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Activation steering becomes a Schrödinger Bridge on the residual-stream hypersphere, deriving the usual density-ratio direction as a special case and making the intervention query-adaptive.

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2026-07-14 11:06 UTC pith:RQBD3HIF

load-bearing objection Clean SB derivation of the density-ratio objective plus a query-adaptive spherical algorithm that actually beats fixed-direction baselines without the usual OOD tax. the 2 major comments →

arxiv 2607.10517 v1 pith:RQBD3HIF submitted 2026-07-12 cs.LG cs.AI

Conditional Optimal Bridge for Riemannian Activation Steering

Seyed Arshan Dalili , Ajay Narayanan Sridhar , Vijaykrishnan Narayanan , Mehrdad Mahdavi This is my paper
classification cs.LG cs.AI
keywords activation steeringSchrödinger Bridgeentropic optimal transportresidual-stream geometryquery-adaptive interventionLLM alignmentout-of-distribution robustness
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Many methods that steer large language models by editing residual-stream activations already behave as if they were climbing the log-density ratio between desired and undesired activations, but they adopt that objective by habit rather than from a well-posed optimization problem, and they usually apply the same fixed direction to every query. This paper recasts the problem as a Schrödinger Bridge between the two activation distributions on the hypersphere of roughly constant residual norm. Solving the bridge with entropic optimal transport and reading off the probability-flow ODE recovers the familiar density-ratio gradient when the dual potentials are uniform, and otherwise yields a direction that depends on the current activation. The resulting algorithm, Cobras, therefore steers more where the query looks like the training contrast and largely abstains where it does not. Across four open models and the axes of helpfulness, truthfulness, and detoxification, the method improves the target metrics while avoiding the sharp drops on unrelated benchmarks that fixed-direction steers often produce.

Core claim

The Schrödinger Bridge that transports undesired residual activations to desired ones on the residual-stream hypersphere has a probability-flow drift equal to the Riemannian gradient of the log-ratio of the two Schrödinger potentials; that drift is exactly the log-density-ratio gradient when the Sinkhorn potentials become uniform, and it is inherently evaluated at the query activation, so the steering direction is conditional rather than fixed.

What carries the argument

The Cobras vector field: a weighted difference of spherical log-maps from the current activation to positive and negative contrastive samples, with weights obtained by extending the Sinkhorn dual potentials through a spherical heat kernel and then taking a softmax; the field is integrated by geodesic Euler steps on the hypersphere and gated by a nearest-neighbor abstain rule for out-of-support queries.

Load-bearing premise

At a fixed layer the residual-stream activations sit tightly enough on a sphere of nearly constant radius that geodesic distances and spherical gradients capture the geometry that matters for steering.

What would settle it

Measure residual-stream norms on the same models and layers used for steering; if the coefficient of variation is large and re-running Cobras with Euclidean rather than spherical geometry (or with a learned manifold) reverses or erases the reported gains on TruthfulQA, UltraFeedback, and RealToxicityPrompts while fixed-direction baselines remain unchanged, the hypersphere premise is false.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper casts LLM residual-stream activation steering as a Schrödinger Bridge between undesired and desired activation distributions on a fixed-radius hypersphere. Solving the bridge by entropic optimal transport (Sinkhorn duals) and extracting the probability-flow ODE yields a query-adaptive Riemannian drift (Eq. 13 / Eq. 19) that recovers the widely used log-density-ratio gradient as the special case of uniform Sinkhorn potentials (Prop. 1). The resulting algorithm (Cobras) combines spherical KDE, geodesic Euler integration, and abstention/vMF gates. Empirically, across four 7–8 B models and three alignment axes (UltraFeedback, TruthfulQA, RealToxicityPrompts), Cobras improves primary metrics over strong baselines while largely preserving MMLU/GSM8K accuracy when trained on TruthfulQA.

Significance. If the derivation and empirical claims hold, the work supplies the first optimization-based justification for the log-density-ratio objective that many existing steering methods already approximate, and simultaneously produces a practical query-adaptive method that mitigates the OOD degradation common to fixed-direction interventions. Strengths include a clean reduction from SB to the probability-flow ODE, an explicit special-case recovery of the density-ratio gradient (Prop. 1), a public code repository, multi-seed evaluation on four models, and an explicit abstention mechanism whose effect is ablated. These contributions are of clear interest to the representation-engineering and controllable-generation communities.

major comments (2)
  1. Sec. 3 / Fig. 1 / Prop. 1: The entire Riemannian construction (geodesic cost Eq. 6, short-time kernel Eq. 5, exp/log maps Eq. 1, Riemannian gradients Eq. 17, geodesic Euler steps in Alg. 1) rests on the claim that residual norms concentrate tightly enough that the relevant geometry is the sphere of constant radius R. Reported CoVs of 2.5–7.3 % leave a non-negligible mass of activations 10–20 % away from the mean radius; after forced projection the induced geodesics and the short-time heat-kernel approximation become approximate. Prop. 1 therefore rigorously recovers ∇_S log(p+/p−) only on the projected surrogate, not necessarily on the original activation space. A short sensitivity study (varying R, or comparing spherical vs. Euclidean KDE on the same samples) would make the load-bearing geometric assumption more secure.
  2. Alg. 1 lines 6–8 and Table 2: Query-adaptivity and OOD preservation are attributed to the Schrödinger potentials, yet the algorithm also multiplies the step size by an explicit abstain gate g and a vMF strength gate s. The OOD ablations in Table 2 vary only the abstention parameters; without a controlled ablation that disables the gates while keeping the SB potentials, it remains unclear how much of the favorable OOD trade-off in Fig. 2 is due to the bridge versus the hand-designed gates. A short “gates-off” column would isolate the contribution of the derived drift.
minor comments (5)
  1. Abstract and Sec. 1 claim “the first principled derivation”; Appendix A already unifies several prior methods under the same log-density-ratio objective. Softening the priority claim to “first derivation from a Schrödinger-Bridge optimization problem” would avoid overstatement.
  2. Fig. 2 caption states Falcon-7B was skipped because original performance was low; a brief quantitative note (or a supplementary panel) would make the OOD comparison complete.
  3. Notation: the same symbol K is used for geodesic Euler steps (Alg. 1) and for the short-time kernel bandwidth; a distinct symbol for the number of integration steps would improve readability.
  4. Default hyper-parameters (σ, T, K=10, abstention percentile 98, k_abs=32) are listed in Appendix C.4 but not in the main text; a short “default settings” paragraph in Sec. 5 would aid reproducibility.
  5. Typos: “modelsatinferencetime”, “andthen”, “log-density-ratioobjective” (missing spaces) appear in the abstract/introduction; a pass for spacing and hyphenation is needed.

Circularity Check

0 steps flagged

No circularity: SB optimization independently yields density-ratio gradient as a special case; empirical claims rest on external benchmarks.

full rationale

The central derivation begins from the well-posed Schrödinger Bridge problem (Eq. 3: KL-minimizing path measure with Brownian reference on the sphere) and obtains the probability-flow drift (Eq. 13) via standard entropic OT / Sinkhorn duals and the probability-flow ODE construction. Proposition 1 then shows that the widely used log-density-ratio gradient emerges only in the uniform-potential limit (c_max → 0). This is a genuine reduction of a known heuristic to a special case of an independent optimization problem; the density-ratio objective is not presupposed as an input. The query-adaptive weights arise by evaluating the same potentials at the current activation (Eqs. 15–19), again without circular definition. Empirical superiority is measured on external, independent benchmarks (TruthfulQA, UltraFeedback, RealToxicityPrompts, MMLU, GSM8K) that do not enter the SB construction. No self-citation is load-bearing for uniqueness or for the derivation itself; citations to prior SB and steering literature supply standard tools, not the target claim. The hypersphere modeling assumption (Sec. 3) is an approximation whose validity can be questioned, but that is an external modeling risk, not circularity of the derivation chain. The paper is therefore self-contained against its own inputs.

Axiom & Free-Parameter Ledger

5 free parameters · 4 axioms · 0 invented entities

The load-bearing claims rest on a geometric modeling choice (activations live on a sphere), a standard but non-unique reference process for the Schrödinger Bridge, a short-time kernel approximation, and a handful of free scalars that control bandwidth, step size and abstention. No new physical entities are postulated; the method re-uses classical SB dual potentials.

free parameters (5)
  • σ (diffusivity / kernel bandwidth)
    Controls the cost matrix and the width of the spherical kernels that define the Schrödinger potentials; chosen by hand / validation.
  • T (time horizon / steering strength)
    Scales the integrated flow; ablated in Fig. 3 and set model-specifically (default 0.65).
  • K (geodesic Euler steps)
    Number of discrete integration steps; default 10, ablated in Table 5.
  • abstention k_abs and percentile p
    Define the OOD gate radius ρ_ref; ablated in Table 2 (defaults k=32, p=98).
  • vMF strength gate s_vMF
    Per-query scalar that attenuates steering when the activation already aligns with the mean contrastive direction.
axioms (4)
  • domain assumption Residual-stream activations at a fixed layer concentrate in Euclidean norm, so the natural state space is the hypersphere of radius R equal to the layer-wise mean norm.
    Stated in Sec. 3 and supported only by the empirical CoV plots in Fig. 1; the entire Riemannian formulation depends on it.
  • domain assumption The reference path measure Q is Brownian motion on the sphere with constant diffusivity σ².
    Chosen in Sec. 4.1; other references would yield different bridges.
  • standard math In the short-time regime the transition kernel of spherical Brownian motion is the geodesic Gaussian K_σ(x,y)=exp(-d_S(x,y)²/(2σ²)).
    Invoked to obtain the static entropic OT problem (Eq. 5–6); standard approximation in the SB literature.
  • domain assumption Finite contrastive samples D± drawn from the desired/undesired behaviors are sufficient to estimate the Schrödinger potentials via Sinkhorn and spherical KDE.
    Implicit throughout Sec. 4; sample sizes are a few thousand pairs.

pith-pipeline@v1.1.0-grok45 · 29283 in / 2842 out tokens · 41677 ms · 2026-07-14T11:06:47.284903+00:00 · methodology

0 comments
read the original abstract

Activation steering offers a lightweight alternative to fine-tuning for controlling large language models at inference time. While many existing methods implicitly optimize a log-density-ratio objective between desired and undesired activation distributions, they do so heuristically rather than deriving it from a principled optimization problem. Moreover, these methods produce query-independent steering directions that can degrade performance on both in-distribution and out-of-distribution (OOD) inputs. We introduce \textsc{Cobras} (Conditional Optimal Bridge for Riemannian Activation Steering), which addresses both limitations by casting activation steering as a Schr\"{o}dinger Bridge on the residual-stream hypersphere. This formulation yields, to our knowledge, the first principled derivation of the log-density-ratio steering objective from a well-posed optimization problem. Solving the bridge via entropic optimal transport and extracting the probability flow ODE recovers the widely used density-ratio gradient as a special case when the Sinkhorn potentials are uniform. Crucially, the Schr\"{o}dinger potentials are evaluated at the current activation, making the resulting steering direction inherently query-adaptive. Empirically, across four models and three alignment axes (helpfulness, truthfulness, and detoxification), \textsc{Cobras} consistently outperforms prior activation steering baselines while avoiding the OOD degradation commonly observed in existing methods. The code can be found at https://github.com/arshandalili/cobras.

Figures

Figures reproduced from arXiv: 2607.10517 by Ajay Narayanan Sridhar, Mehrdad Mahdavi, Seyed Arshan Dalili, Vijaykrishnan Narayanan.

Figure 1
Figure 1. Figure 1: Normalized last-token residual-stream norms across layers on TruthfulQA for four LLMs. Each [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Performance of steering methods in OOD tasks: MMLU [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Cobras performance as a function of steering strength T. Moderate values of T generally yield the best TruthfulQA performance, while overly large values degrade both TruthfulQA and UltraFeedback performance for several models. Results are averaged over three runs with different seeds. For Falcon-7B, larger K increases informativeness, but also reduces the True score, leading to only a modest gain in True ×… view at source ↗
Figure 4
Figure 4. Figure 4: Performance vs. steering strength (T) on TruthfulQA and UltraFeedback across four models for selected steering methods. D.4 Performance vs Steering Strength for baselines To select the steering strength for each baseline, we sweep over model-specific ranges: T = 20–23 for Falcon-7B, T = 2–5 for Mistral-7B, T = 3–6 for LLaMA3.1-8B, and T = 13–16 for Qwen2.5-7B [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗

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Reference graph

Works this paper leans on

31 extracted references · 19 linked inside Pith

  1. [1]

    Understanding intermediate layers using linear classifier probes

    Guillaume Alain and Yoshua Bengio. Understanding intermediate layers using linear classifier probes. arXiv preprint arXiv:1610.01644,

    Pith/arXiv arXiv

  2. [2]

    The falcon series of open language models.arXiv preprint arXiv:2311.16867,

    Ebtesam Almazrouei, Hamza Alobeidli, Abdulaziz Alshamsi, Alessandro Cappelli, Ruxandra Cojocaru, Mérouane Debbah, Étienne Goffinet, Daniel Hesslow, Julien Launay, Quentin Malartic, et al. The falcon series of open language models.arXiv preprint arXiv:2311.16867,

    Pith/arXiv arXiv

  3. [3]

    Understanding (un) reliability of steering vectors in language models.arXiv preprint arXiv:2505.22637,

    Joschka Braun, Carsten Eickhoff, David Krueger, Seyed Ali Bahrainian, and Dmitrii Krasheninnikov. Understanding (un) reliability of steering vectors in language models.arXiv preprint arXiv:2505.22637,

    Pith/arXiv arXiv

  4. [4]

    Likelihood training of schr\" odinger bridge using forward-backward sdes theory.arXiv preprint arXiv:2110.11291,

    Tianrong Chen, Guan-Horng Liu, and Evangelos A Theodorou. Likelihood training of schr\" odinger bridge using forward-backward sdes theory.arXiv preprint arXiv:2110.11291,

    Pith/arXiv arXiv

  5. [5]

    Deep multi-marginal momentum schrödingerbridge

    Tianrong Chen, Guan-horng Liu, Molei Tao, and Evangelos A Theodorou. Deep multi-marginal momentum schrödingerbridge. InProceedings of the 37th International Conference on Neural Information Processing Systems, pages 57058–57086, 2023a. Xin Chen, Xue-Mei Li, and Bo Wu. Logarithmic heat kernel estimates without curvature restrictions. The Annals of Probabil...

    Pith/arXiv arXiv

  6. [6]

    Training verifiers to solve math word problems.arXiv preprint arXiv:2110.14168,

    Karl Cobbe, Vineet Kosaraju, Mohammad Bavarian, Mark Chen, Heewoo Jun, Lukasz Kaiser, Matthias Plappert, Jerry Tworek, Jacob Hilton, Reiichiro Nakano, Christopher Hesse, and John Schulman. Training verifiers to solve math word problems.arXiv preprint arXiv:2110.14168,

    Pith/arXiv arXiv

  7. [7]

    Ultrafeedback: Boosting language models with scaled ai feedback

    Ganqu Cui, Lifan Yuan, Ning Ding, Guanming Yao, Bingxiang He, Wei Zhu, Yuan Ni, Guotong Xie, Ruobing Xie, Yankai Lin, et al. Ultrafeedback: Boosting language models with scaled ai feedback. arXiv preprint arXiv:2310.01377,

    Pith/arXiv arXiv

  8. [8]

    Realtoxicityprompts: Evaluating neural toxic degeneration in language models.arXiv preprint arXiv:2009.11462,

    10 Samuel Gehman, Suchin Gururangan, Maarten Sap, Yejin Choi, and Noah A Smith. Realtoxicityprompts: Evaluating neural toxic degeneration in language models.arXiv preprint arXiv:2009.11462,

    Pith/arXiv arXiv 2009

  9. [9]

    Measuring massive multitask language understanding.arXiv preprint arXiv:2009.03300,

    Dan Hendrycks, Collin Burns, Steven Basart, Andy Zou, Mantas Mazeika, Dawn Song, and Jacob Steinhardt. Measuring massive multitask language understanding.arXiv preprint arXiv:2009.03300,

    Pith/arXiv arXiv 2009

  10. [10]

    Albert Q. Jiang, Alexandre Sablayrolles, Arthur Mensch, Chris Bamford, Devendra Singh Chaplot, Diego de las Casas, Florian Bressand, Gianna Lengyel, Guillaume Lample, Lucile Saulnier, Lélio Re- nard Lavaud, Marie-Anne Lachaux, Pierre Stock, Teven Le Scao, Thibaut Lavril, Thomas Wang, Timothée Lacroix, and William El Sayed. Mistral 7b.arXiv preprint arXiv:...

    Pith/arXiv arXiv

  11. [11]

    URL https://arxiv.org/abs/2310.06825

    doi:10.48550/arXiv.2310.06825. URL https://arxiv.org/abs/2310.06825. Jigsaw and Google. Perspective api. https://perspectiveapi.com/,

    Pith/arXiv arXiv doi:10.48550/arxiv.2310.06825

  12. [12]

    Anton Korznikov, Andrey Galichin, Alexey Dontsov, Oleg Y Rogov, Ivan Oseledets, and Elena Tutubalina

    Accessed: 2026-05-03. Anton Korznikov, Andrey Galichin, Alexey Dontsov, Oleg Y Rogov, Ivan Oseledets, and Elena Tutubalina. The rogue scalpel: Activation steering compromises llm safety.arXiv preprint arXiv:2509.22067,

    arXiv 2026

  13. [13]

    Rewardbench: Evaluating reward models for language modeling

    Nathan Lambert, Valentina Pyatkin, Jacob Morrison, LJ Miranda, Bill Yuchen Lin, Khyathi Chandu, Nouha Dziri, Sachin Kumar, Tom Zick, Yejin Choi, et al. Rewardbench: Evaluating reward models for language modeling. InFindings of the Association for Computational Linguistics: NAACL 2025, pages 1755–1797,

    2025

  14. [14]

    Truthfulqa: Measuring how models mimic human falsehoods.arXiv preprint arXiv:2109.07958,

    Stephanie Lin, Jacob Hilton, and Owain Evans. Truthfulqa: Measuring how models mimic human falsehoods.arXiv preprint arXiv:2109.07958,

    Pith/arXiv arXiv

  15. [15]

    The linear representation hypothesis and the geometry of large language models.arXiv preprint arXiv:2311.03658,

    Kiho Park, Yo Joong Choe, and Victor Veitch. The linear representation hypothesis and the geometry of large language models.arXiv preprint arXiv:2311.03658,

    Pith/arXiv arXiv

  16. [16]

    Householder pseudo-rotation: A novel approach to activation editing in llms with direction-magnitude perspective.arXiv preprint arXiv:2409.10053,

    Van-Cuong Pham and Thien Huu Nguyen. Householder pseudo-rotation: A novel approach to activation editing in llms with direction-magnitude perspective.arXiv preprint arXiv:2409.10053,

    Pith/arXiv arXiv

  17. [17]

    The necessity for intervention fidelity: Unintended side effects when steering llms

    Jonas B Raedler, Weiyue Li, Alyssa Mia Taliotis, Manasvi Goyal, Siddharth Swaroop, and Weiwei Pan. The necessity for intervention fidelity: Unintended side effects when steering llms. InICML 2025 Workshop on Methods and Opportunities at Small Scale,

    2025

  18. [18]

    doi:10.18653/v1/2024.acl-long.828

    Association for Computational Linguistics. doi:10.18653/v1/2024.acl-long.828. URL https://aclanthology.org/2024.acl-long.828/. Pau Rodriguez, Arno Blaas, Michal Klein, Luca Zappella, Nicholas Apostoloff, marco cuturi, and Xavier Suau. Controlling language and diffusion models by transporting activations. InThe Thirteenth International Conference on Learni...

    doi:10.18653/v1/2024.acl-long.828 2024

  19. [19]

    Representation surgery: Theory and practice of affine steering.arXiv preprint arXiv:2402.09631,

    11 Shashwat Singh, Shauli Ravfogel, Jonathan Herzig, Roee Aharoni, Ryan Cotterell, and Ponnurangam Kumaraguru. Representation surgery: Theory and practice of affine steering.arXiv preprint arXiv:2402.09631,

    Pith/arXiv arXiv

  20. [20]

    Score-based generative modeling through stochastic differential equations.arXiv preprint arXiv:2011.13456,

    Yang Song, Jascha Sohl-Dickstein, Diederik P Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations.arXiv preprint arXiv:2011.13456,

    Pith/arXiv arXiv 2011

  21. [21]

    Tian Xie, Haoming Luo, Haoyu Tang, Yiwen Hu, Jason Klein Liu, Qingnan Ren, Yang Wang, Wayne Xin Zhao, Rui Yan, Bing Su, et al

    URL https://openreview.net/forum?id=7TDnfx5s14. Tian Xie, Haoming Luo, Haoyu Tang, Yiwen Hu, Jason Klein Liu, Qingnan Ren, Yang Wang, Wayne Xin Zhao, Rui Yan, Bing Su, et al. Controlled llm training on spectral sphere.arXiv preprint arXiv:2601.08393,

    arXiv

  22. [22]

    A spectral condition for feature learning.arXiv preprint arXiv:2310.17813,

    Greg Yang, James B Simon, and Jeremy Bernstein. A spectral condition for feature learning.arXiv preprint arXiv:2310.17813,

    Pith/arXiv arXiv

  23. [23]

    Spherical steering: Geometry-aware activation rotation for language models.arXiv preprint arXiv:2602.08169,

    Zejia You, Chunyuan Deng, and Hanjie Chen. Spherical steering: Geometry-aware activation rotation for language models.arXiv preprint arXiv:2602.08169,

    Pith/arXiv arXiv

  24. [24]

    Odesteer: A unified ode-based steering framework for llm alignment.arXiv preprint arXiv:2602.17560,

    Hongjue Zhao, Haosen Sun, Jiangtao Kong, Xiaochang Li, Qineng Wang, Liwei Jiang, Qi Zhu, Tarek Abdelzaher, Yejin Choi, Manling Li, et al. Odesteer: A unified ode-based steering framework for llm alignment.arXiv preprint arXiv:2602.17560,

    arXiv

  25. [25]

    Representation engineering: A top-down approach to ai transparency.arXiv preprint arXiv:2310.01405,

    Andy Zou, Long Phan, Sarah Chen, James Campbell, Phillip Guo, Richard Ren, Alexander Pan, Xuwang Yin, Mantas Mazeika, Ann-Kathrin Dombrowski, et al. Representation engineering: A top-down approach to ai transparency.arXiv preprint arXiv:2310.01405,

    Pith/arXiv arXiv

  26. [26]

    Specifically, CAA simplifies this trajectory by optimizing over a flat Euclidean space M = Rd via a single discrete step (K = 1)

    Contrastive Activation Addition (CAA) [Rimsky et al., 2024]The CAA algorithm can be framed as a specific, single-step optimization instantiation of a continuous global flow over the hidden state manifold. Specifically, CAA simplifies this trajectory by optimizing over a flat Euclidean space M = Rd via a single discrete step (K = 1). By modeling the factua...

    2024

  27. [27]

    13 Table 3: Summary of Parameterizations under the Unified Global Flow Framework

    presents a related unification of several of these methods, but their analysis is confined to Euclidean space; by treating the manifoldM as an explicit design choice, however, our framework additionally captures geometry-aware methods such as SS and HPR. 13 Table 3: Summary of Parameterizations under the Unified Global Flow Framework. Method Manifold (M) ...

    2024

  28. [28]

    B.2 Derivation of the Riemannian Gradient oflog ˆψandlog ˆϕ We derive Eq.(17) which is the closed-form Riemannian gradient of the log backward Schrödinger factor onS

    The numerator and denominator are spherical kernel density estimates ofp + andp −, which converge to the true densities asN± → ∞[Pelletier, 2005]. B.2 Derivation of the Riemannian Gradient oflog ˆψandlog ˆϕ We derive Eq.(17) which is the closed-form Riemannian gradient of the log backward Schrödinger factor onS. The derivation for∇ S log ˆφis identical wi...

    2005

  29. [29]

    By the chain rule and the fact that ∇S dS(x,h) 2 =−2 log x(h): ∇S Kσ(x,h) =K σ(x,h)· −1 2σ ∇S dS(x,h) 2 =K σ(x,h)· −1 2σ · −2 log x(h) = 1 σ Kσ(x,h) log x(h).(24) B.4 Gradient of the Log Backward Potential Define the backward potential atx∈ S: ˆψ(x) = N+X i=1 ˜ψi(x)K σ2(x,h + i ),(25) where the extended potentials˜ψi(x)are also treated as constants with r...

    2026

  30. [30]

    The judge predicts whether a response is informative or helpful using prompts of the form: Q: {question} \n A: {answer}

    using a publicly available fine-tuned LLaMA2-7B judge9. The judge predicts whether a response is informative or helpful using prompts of the form: Q: {question} \n A: {answer}. \n Helpful: We report the percentage of generations classified as informative. • Truthfulness × Informativeness (%).Following prior work, we report the percentage of responses that...

    2025

  31. [31]

    Lower values therefore indicate less toxic generations

    The toxicity score is a probability-like score indicating the likelihood that a reader would perceive the generated text as toxic. Lower values therefore indicate less toxic generations. • Perplexity.We evaluate generation fluency using perplexity under GPT-2-XL Radford et al. [2019]. Given a generated responsey = (y1, . . . , yT ), GPT-2-XL assigns a nex...

    2019