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Naturalness-guided Manifold Flow Matching for Sign Language Production

Jiayi He, Shengeng Tang, Sisi You, Yanbin Hao, Lechao Cheng, Richang Hong

arXiv:2609.33339v1cs.AI

TL;DR

Existing flow-matching SLP methods use linear interpolation that leaves the manifold of valid joint rotations and ignores the motion distribution. SignNMFlow learns conditional paths on the motion manifold by minimizing kinetic energy under a prototype-based naturalness measure, coupling a closed-form geodesic with a learnable deviation. SignNMFlow achieves hand DTW-PA-JPE values of 1.07, 1.30, and 2.07.

  • Problem

    Existing flow-matching SLP methods use linear interpolation that leaves the manifold of valid joint rotations and ignores the motion distribution.

    Source: §2.1 · p. 3

  • Method

    SignNMFlow learns conditional paths on the motion manifold by minimizing kinetic energy under a prototype-based naturalness measure, coupling a closed-form geodesic with a learnable deviation.

    Source: §1 · p. 2, §3.3 · p. 5, §5 · p. 9

  • Results

    SignNMFlow achieves hand DTW-PA-JPE values of 1.07, 1.30, and 2.07.

    Source: §1 · p. 2

  • Takeaways & Limitations

    SignNMFlow provides geometrically efficient, motion-distribution-aware conditional paths for generating higher-fidelity sign motions.

    Source: §1 · p. 2, §5 · p. 9

Abstract

from arXiv · show

Sign Language Production (SLP) aims to generate sign motions from text. Conditional Flow Matching methods have achieved strong performance in SLP by constructing conditional paths that transform a source distribution into a target distribution. However, existing methods construct these paths via linear interpolation, whereas the rotational geometry of human joints confines valid joint rotations to a manifold embedded in Euclidean space. Consequently, linear interpolation between two sign motions leaves this manifold and ignores the motion distribution on it. In this paper, we revisit SLP from the perspective of manifold transport and propose a Naturalness-guided Manifold Flow Matching framework, termed \textbf{SignNMFlow}, which constructs conditional paths directly on the motion manifold by jointly considering geometric efficiency and the motion distribution. Specifically, we exploit the intrinsic geometry of the manifold and introduce a motion naturalness measure to characterize the motion distribution. By minimizing the kinetic energy under this measure, we learn a naturalness-guided interpolation that couples a closed-form geodesic, which provides geometrically efficient transport, with a learnable deviation that incorporates the motion distribution, thereby significantly improving the fidelity of generated sign motions. Extensive qualitative and quantitative evaluations demonstrate the effectiveness of this work.

1 INTRODUCTION

Sign Language Production must generate temporally coordinated, semantically consistent, visually natural motion, but existing flow-matching paths use linear interpolation that conflicts with joint-rotation geometry and ignores the motion distribution. SignNMFlow constructs paths on the motion manifold by combining geometric efficiency with distributional guidance.

1 INTRODUCTION

  • SLP has received relatively little attention compared with recognition and translation, and requires semantically consistent, temporally coordinated, visually natural motion.Recognition and translation decode existing sign content, whereas production generates motion from text.Source: §1 · p. 1
  • Existing flow-matching SLP methods use linear interpolation, which can produce invalid intermediate rotations and ignores the distribution of sign motions.The mismatch arises because valid joint rotations lie on a structured manifold, while the interpolation is formed in ambient Euclidean space.Source: §1 · p. 1
  • The desired conditional path should remain on the manifold while avoiding unnecessary detours and reflecting the motion distribution.The paper identifies geometric efficiency and distributional consistency as the two requirements for improving motion fidelity.Source: §1 · p. 2
  • SignNMFlow constructs on-manifold conditional paths by combining a closed-form geodesic with a learnable, naturalness-guided deviation.The interpolation is learned by minimizing kinetic energy under a motion naturalness measure built to characterize the motion distribution.Source: §1 · p. 2
  • The paper reports that How2Sign dataset evaluations demonstrate SignNMFlow’s effectiveness, with hand DTW-PA-JPE values of 1.07, 1.30, and 2.07, respectively.Source: §1 · p. 2

2 PRELIMINARIES

The preliminaries establish the geometric mismatch between Euclidean linear interpolation and valid sign-motion rotations, then introduce flow matching and its conditional training formulation.

2.1 GEOMETRIC MISMATCH

  • Each sign-motion frame comprises J rotations, so valid motions occupy the product manifold SO(3)^J embedded in R^9J rather than the full Euclidean space.Linear interpolation of rotation representations leaves the manifold for intermediate states, motivating on-manifold conditional paths.Source: §2.1 · p. 3

2.2 FLOW MATCHING

  • Flow matching learns a velocity field transporting samples from a source distribution to a target distribution along a predefined probability path.It directly minimizes discrepancy between a learned vector field and a target velocity field.Source: §2.2 · p. 3
  • Conditional Flow Matching makes training tractable by conditioning paths and velocity fields on individual source-target pairs.Its conditional objective yields the same parameter gradients as the flow-matching objective and supports per-sample path-based supervision.Source: §2.2 · p. 3

3 NATURALNESS-GUIDED MANIFOLD FLOW MATCHING

SignNMFlow represents sign motions on a rotation manifold and learns conditional paths that combine intrinsic geometric efficiency with the distribution of observed motions.

3.1 MOTION REPRESENTATION

  • Each sign motion is represented as J unit quaternions on the product manifold (S3)^J, with tangent spaces and exponential and logarithmic maps defining its geometry.The representation removes the sign ambiguity between q and −q; sequence-level definitions extend the single-motion geometry frame by frame.Source: §3.1 · p. 4

3.2 MOTION NATURALNESS MEASURE

  • The motion naturalness measure uses K-means prototypes, frequency- and proximity-weighted contributions, and local prototype scatter to characterize motion distribution on the manifold.The measure is lifted from individual points to paths so distributional information can guide conditional-path construction.Source: §3.2 · p. 4

3.3 MOTION NATURALNESS LEARNING

  • When the naturalness measure is uniform, the minimum-energy path is the geodesic; with a non-uniform distribution, a learnable correction deviates from that reference.The exact minimization over all admissible paths is intractable, so the interpolation is parameterized around the geodesic with a correction network.Source: §3.3 · p. 5
  • SignNMFlow’s learned interpolation combines a closed-form geodesic with a learnable deviation, minimizing kinetic energy under the naturalness measure while preserving endpoint conditions.The geodesic supplies geometrically efficient transport; the deviation incorporates the non-uniform motion distribution.Source: §3.3 · p. 5
  • After learning the paths, the correction network is frozen and the generation network is trained with target prediction, projecting each predicted joint motion onto the manifold.At sampling time, each solver step re-predicts the target and advances to the next path point; facial expressions are generated in Euclidean space.Source: §3.3 · pp. 5–6

4 EXPERIMENTS

Experiments evaluate SignNMFlow on three sign-language datasets and compare it with existing methods and ablated variants. The ablations examine manifold geometry, naturalness-measure settings, and training objectives.

4.1 EXPERIMENTAL SETTINGS

  • Evaluation uses Phoenix-2014T, CSL-Daily, and How2Sign, with DTW-JPE and DTW-PA-JPE for motion accuracy and back-translated BLEU-4 for semantic intelligibility.The datasets cover German, Chinese, and American sign language; the multilingual model is trained on their combination.Source: §4.1 · p. 6

4.2 COMPARISON WITH STATE-OF-THE-ART METHODS

  • SignNMFlow generates more realistic, expressive motions with higher pose fidelity and spatial consistency than SOKE; SignNMFlow-M is also described as competitive.The qualitative comparison covers Phoenix-2014T, CSL-Daily, and How2Sign test sets.Source: Figure 2 · p. 7, §4.2 · p. 8

4.3 ABLATION STUDY

  • For flow-matching SLP variants, quaternion MR with GE lowers DTW-JPE All from 21.25 to 18.31 and DTW-PA-JPE All from 11.26 to 9.58; adding NG further lowers them to 16.86 and 8.92.The 6D baseline scores 20.08 and 10.78 on these respective metrics; the NG results are 16.04% and 17.25% below baseline.Source: §4.3 · p. 8
  • For the naturalness measure, performance peaks around K = 500; DTW-JPE All rises to 19.41 at σ = 0.25 and 17.58 at σ = 0.75, versus 16.86 at σ = 0.50.The authors attribute the larger degradation at σ = 0.25 to reliance on the nearest prototype and the milder degradation at σ = 0.75 to over-smoothing.Source: §4.3 · pp. 8–9, Table 4 · p. 8
  • Under the CFM baseline, target prediction reduces DTW-JPE All error from 21.62 to 20.08.The passage also reports that DTW-PA-JPE All remains well scaled across timesteps.Source: §4.3 · p. 9, Table 5 · p. 9

4.4 USER STUDY

  • Figure 4 presents a CSL-Daily user study comparing SignNMFlow with state-of-the-art methods.Source: Figure 4 · p. 9

5 CONCLUSIONS

SignNMFlow constructs conditional paths on the motion manifold by combining a geodesic with a learnable, naturalness-guided deviation. The paper situates this approach among SLP and flow-matching methods and points to its benchmarks, reproducibility materials, and derivation appendices.

5 CONCLUSIONS

  • SignNMFlow couples a closed-form geodesic with a learnable deviation, using a prototype-based naturalness measure and kinetic-energy minimization to construct manifold paths.The paper also reports a target-prediction parameterization that provides well-scaled supervision without extra losses, and evaluations on Phoenix-2014T, CSL-Daily, and How2Sign.Source: §5 · p. 9
  • The reproducibility statement locates datasets and evaluation protocols in Section 4.1, training and sampling procedures in Algorithms 1 and 2, and hyperparameters in Appendix D.Appendix B contains the full derivation of the geometric mismatch; comparisons with prior work follow the SOKE protocol.Source: REPRODUCIBILITY STATEMENT · p. 10

Appendix

  • The appendix contents list experimental details and additional experimental results as separate sections.The supplied passage gives their locations as D.1 and D.2, respectively.Source: Appendix · p. 13

A.1 SIGN LANGUAGE PRODUCTION

  • The related-work review identifies linear interpolation in Euclidean space as the path construction used by SignFlow and SignSparK, despite the rotational geometry and motion distribution of sign motions.It distinguishes GARD’s geodesic refinement of inter-gloss transitions from the geometry of generative transport.Source: §A.1 · p. 14, §A.2 · p. 14

B DETAILED DERIVATION OF PROPERTY 1

  • The geometric-mismatch derivation examines SO(3), then extends the off-manifold deviation to 6D, quaternion, and Euler-angle rotation representations.The section states that linear interpolation either leaves the representation manifold or distorts the rotation path, while exponential-map paths avoid both failure modes.Source: B DETAILED DERIVATION OF PROPERTY 1 · p. 14

B.1 DERIVATION ON SO(3)

The derivation shows that Euclidean linear interpolation between distinct valid rotations leaves the SO(3) manifold. The same off-manifold deviation holds for the 6D representation used by existing flow-matching SLP methods.

B.1 DERIVATION ON SO(3)

  • For distinct rotations, every interior point of their linear interpolation violates SO(3)’s orthogonality and determinant constraints.The derivation shows the squared Frobenius norm falls below 3 and the determinant below 1 for interior interpolation times.Source: §B.1 · pp. 14–15

B.2 DERIVATION ON THE 6D ROTATION REPRESENTATION

  • Linear interpolation between distinct valid 6D rotation representations leaves the 6D manifold because at least one component violates its unit-norm constraint.The passage notes that existing flow-matching SLP methods use this representation and that the off-manifold deviation persists under its linear coordinate map.Source: §B.2 · pp. 15–16

B.3 DERIVATION ON THE QUATERNION REPRESENTATION

The quaternion derivation shows that Euclidean linear interpolation between distinct unit quaternions leaves the unit sphere, including when applied joint by joint to motions. Renormalization restores the constraint but not constant-speed geodesic motion.

B.3 DERIVATION ON THE QUATERNION REPRESENTATION

  • Linear interpolation between distinct unit quaternions violates the unit-norm constraint at every interior time, so motion states leave the product of quaternion spheres joint by joint.The paper identifies its exponential-map construction as restoring manifold validity.Source: §B.3 · pp. 16–17
  • Renormalized linear interpolation follows the same great-circle arc as spherical interpolation but at state-dependent speed, departing from the constant-speed geodesic.The distortion grows without bound as the quaternion representatives approach antipodes.Source: §B.3 · p. 17

B.4 DERIVATION ON THE EULER-ANGLE REPRESENTATION

The Euler-angle analysis distinguishes off-manifold interpolation of a faithful torus embedding from distortion when raw angles are interpolated as Euclidean vectors. It also describes the paper’s quaternion sign convention and prototype extraction procedure.

B.4 DERIVATION ON THE EULER-ANGLE REPRESENTATION

  • Linear interpolation in the circle embedding of Euler angles leaves the torus whenever endpoint representations differ, with norm contraction greatest for angle increments that are odd multiples of π.For raw angle vectors, all intermediate triples remain valid parameterizations, but the induced rotation path can be state-dependent, rank-deficient, and non-injective.Source: §B.4 · pp. 17–18

C.1 QUATERNION SIGN CONVENTION

  • The paper resolves quaternion sign ambiguity by selecting the representative whose first nonzero component is positive, giving a unique representative for each rotation.The selected representatives lie in a closed hemisphere, so the logarithmic maps are evaluated between non-antipodal quaternions.Source: §C.1 · pp. 18–19

C.2 PROTOTYPE EXTRACTION

  • Prototype extraction applies per-joint K-means to training-set rotations and associates each prototype with its empirical occurrence probability.The normalized prototype coincides with the spherical K-means centroid and approximates the Riemannian mean when the cluster is concentrated.Source: §C.2 · p. 19

C.3 VALIDITY OF THE MOTION NATURALNESS MEASURE

The motion naturalness measure is shown to be well-defined, density-consistent, and distribution-aware, while recovering standard geodesic transport under a uniform motion distribution. The interpolation remains on the motion manifold, whose product-sphere geometry is described through exponential and logarithmic maps.

C.3 VALIDITY OF THE MOTION NATURALNESS MEASURE

  • The regularized measure is positive definite everywhere, making the naturalness norm and kinetic-energy objective well-defined across the manifold.Its density factor is positive, while the scatter term plus regularization is positive definite.Source: §C.3 · p. 19
  • The density factor is a prototype-compressed kernel density estimate of the motion distribution, with compression error controlled by the number of prototypes.Its consistency follows from manifold KDE as sample size grows and bandwidth shrinks.Source: §C.3 · pp. 19–20
  • At fixed local dispersion, the estimated density lowers transport cost in dense motion regions through the inverse-density factor.The metric also reflects prototype dispersion, making transit through sparse or dispersed regions expensive.Source: §C.3 · p. 20
  • Under a uniform motion distribution, the measure becomes a constant multiple of the identity, and the energy minimizer is a constant-speed minimizing geodesic.The learned correction is zero in this regime.Source: §C.3 · pp. 20–21

C.4 EXACTNESS OF THE NATURALNESS-GUIDED INTERPOLATION

  • The naturalness-guided interpolation stays on the motion manifold for all times, unlike linear interpolation, which leaves the rotation manifold at interior times.The exactness result holds for any raw correction-network output when its correction is projected onto the tangent space.Source: §C.4 · p. 22

C.4.1 CLOSED FORMS ON THE MANIFOLD

  • The motion manifold is a product of unit 3-spheres, with joint-wise exponential maps and logarithmic maps defined away from antipodal points.The sign disambiguation excludes the antipodal case.Source: §C.4.1 · p. 22

C.4.2 DERIVATION OF THE EXACTNESS RESULT

The derivation shows that tangent-projecting the correction keeps the learned interpolation on the manifold. This guarantee holds regardless of the raw network output.

C.4.2 DERIVATION OF THE EXACTNESS RESULT

  • Orthogonal projection places any raw correction-network output in the source tangent space, without requiring the raw output itself to lie on the manifold.Combining that tangent correction with the logarithmic-map direction keeps the path argument tangent; the exponential map then returns points on the manifold.Source: §C.4.2 · p. 22
  • The generation algorithm trains along the naturalness-guided path and renormalizes predicted poses per joint onto the manifold before computing the target-prediction loss.At sampling time, it re-predicts the target and re-plans transport in five Euler steps.Source: §C.4.2 · p. 23

C.5 TRAINING AND SAMPLING THE GENERATION NETWORK

The generation network is trained to predict target poses and expressions from states along the learned path. At inference, it re-predicts the target at each step and re-plans transport.

Training

  • Training samples states along the naturalness-guided path and trains the generation network to predict the target pose and expression.Predicted poses are normalized per joint onto the manifold before the loss is evaluated.Source: §C.5 · p. 23

Inference

  • At each inference step, the network re-predicts the target motion from the current state and re-plans the path from the retained source.The frozen correction is projected onto the source tangent space when evaluating the path.Source: §C.5 · p. 23

Expression channel

  • Facial expressions use an auxiliary Euclidean channel with a linear path, while joint rotations use the manifold construction.The pose and expression loss weights are 10 and 0.1, respectively.Source: §C.5 · p. 23

D.1 EXPERIMENTAL DETAIL

The experimental setup uses three sign-language datasets and SMPL-X motion parameters. The paper specifies separate correction and generation networks, training settings, and sampling hyperparameters.

Dataset Details

  • Experiments use Phoenix-2014T, CSL-Daily, and How2Sign, with SMPL-X parameters sourced from NSA for How2Sign and SOKE for the other datasets.The datasets cover German, Chinese, and American Sign Language, respectively.Source: §D.1 · p. 24

Implementation Details

  • The correction network is a three-layer MLP trained for 100 epochs, while the eight-layer transformer generation network is trained for 300 epochs.Both use AdamW and a cosine learning-rate schedule beginning at 2 × 10^-4; mBART-large-cc25 provides the text condition.Source: §D.1 · p. 24
  • The setup uses 500 prototypes per joint, bandwidth σ = 0.50, regularization ρ = 10^-6, and five-step explicit Euler sampling.The source pose is uniform on the sign-convention hemisphere; facial-expression noise is standard Gaussian.Source: §D.1 · p. 24

D.2 ADDITIONAL EXPERIMENTAL RESULTS

SignNMFlow is the most efficient of the compared methods, and its sampling quality is robust to the number of function evaluations; five evaluations are used by default.

D.2 ADDITIONAL EXPERIMENTAL RESULTS

  • On CSL-Daily, increasing NFE from 5 to 20 changes DTW-JPE All error from 16.86 to 16.98 and DTW-PA-JPE All error from 8.92 to 9.11.The passage reports consistent behavior on body and hand joints.Source: §D.2 · p. 25, Table 8 · p. 25
  • The re-planned sampling scheme places each state at the corresponding path point, and five function evaluations already saturate sampling quality, so NFE=5 is the default.The authors report that the scheme's target re-prediction at each solver step leaves discretization with little influence on generated motions.Source: §D.2 · p. 25
  • SignNMFlow uses 233.20M parameters and 0.0103 seconds per motion, yielding 8.3× and 26.1× speedups over SignFlow and SOKE, respectively.SignFlow uses 261.04M parameters and 0.0860 seconds; SOKE uses 430.40M parameters and 0.2687 seconds.Source: §D.2 · p. 25

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