NN graph $k$	$5d^{*}$
	Metric regularization $\epsilon$	$10^{-4}$

Appendix

Table A.1 : Hyperparameters for PersonaManifold .

You are a psychometrician designing behavioral assessments.

Original item from [INVENTORY]: "[ITEM_TEXT]"

Rewrite this as a situational question that:

1. Describes a specific, concrete scenario the respondent is placed in.

2. Asks "What do you do?" or "How do you respond?" --- never "How much do you agree?".

3. Does NOT mention the trait name or any psychological jargon.

4. Has no obvious "correct" answer --- any reasonable adult could plausibly choose different actions.

Appendix

Table B.1 : Prompt template used to rephrase personality inventory items into situational behavioral probes.

#	Facet (Source)	Probe Question
1	Extraversion (B)	You are at a networking event where you know no one. The person next to you seems absorbed in their phone. What do you do?
2	Extraversion (H)	Your team just finished a major project. Someone suggests a celebratory dinner at a noisy restaurant. You were planning a quiet evening at home. How do you handle the situation?
3	Extraversion (T)	You arrive at a party where you recognize only the host. The host is busy greeting other guests. Describe your first 15 minutes.
4	Agreeableness (B)	A colleague takes credit for an idea you shared in a meeting last week. Your manager asks for your opinion in a follow-up email. How do you respond?
5	Agreeableness (H)	A friend asks you to review their business plan. You think it has fundamental flaws. They seem excited and have already invested personal savings. What do you say?
6	Agreeableness (T)	A customer at a store is berating a cashier over a minor policy issue. You are next in line. What do you do?

Appendix

Table B.2 : Representative subset of the 40 probe questions (15 of 40 shown; full list in supplementary code). Source: B = BFI-44, H = HEXACO-60, T = TIPI.

You are a psychometrician designing behavioral assessment questions grounded in Moral Foundations Theory (Graham et al., 2013).

Moral Foundations Theory identifies five moral foundations:

- Care/Harm: sensitivity to suffering, compassion, nurturance

- Fairness/Cheating: proportionality, justice, reciprocity

- Loyalty/Betrayal: group solidarity, patriotism, self-sacrifice for the group

- Authority/Subversion: respect for hierarchy, duty, obedience

- Sanctity/Degradation: purity, disgust, bodily integrity

Appendix

Table C.1 : Full BST generation prompt for Moral Foundations Theory (forced-choice format).

You are designing behavioral assessment questions grounded in the Domain-Specific Risk-Taking scale (DOSPERT; Blais & Weber, 2006).

DOSPERT measures risk attitudes across five domains:

- Financial (investment/gambling): willingness to risk money

- Health/Safety: tolerance for physical danger

- Recreational: interest in thrill-seeking activities

- Ethical: willingness to bend rules for personal gain

- Social: comfort with social disapproval or rejection

Appendix

Table C.2 : Full BST generation prompt for DOSPERT (open-ended format).

Construct	Example Question
Moral Foundations (forced-choice)	You witness a stranger cutting in line at a busy coffee shop. Do you: (A) Say nothing, (B) Politely point out the line, (C) Loudly call them out, (D) Tell the barista?
DOSPERT (open-ended)	You are offered a chance to invest 30% of your savings in a promising but unproven startup founded by a friend. They need an answer by tomorrow. What do you do?
Interpersonal Circumplex (forced-choice)	You are assigned to co-lead a project with someone who has a very different working style. They prefer strict schedules; you prefer flexibility. Do you: (A) Propose a compromise structure, (B) Adapt to their style entirely, (C) Suggest you divide tasks and work independently, (D) Raise the tension openly and discuss it?
Decision-Making (open-ended)	You need to choose a new apartment by Friday. You have two options with different trade-offs and no clear winner. Your partner likes Option A; your gut says Option B. How do you make the final call?
Schwartz Values (forced-choice)	Your employer asks you to relocate to a country with better career prospects but fewer personal freedoms. Your family prefers to stay. Do you: (A) Accept for the career growth, (B) Decline to keep the family stable, (C) Negotiate remote work, (D) Accept temporarily for one year?
Communicator Style (open-ended)	You need to deliver negative performance feedback to a team member who is also a close friend. The feedback is accurate but will likely hurt them. Walk through exactly how you prepare for and conduct the conversation.

Appendix

Table C.3 : Representative BST questions by construct (after discriminativeness filtering).

Task: Behavioral Similarity Judgment

You will read the behavioral responses of three personas (A, P, N) to the same 10 situational questions. Your task is to judge which of P or N is more behaviorally similar to A.

Guidelines:

– Focus on what each persona does in each situation, not their writing style or vocabulary.

– Consider the overall pattern across all 10 questions, not just one.

– There are no “correct” answers. Use your best judgment.

– If you genuinely cannot decide, select “Cannot tell.”

Appendix

Table D.1 : Annotation instruction sheet provided to human evaluators.

Construct	Fleiss $\kappa$	Human–System (%)	“Cannot tell” rate (%)
Moral Foundations	0.83	87.5	4.2
DOSPERT	0.88	91.0	2.1
Interpersonal Circumplex	0.79	84.5	6.8
Decision-Making Style	0.82	86.0	5.3
Schwartz Values	0.74	82.0	8.5
Communicator Style	0.80	85.5	5.0

Appendix

Table D.2 : Per-construct inter-annotator agreement and human–system agreement.

	Llama-3.1-8B		Qwen2.5-7B		Mistral-7B
Metric	Rare	Common	Rare	Common	Rare	Common
Euclidean	0.43	0.65	0.38	0.63	0.40	0.61
Cosine	0.41	0.63	0.37	0.62	0.39	0.62
Mahalanobis	0.50	0.69	0.44	0.66	0.46	0.65
Geodesic (ours)	0.57	0.71	0.54	0.67	0.52	0.65
$\Delta$ vs. Euclid.	+0.14	+0.06	+0.16	+0.04	+0.12	+0.04

Appendix

Table F.1 : Few-shot persona retrieval (MRR) on PersonaGym targets. Rare: bottom-25% local density; Common: top-25%. Best in bold . Precision@10 shows consistent patterns.

		Llama			Qwen			Mistral
		E	G	$\Delta$	E	G	$\Delta$	E	G	$\Delta$
Type	Professional	63.5	68.8	+5.3	62.8	67.5	+4.7	62.1	66.9	+4.8
	Demographic	62.0	67.6	+5.6	61.4	66.8	+5.4	60.5	65.0	+4.5
	Fictional	60.8	69.1	+8.3	59.9	67.8	+7.9	58.9	66.7	+7.8
Quartile	Q1 (flat)	63.8	65.2	+1.4	63.1	64.3	+1.2	62.3	63.4	+1.1
	Q2	62.5	66.3	+3.8	61.8	66.0	+4.2	61.0	65.3	+4.3

Appendix

Table K.1 : TCR-combined (%) by persona type and

d_{g}/d_{E}

quartile. Euclidean (E) vs. Geodesic (G).

\Delta

: absolute improvement.

	Llama-3.1-8B			Qwen2.5-7B			Mistral-7B
Method	IC	TS	MA	IC	TS	MA	IC	TS	MA
Euclidean-Linear	0.68	0.65	0.59	0.66	0.63	0.57	0.64	0.61	0.55
SLERP	0.69	0.66	0.60	0.67	0.64	0.58	0.66	0.64	0.56
Isomap-Euclid.	0.69	0.66	0.61	0.67	0.64	0.60	0.65	0.62	0.57
Graph-NN Chain	0.68	0.63	0.61	0.66	0.61	0.59	0.64	0.60	0.58
Geodesic (ours)	0.70	0.67	0.62	0.68	0.65	0.60	0.65	0.63	0.58

Appendix

Table K.2 : Steering quality on low-deviation pairs (bottom-25% by

d_{g}/d_{E}

). Differences between methods are small, as expected on near-flat manifold regions.

	Llama-3.1-8B			Qwen2.5-7B			Mistral-7B
Method	PS	BFI-M	TES	PS	BFI-M	TES	PS	BFI-M	TES
Euclidean-Linear	3.12	58.4	61.3	2.98	55.1	58.7	2.85	53.8	56.2
SLERP	3.25	61.2	64.0	3.11	58.3	61.5	3.01	56.5	59.1
Isomap-Euclid.	3.48	67.5	70.2	3.35	64.0	67.3	3.22	62.1	64.8
Graph-NN Chain	3.40	64.8	68.5	3.28	61.5	65.7	3.18	60.3	63.0
Geodesic (ours)	3.71	72.8	76.5	3.58	69.4	73.1	3.42	66.7	70.3

Appendix

Table L.1 : External benchmark results for steering on high-deviation pairs. PS: PersonaGym PersonaScore (1–5), BFI-M: BFI-44 Monotonicity (%), TES: Trait Expression Score (0–100). Best in bold .

Manifold	Spearman $\rho$	Kendall $\tau$ (10% subsample)	$d^{*}$ recovered
Swiss Roll ( $\sigma=0.1$ , $\mathbb{R}^{3}$ )	0.94	0.82 $\pm$ 0.04	2.0
Swiss Roll ( $\sigma=0.3$ , $\mathbb{R}^{3}$ )	0.89	0.76 $\pm$ 0.05	2.3
Swiss Roll ( $\sigma=0.1$ , $\mathbb{R}^{10}$ )	0.86	0.74 $\pm$ 0.06	2.4
$S^{2}$ Sphere ( $\mathbb{R}^{3}$ )	0.92	0.85 $\pm$ 0.03	2.0

Appendix

Table M.1 : Geodesic estimation accuracy on synthetic manifolds.

Method	$t=0.0$	$t=0.25$	$t=0.5$	$t=0.75$	$t=1.0$
Linear	0.72	0.54	0.43	0.57	0.72
Geodesic	0.72	0.69	0.66	0.68	0.72

Appendix

Table N.1 : Mean IC at different interpolation points for high-deviation pairs (averaged over 50 pairs). The linear path shows a coherence dip at

t=0.5

; the geodesic path maintains coherence.

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