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Schrödinger Bridges on Lie Group Manifolds for Probabilistic Intrinsic Generation

Shizhe Zhang, Mingyang Zhao, Lei Ma

arXiv:2609.02196v1stat.MLcs.AIcs.LG

TL;DR

Generative modeling on non-Euclidean data requires methods that preserve intrinsic geometry, while Schrödinger bridges provide entropy-regularized transport between prescribed endpoint laws. This paper develops observed-endpoint kinetic bridges on Lie groups, with WKBC for compact Abelian groups and RCCBM for compact non-Abelian groups, and reports validation across manifold and scientific datasets.

  • Problem

    Existing generative modeling approaches must account for intrinsic geometry, while scientific settings may observe only part of a kinetic Lie group state.

  • Method

    The paper develops observed-endpoint Schrödinger bridges on G×g and implements them with wrapped-kernel calibration or reciprocal conditional-control matching.

  • Results

    Experiments across protein and RNA torsions, SO(3), U(n), and protein conformational pathways demonstrate effectiveness across compact Abelian and non-Abelian settings.

  • Takeaways & Limitations

    The framework provides a unified probabilistic foundation with structure-adapted computational realizations for distribution-to-distribution transport on Lie groups.

  • Takeaways & Limitations

    Evaluation remains concentrated on moderate-dimensional groups and selected tasks, while genuinely noncompact Lie groups and more general geometric state spaces remain outside scope.

Abstract

from arXiv · show

Generative modeling directly on geometric manifolds can avoid errors introduced by flattening non-Euclidean data, repeated ambient projection, and coordinate inconsistency in Euclidean representations. Schrodinger bridges provide a probabilistic generative framework for entropy-regularized transport between prescribed endpoint distributions. We study Schrodinger bridges for kinetic dynamics on Lie group manifolds with state X_t = (g_t, xi_t) in G x g, allowing endpoint observations to constrain only the variables that are actually measured. In particular, the entropy projection determines the conditional law of the unobserved endpoint velocities. For the same observed endpoint bridge, we develop two computational realizations: Wrapped-Kernel Bridge Calibration (WKBC) uses an explicit periodized kinetic kernel on compact Abelian groups, whereas Reciprocal Conditional-Control Bridge Matching (RCCBM) handles compact non-Abelian groups through two-sided endpoint calibration and mollified conditional-control matching. The canonical teacher-mixture path law is itself a Markov reciprocal law, so forward generation uses a calibrated initial law and one learned Doob controller. Moreover, we establish a modular error bound in the bounded-Lipschitz path metric that provides a clean separation of errors due to endpoints, control regression, initialization, discretization, and related approximations. Experiments on multiple Lie group manifold datasets validate the feasibility and consistency of our proposed method, covering protein and RNA torsions, SO(3), U(n), and the Protein Conformational Transition Pathway Generation task using mdCATH trajectories in a compact reduced representation. The source code is publicly available at https://github.com/cafferyzhang12/Schr-dinger_Bridge_on_LieGroup.

1 INTRODUCTION

The paper formulates observed-endpoint kinetic Schrödinger bridges on Lie group manifolds, preserving intrinsic geometry while inferring latent endpoint velocities. It develops WKBC and RCCBM as structure-adapted realizations of the same bridge and validates them across several manifold datasets.

  • Motivation: Intrinsic manifold modeling preserves the geometry of periodic torsions, rotations, and rigid configurations throughout generative dynamics.Euclidean parameterizations can require repeated projection or tangent-space transport, introducing geometric overhead and numerical errors.
  • Motivation: Schrödinger bridges formulate stochastic optimal transport between prescribed endpoint laws while remaining close in relative entropy to reference dynamics.This distribution-to-distribution formulation retains stochastic path information for constrained generation.
  • Observed-endpoint bridge: The observed-endpoint formulation constrains only measured variables, while entropy projection determines latent endpoint velocity conditionals and the intermediate path law.Two-sided Schrödinger scaling links endpoint coupling, latent conditionals, and trajectory dynamics.
  • Observed-endpoint bridge: The kinetic state uses X_t=(g_t,ξ_t), reconstructing group coordinates intrinsically while applying stochastic forcing and controls in the Lie algebra.This separates geometric configuration from velocity-like variables without relaxing the constraint on g_t.
  • Computational realizations: WKBC uses an explicit wrapped kinetic kernel on compact Abelian groups, whereas RCCBM uses endpoint calibration and conditional-control matching when kernels are unavailable on compact non-Abelian groups.Both procedures target the same observed-endpoint bridge; rigid frame data are handled after compact reduction.
  • Validation: Experiments cover protein and RNA torsions, SO(3), U(n), and mdCATH-based protein conformational transition pathways, with ablations and numerical consistency checks.The reported validation is quantitative across the evaluated settings.

2 RELATED WORK

Related work spans Schrödinger bridges, reciprocal matching, intrinsic manifold generation, Lie group dynamics, and geometric-state-space bridges. The paper positions itself as an observed-endpoint kinetic bridge on G×g with structure-adapted computational methods.

  • Schrödinger bridges and reciprocal matching: Schrödinger bridge research began with entropy projection onto prescribed endpoint laws and later connected the problem to optimal transport, stochastic control, and generative modeling.Related computational approaches include iterative proportional fitting and simulation-based bridge methods.
  • Manifold and Lie group generative modeling: Intrinsic generative modeling introduced distributions, flows, and continuous-time formulations for data supported on non-Euclidean manifolds such as Lie groups, tori, and spheres.These methods provide geometric tools for modeling manifold-valued observations.
  • Kinetic Lie group dynamics: Kinetic Lie group methods evolve momentum in the Lie algebra while updating configurations intrinsically on the group.Subsequent work uses fixed Lie algebra auxiliary variables and generalized score matching for group-valued generation.
  • Position of this work: Prior manifold Schrödinger bridges address Riemannian or Brownian settings, whereas this work uses kinetic state space G×g and Lie algebra velocity control.The formulation also permits endpoint observations of only part of the state.

3 METHODOLOGY

The methodology defines an observed-endpoint kinetic Schrödinger system on Lie groups and derives two computational regimes. WKBC exploits wrapped kernels on compact Abelian groups, while RCCBM calibrates endpoints and learns Doob control through conditional teachers on compact non-Abelian groups.

  • Kinetic generator: The kinetic generator evolves g intrinsically through ξ and applies drift and diffusion to ξ in the Lie algebra.The compact-group experiments use a kinetic Ornstein–Uhlenbeck specialization.
  • Observed-endpoint Schrödinger system: Under the observed endpoint kernel condition, the endpoint problem becomes a bounded two-sided scaling problem with a well-posed Schrödinger system.The condition is verified for compact connected groups under continuous strictly positive densities in the stated setting.
  • Compact Abelian groups: WKBC: WKBC periodizes the Gaussian lift of kinetic dynamics on a torus to obtain an exact wrapped kinetic kernel and explicit observation kernel.The wrapped kernel supports endpoint calibration and computation of the full kinetic marginal score and forward Doob control.
  • Compact non-Abelian groups: RCCBM: RCCBM targets the same bridge without a closed-form kernel by calibrating the two-sided endpoint law and constructing finite-energy mollified conditional-control teachers.Its stages include endpoint calibration and conditional-control matching on compact non-Abelian groups.
  • RCCBM calibration: Endpoint calibration has population dual gap D(β*)−D(β)=KL(Γ*∥Γβ)=KL(P SB∥P β), making it an exact path-law error measure.Thus calibration is tied directly to bridge accuracy rather than serving only as a heuristic.
  • RCCBM control learning: The conditional teacher mixture is a Markov reciprocal law, allowing conditional regression to identify the Markov Doob control without changing the population target.Mollification avoids singular point-pinned controls near the terminal time, while implementation devices approximate the population objective.
  • Error propagation: RCCBM control error satisfies Δctrl,N ≤ εapp,N + εopt,N + 4εgen,N + 4εteach,N + 2εiw,N + 2εclip,N + δref,N.The bound separates approximation, optimization, generation, teacher, importance-weighting, clipping, and refinement contributions under the stated assumptions.

4 EXPERIMENTS

Experiments evaluate WKBC and RCCBM across compact Abelian torsions, non-Abelian rotations, protein transitions, ablations, and numerical sensitivity. Results support accurate endpoint and path modeling, bridge recovery, and intrinsic geometric fidelity across these settings.

  • Compact Abelian torsion bridges: WKBC attains the lowest NLL on all five torsion datasets, with absolute reductions over TDM ranging from 0.10 to 2.00.The reductions are 0.10, 0.20, 0.42, 0.35, and 2.00 on General, Glycine, Proline, Pre-Pro, and RNA, respectively.
  • Compact Abelian torsion bridges: WKBC improves Sinkhorn distance, Ramachandran JSD, ValidRate, and path energy over TDM on all five torsion tasks.GeoRMSE is lower on four protein subsets but slightly higher on RNA: 0.854 versus 0.843.
  • Compact Abelian torsion bridges: WKBC-generated torsion samples closely overlap reference distributions and reproduce their dominant multimodal structures in wrapped angular coordinates.The visualizations assess agreement of periodic endpoint support across General, Glycine, Pre-Pro, Proline, and RNA tasks.
  • Verification against an explicit wrapped bridge: RCCBM remains on the same numerical scale as a high-accuracy WKBC reference and provides a close approximation to its optimal Doob control.The comparison reports lower discrepancies across all four verification metrics for WKBC relative to RCCBM.
  • Compact non-Abelian SO(3) and U(n) bridges: RCCBM attains the highest log-likelihood on every SO(3) benchmark, with margins of 0.036, 0.193, 0.088, and 0.406 over the second-best method.The benchmarks are SO(3)-GMM32, SO(3)-GMM64, SO(3)-GMM128, and SO(3)-RingBand, respectively.
  • Protein conformational transition pathway generation: On mdCATH protein transitions, RCCBM yields a 95.06% directness gain and a 99.21% roughness reduction over reference MD segments across 15 held-out cases.The reduced-representation pathways retain the endpoint constraint while becoming less tortuous and less locally oscillatory.

5 LIMITATIONS AND FUTURE WORK

The framework is validated on selected moderate-dimensional molecular and geometric problems, but higher-dimensional, larger-scale, heterogeneous, and noncompact settings remain outside its demonstrated scope. Future work targets broader experiments and extensions beyond compact or compact-reduced Lie groups.

  • The empirical evaluation remains concentrated on moderate-dimensional groups and selected molecular and geometric tasks.Behavior in substantially higher-dimensional groups, larger systems, and more heterogeneous scientific settings remains to be characterized.
  • The current treatment focuses on compact Abelian, compact non-Abelian, and compact-reduced rigid-frame settings.Genuinely noncompact Lie groups and more general geometric state spaces are outside the current scope.
  • Future experiments could examine higher-dimensional groups, larger molecular systems, and more diverse scientific datasets.These studies are proposed to clarify scalability and robustness.
  • Extending the analysis to noncompact Lie groups, quotient manifolds, and homogeneous spaces would broaden its geometric applicability.

6 CONCLUSIONS

The paper develops an observed-endpoint kinetic Schrödinger bridge framework on Lie group state spaces, with structure-adapted realizations for compact Abelian and non-Abelian groups. Its analysis and experiments support a modular, stable, and practical approach to intrinsic generation across several scientific datasets.

  • Contributions: The framework constrains observed endpoint variables while entropy projection determines conditional laws for latent Lie algebra velocities.Under an endpoint-kernel condition, it admits a two-sided Schrödinger factorization and kinetic Doob representation.
  • Contributions: WKBC uses explicit wrapped kinetic kernels for compact Abelian groups, while RCCBM uses reciprocal calibration and mollified conditional teachers for compact non-Abelian groups.
  • Contributions: The conditional consistency analysis separates endpoint, mollification, teacher, control-regression, initialization, and time-discretization errors in bounded-Lipschitz path distance.
  • Experiments: Experiments across protein and RNA torsions, SO(3), U(n), and protein conformational pathways support the framework’s effectiveness across compact Abelian and non-Abelian settings.Ablations, theory-facing diagnostics, and sensitivity analyses support the roles of the principal components and stable numerical settings.
  • Implementation: Forward sampling uses calibrated initial laws and learned controls, while numerical procedures include endpoint calibration, quadrature, Sinkhorn scaling, regression, and controlled propagation.The supplied implementation passages describe these computational stages and their practical realization.

A.4 TDM-ALIGNED LIKELIHOOD EVALUATION

The likelihood evaluator aligns bridge generators with a TDM-style probability-flow calculation on compact groups. Exact scores preserve benchmark marginals, while RCCBM obtains its evaluation score through post hoc implicit score matching.

  • Benchmark setup: The likelihood benchmark uses a Haar-Gaussian prior and full-state endpoint augmentation, with independent Gaussian Lie algebra velocities paired to group-valued data.
  • Probability-flow evaluation: The probability-flow ODE has the same one-time marginals as the stochastic generator when the marginal velocity score is exact.
  • Score construction: WKBC obtains the full marginal score from propagated wrapped-kernel factors, whereas RCCBM fits an auxiliary score network on fresh frozen-controller trajectories.
  • Evaluation procedure: The evaluator integrates probability-flow characteristics backward, accumulates divergence terms, and reports held-out LL or NLL under the common benchmark protocol.
  • Consistency: Under the numerical approximation conditions, the numerical-bridge and regression terms in the path-space decomposition vanish as discretization and bridge-weighted risk errors decrease.

B.4 DETAILED RCCBM CALIBRATION, TEACHERS, AND LEARNING ASSUMPTIONS

RCCBM calibration combines two-sided endpoint scaling with conditional-control teachers, mollification, empirical matching, and constrained refinement. Its assumptions and propositions connect these stages to convergence, feature-based identification, and controlled path approximations.

  • Endpoint calibration: Alternating KL projections recover both endpoint scaling factors, including the source corrector omitted by terminal-only constructions.
  • Endpoint calibration: Endpoint calibration assumes bounded centered function classes, empirical-dual convergence, sieve approximation, and optimization consistency, with effective sample size as a finite-sample diagnostic.
  • Teachers and learning: The conditional-control stage uses mollified endpoint factors, finite teacher objectives, replay correction, direct-control fitting, and constrained refinement.
  • Identification: Feature-space MMD identifies weak convergence under an injective map, and the production embeddings for SO(3), U(n), and the compact reduced product group satisfy the required injectivity condition.
  • Reduced representation: The reduced representation requires a smooth reconstruction map for exact ambient statements; without injectivity, only a data-processing statement is asserted.

B.8 WKBC NUMERICAL ERROR DECOMPOSITION

The WKBC analysis decomposes numerical bridge error into endpoint, teacher, control, initialization, and discretization components, then verifies conditions under which all terms vanish.

  • Error decomposition: The numerical error bound separates lattice, quadrature, Sinkhorn, factor-interpolation, and numerical initial-law errors from amortized-control regression error.The regression term is measured under the numerical bridge occupation law, without an occupation-density-ratio constant.
  • Endpoint calibration: For the torus specialization, wrapped Gaussian kernels provide smooth, strictly positive endpoint densities and regular transition derivatives needed for endpoint calibration.Compactness and positive continuous source, target, and reference densities yield positive endpoint factors with C^2 log-potentials.
  • Control regression: Trigonometric, spline, and polynomial control sieves approximate the mollified regression target, while velocity truncation and clipping contributions vanish under the stated growth schedules.The argument uses smoothness on compact truncated domains and Gaussian tails of the Ornstein–Uhlenbeck reference with bounded endpoint tilts.
  • Teacher approximation: WKBC can sample directly from the canonical teacher measure, making the replay importance-weight error zero and eliminating auxiliary CondSOC discretization in this verification regime.The stated specialization permits εiw,N = 0, λS,N = 0, RS,N = 0, and εteach,N = 0.
  • Final convergence: Bounded-weight initialization laws and Lie–Trotter convergence ensure initialization and discretization errors tend to zero, so the complete bounded-Lipschitz path-law error vanishes in probability.The conclusion follows after all terms on the right-hand side of Eq. (3.22) vanish in probability.

B.12 FEATURE IDENTIFICATION USED IN REFINEMENT

The refinement analysis uses a continuous injective feature map to identify group-valued distributions, while auxiliary descriptors may enrich neural-controller inputs without replacing that identification role.

  • Identification condition: The MMD identification condition requires only a continuous injective feature map for the production group.This feature map is distinguished from auxiliary controller inputs and descriptors.
  • Auxiliary features: Spectral descriptors used in the U(n) experiments are included as auxiliary controller features rather than as the stated basis of the MMD identification result.The identification result is established through the continuous injective map Φid_G.
  • Production embedding: The production groups use continuous injective embeddings based on matrix entries for SO(3) and U(n), together with periodic sine-cosine coordinates for torus factors.Translation by target statistics and multiplication by an invertible whitening matrix preserve continuity and injectivity.
  • MMD identification: The identification map is a homeomorphism onto its Euclidean image, allowing a characteristic Gaussian-kernel MMD to metrize weak convergence there.Compactness of the groups and invertibility of the whitening transform support this conclusion.

C.1 PROOF OF THEOREM 3.2

The proof constructs a Schrödinger scaling pair through a projective contraction and establishes its uniqueness, continuity, and prescribed marginals. Entropy decomposition then yields uniqueness of the path-law minimizer.

  • Birkhoff contraction makes the composite scaling map contractive on projective classes, with coefficient at most τ^2.The contraction yields Cauchy projective iterates, enabling Banach’s fixed-point argument.
  • Banach’s fixed-point theorem produces a unique projective fixed point, which defines the scaling function f0 after normalization.The proof sets f0 = (KgT)^−1 after obtaining the fixed point [gT].
  • The resulting scaling functions are bounded and continuous on compact spaces because the integral operators map bounded functions to continuous functions.The proof also derives explicit bounds for f0 and gT from kernel bounds.
  • Defining Γ∗ = f0gT R0T gives the prescribed marginals and identifies the entropy minimizer among couplings with finite entropy.The marginal verification follows from Equation (C.1), while the entropy comparison is completed by the subsequent decomposition.
  • The minimizer Γ∗ is unique, and a second scaling pair can differ only by reciprocal gauge.Disintegration with respect to the observed endpoint map then gives path-law uniqueness.

C.2 DERIVATION FOR LEMMA B.2

The derivation establishes smooth, strictly positive kinetic transition densities on connected compact Lie groups and verifies the uniform endpoint-kernel bounds required for the observed endpoint bridge.

  • Hypoellipticity: The global Hörmander bracket condition yields a smooth transition density p_t(x,x′) for every t > 0.The diffusion is nonexplosive because the group is compact and the velocity drift is linear.
  • Controllability: Every terminal state is reachable at any prescribed positive time, and the fixed-time endpoint map is a submersion at a steering control.A C2 group path with prescribed endpoint velocities is converted into an L2 control solving the deterministic kinetic system.
  • Positivity: Strict positivity follows globally: p_t(x0,x1) > 0 for every t > 0 and every pair of states.The support theorem and positivity criterion combine controllability, endpoint-map surjectivity, and the Hörmander condition.
  • Observed endpoints: Integrating out latent initial and terminal velocities preserves strict positivity of the observed group transition density.Compactness and continuity then upgrade pointwise positivity to a uniform positive lower bound.
  • Entropy projection: The entropy chain rule uniquely assigns the reference conditional law to unobserved velocities, while two-sided scaling determines endpoint coupling and intermediate trajectories.A terminal-only correction cannot generally reproduce the two-sided optimum because the first projection changes the source factor.

C.17 PROOF OF THEOREM 3.8

The proof decomposes learned path-law error into endpoint, control, mollification, initialization, and discretization terms, then establishes convergence through strong numerical approximation and geometry-preserving reconstruction.

  • Control approximation: The finite-sample control-risk bound contains approximation, optimization, generalization, teacher, importance-weighting, clipping, and reference terms.Under the stated assumptions, every component in the bound converges to zero in probability.
  • Path-law convergence: The resulting bounded-Lipschitz path-law error converges in probability because endpoint, mollification, control, initialization, and discretization errors all vanish.The proof applies a triangle inequality across continuous and numerical controlled laws.
  • Geometry preservation: The Lie–Trotter reconstruction preserves the matrix Lie group exactly through exponential updates.For unitary groups, exp(H) remains in U(n), so each reconstructed state stays on the manifold.
  • Numerical scheme: The continuous interpolation couples exact and numerical processes with the same initial state and Brownian motion, while frozen-control velocity updates are solved exactly.This coupling supports uniform moment estimates and the subsequent strong path comparison.
  • Discretization: The noncommutative reconstruction defect contributes O(h) in mean square over the full time horizon.The defect arises because the reconstruction uses the endpoint velocity while the Ornstein–Uhlenbeck velocity varies within each step.
  • Reduced geometry: The reduced-state lift is a Borel isomorphism, and the reduced learning problem therefore falls in the compact non-Abelian RCCBM regime.This conclusion assumes the endpoint and reference conditions are verified for the reduced laws.

D ADDITIONAL EXPERIMENTAL RESULTS

Additional experiments provide marginal, qualitative, implementation, and evaluation diagnostics for WKBC and RCCBM across torsion and unitary-group tasks.

  • Torsion experiments: RNA torsion marginals are heterogeneous, ranging from sharply concentrated modes to broader asymmetric profiles across seven periodic coordinates.The marginal plots complement the quantitative endpoint and path results.
  • Unitary-group experiments: Projected RCCBM samples closely agree with reference support geometry across oscillator-induced U(4), U(6), U(8), and TFIM-induced U(4), U(8) tasks.The comparisons include annular, spiral, crescentlike, and strongly anisotropic structures.
  • Experimental protocol: RCCBM experiments use endpoint calibration, reciprocal training, control matching, refinement, replay, and calibrated-initial-state settings recorded in configuration files.The reproducibility record includes seeds, data splits, checkpoint rules, sampler settings, and evaluator settings.
  • Diagnostics: Endpoint, teacher, and control diagnostics are reported separately, with validation MSE, relative MSE, and cosine recomputed after terminal refinement.This separates calibration, conditional-teacher, and direct-control-matching behavior.
  • Evaluation: The fixed periodic and intrinsic evaluation procedures use common grids, predetermined pairings, and fixed validity regions across relevant comparisons.GeoRMSE measures geometric error under a pairing fixed before training, while Sinkhorn settings remain fixed across methods.

E.2.4 PROTEIN CONFORMATIONAL TRANSITION PATHWAY EVALUATION

The pathway evaluation measures geometric detour and local roughness for RCCBM trajectories in a compact reduced representation, using aligned temporal resolution and target-aware metrics.

  • Reduced representation: The production representation contains nR = 63 rotation factors and K = 192 torus coordinates in the reduced state space.The evaluator uses the same reduced geometry as the generator and evaluates mdCATH domain 1jvmB00 trajectories.
  • Temporal alignment: MD and RCCBM paths are evaluated on the same 101-point normalized-time grid, with intrinsic interpolation for rotations and wrapped angular interpolation for torus coordinates.The RCCBM sampler itself uses 128 stochastic integration steps.
  • Tortuosity: Target-aware tortuosity penalizes trajectories that fail to reach the prescribed target and satisfies τT ≥ 1.Values closer to one indicate less geometric detour; the terminal residual vanishes for the observed MD segment.
  • Path roughness: Normalized roughness measures changes in successive local displacement vectors relative to total local motion.Persistent local direction and amplitude produce small roughness, whereas reversals, abrupt changes, and oscillations increase it.
  • Aggregation: Each endpoint pair contributes 32 RCCBM trajectories summarized by mean and standard deviation across a prespecified list of 15 held-out pairs.Aggregate percentages are computed from pair-level means.
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