Paper deep dive

Feynman Kac Reweighted Schrödinger Bridge Matching for Surface-Based Tau PET Harmonization

Jianwei Zhang, Xinyu Nie, Jiaxin Yue, Yonggang Shi

Year: 2026Venue: arXiv preprintArea: eess.IVType: PreprintEmbeddings: 59

Intelligence

Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 96%

Last extracted: 6/20/2026, 9:49:25 AM

Summary

The paper proposes the Feynman–Kac Reweighted Schrödinger Bridge Matching (FKRSBM) model for harmonizing surface-based tau PET imaging data. Traditional unpaired harmonization methods often conflate site-induced shifts with biological variation (e.g., tau-positivity status) when source and target cohorts differ in subgroup composition. FKRSBM addresses this by incorporating a subgroup-aware endpoint proposal via Feynman–Kac reweighting of the reference bridge measure. This allows the model to enforce biologically consistent transport by penalizing implausible endpoint pairings without changing the underlying bridge-matching architecture. The method uses a spherical convolutional backbone for vertex-level harmonization on cortical meshes and is evaluated on harmonizing PI-2620 data from the HABS-HD cohort into the AV-1451 domain of ADNI, demonstrating superior distributional alignment and subgroup consistency compared to baselines like ComBat and CycleGAN.

Entities (10)

ADNI · cohortdataset · 100%AV-1451 · radiotracer · 100%ComBat · baseline-method · 100%CycleGAN · baseline-method · 100%Diffusion Schrödinger Bridge Matching · baseline-method · 100%Feynman–Kac Reweighted Schrödinger Bridge Matching · modelalgorithm · 100%HABS-HD · cohortdataset · 100%PI-2620 · radiotracer · 100%tau PET · imaging-modality · 100%University of Southern California · organization · 100%

Relation Signals (5)

HABS-HD → contains → PI-2620

confidence 100% · harmonizing PI-2620 data from the HABS-HD cohort

ADNI → contains → AV-1451

confidence 100% · into the AV-1451 domain of ADNI.

Feynman–Kac Reweighted Schrödinger Bridge Matching → uses → spherical convolutional backbone

confidence 100% · For surface-based neuroimaging, FKRSBM employs a spherical convolutional backbone operating on cortical meshes

Feynman–Kac Reweighted Schrödinger Bridge Matching → addresses → tau PET harmonization

confidence 95% · We propose the Feynman–Kac Reweighted Schrödinger Bridge Matching (FKRSBM) model to address this problem [tau PET harmonization].

Feynman–Kac Reweighted Schrödinger Bridge Matching → outperforms → ComBat

confidence 90% · Compared against ComBat... FKRSBM achieves superior distributional alignment

Cypher Suggestions (0)

No Cypher suggestions yet.

Abstract

Abstract:Tau PET imaging is central to tracking Alzheimer's disease progression, but systematic differences between scanners, protocols, and radiotracers across sites introduce nonbiological variability that inflates biomarker variance, reduces sensitivity to disease effects, and can bias downstream clinical assessments. Harmonization methods aim to remove these site-induced shifts while preserving biologically meaningful signal, yet existing approaches struggle when source and target cohorts differ in subgroup composition, risking conflation of site effects with biological variation such as tau-positivity status. We propose the Feynman Kac Reweighted Schröodinger Bridge Matching (FKRSBM) model to address this problem. Rather than routing data through a Gaussian noise prior as in diffusion-based methods, FKRSBM learns a direct stochastic transport process between source and target distributions via entropy-regularized optimal transport. To enforce biologically consistent transport, FKRSBM incorporates a subgroup-aware endpoint proposal derived from a Feynman Kac reweighting of the reference bridge measure, implemented entirely through stratified importance sampling at the data level and requiring no changes to the underlying bridge-matching solver or network architecture. For surface-based neuroimaging, FKRSBM employs a spherical convolutional backbone operating on cortical meshes to perform vertex-level harmonization. We evaluate the method on tau PET SUVR maps, harmonizing PI-2620 data from the HABS-HD cohort into the AV-1451 domain of ADNI. Compared against ComBat, CycleGAN, a diffusion-based method (DF), and unregularized Diffusion Schröodinger Bridge Matching (DSBM), FKRSBM achieves superior distributional alignment, reduced tau-positivity sign mismatch, stronger APOE subgroup alignment, and improved downstream disease classification performance.

PDF

Open source PDF →Open local PDF →

Trouble viewing inline? Open PDF directly →

Full Text

58,242 characters extracted from source content.

Expand or collapse full text

Feynman–Kac Reweighted Schrödinger Bridge Matching for Surface-based Tau PET Harmonization Jianwei Zhang Xinyu Nie Jiaxin Yue and Yonggang Shi This work was supported by the National Institutes of Health (NIH) under grants R01EB022744, RF1AG077578, RF1AG064584, U19AG078109, P30AG066530 and S10OD032285.Jianwei Zhang, Xinyu Nie, Jiaxin Yue, and Yonggang Shi are with the Stevens Neuroimaging and Informatics Institute, University of Southern California, Los Angeles, CA 90089 USA. Jianwei Zhang, Jiaxin Yue, and Yonggang Shi are with the Ming Hsieh Department of Electrical and Computer Engineering of Viterbi School of Engineering, University of Southern California, Los Angeles, CA 90089 USA. Yonggang Shi is also with the Alfred E. Mann Department of Biomedical Engineering of Viterbi School of Engineering, University of Southern California, Los Angeles, CA, USA.Corresponding author: Yonggang Shi. Email: yshi@loni.usc.edu. Abstract Tau PET imaging is central to tracking Alzheimer’s disease progression, but systematic differences between scanners, protocols, and radiotracers across sites introduce nonbiological variability that inflates biomarker variance, reduces sensitivity to disease effects, and can bias downstream clinical assessments. Harmonization methods aim to remove these site-induced shifts while preserving biologically meaningful signal, yet existing approaches struggle when source and target cohorts differ in subgroup composition, risking conflation of site effects with biological variation such as tau-positivity status. We propose the Feynman–Kac Reweighted Schrödinger Bridge Matching (FKRSBM) model to address this problem. Rather than routing data through a Gaussian noise prior as in diffusion-based methods, FKRSBM learns a direct stochastic transport process between source and target distributions via entropy-regularized optimal transport. To enforce biologically consistent transport, FKRSBM incorporates a subgroup-aware endpoint proposal derived from a Feynman–Kac reweighting of the reference bridge measure, implemented entirely through stratified importance sampling at the data level and requiring no changes to the underlying bridge-matching solver or network architecture. For surface-based neuroimaging, FKRSBM employs a spherical convolutional backbone operating on cortical meshes to perform vertex-level harmonization. We evaluate the method on tau PET SUVR maps, harmonizing PI-2620 data from the HABS-HD cohort into the AV-1451 domain of ADNI. Compared against ComBat, CycleGAN, a diffusion-based method (DF), and unregularized Diffusion Schrödinger Bridge Matching (DSBM), FKRSBM achieves superior distributional alignment, reduced tau-positivity sign mismatch, stronger APOE ε 4 subgroup alignment, and improved downstream disease classification performance. IEEEkeywords Neuroimaging harmonization, Schrödinger bridge, Feynman–Kac reweighting, tau PET, cortical surface, entropy-regularized optimal transport, unpaired domain translation. 1 Introduction Aggregating neuroimaging data across multiple sites is essential for increasing statistical power and capturing biological heterogeneity in large-scale brain studies. However, multi-site studies are confounded by scanner effects, systematic variations in image intensity, contrast, and noise caused by differences in hardware, field strengths, radiotracers, and acquisition protocols [1]. These non-biological variances inflate biomarker variance and reduce sensitivity to subtle disease effects [2, 3], and can introduce systematic bias when site distributions covary with biological variables [4]. The problem is compounded in machine learning pipelines, where models risk learning site identity rather than clinically relevant signal [5]. These challenges are especially acute for surface-based analyses, where the high sensitivity of cortical metrics to tissue contrast makes them particularly vulnerable to scanner-induced bias [2]. In practice, harmonization typically needs to operate in an unpaired setting, since acquiring large traveling cohorts of subjects scanned at every site under every protocol is logistically and financially challenging. Most multi-site datasets consequently share few or no common subjects, favoring methods that can align entire distributions without relying on subject-level correspondences. Several families of methods have been proposed for unpaired harmonization. Statistical methods such as ComBat [3, 2] model scanner effects as linear transformations. While effective for low-dimensional summaries, they assume Gaussian effects, treat features independently, and cannot capture nonlinear interactions between the source of variation and the underlying data structure [6, 7]. Recently, Deep learning methods have lifted the linearity restriction but have introduced new issues. CycleGANs [8, 9, 10] enforce cycle-consistency for unpaired translation but suffer from training instability and mode collapse that can suppress input variability. Disentanglement-based approaches [11, 12, 13, 14] seek to separate content from style, but true unsupervised disentanglement is theoretically unattainable without strong inductive biases [15], and the ELBO objective trades reconstruction fidelity for latent regularity [16]. Diffusion-based translation via SDEdit [17] relies on noise injection to erase domain-specific characteristics before denoising into the target domain, but calibrating the noise level presents an inherent tradeoff between domain adaptation and content preservation. Unpaired harmonization can be formulated as aligning a source distribution π0 _0 with a target distribution π1 _1 without paired samples. We adopt the Schrödinger Bridge (SB) framework, an entropy-regularized optimal transport formulation over stochastic processes, to learn a direct source-to-target harmonization path between the two distributions [18]. In contrast to cycle-consistency or noise-routing approaches, SB provides a principled distributional alignment objective while preserving the source image as a structural prior throughout the transport process [19, 20]. Detailed comparisons with related translation frameworks are provided in the Method section. Despite these strengths, two open problems remain before SB can be applied to surface-based neuroimaging harmonization. First, SB-based harmonization on cortical surface data remains underexplored, where the non-Euclidean mesh topology demands geometry-aware architectures. Second, and more fundamentally, common DSBM-style deep learning implementations pair source and target endpoints by sampling independently from the product coupling π0⊗π1 _0 _1, treating every cross-site pairing as equally plausible. This agnostic coupling ignores covariate structure. When the two sites differ in subgroup composition (e.g., different proportions of tau-positive and tau-negative subjects), the transport can map samples from one clinical subgroup toward a different subgroup in the target simply because the latter is more prevalent. The resulting harmonized data would conflate tracer differences with biological differences, defeating the purpose of harmonization. What is needed is a mechanism to constrain the bridge endpoints so that the transport respects known covariate labels. Recent theoretical work on soft-constrained Schrödinger bridges [21] shows that replacing hard terminal constraints with penalty functions yields well-posed problems with quantitative convergence guarantees, providing a principled foundation for such constraints. In this work, we propose the Feynman–Kac Reweighted Schrödinger Bridge Matching (FKRSBM) model, a framework that addresses both gaps. Our contributions are twofold: 1. Schrödinger Bridge on the cortical surface domain. We introduce an SB-based harmonization framework operating on cortical surface meshes. A spherical convolutional backbone [22] respects the non-Euclidean geometry of the cortex, preserving surface topology. 2. Endpoint-constrained transport. We formulate an endpoint-regularized SB objective that augments the standard KL minimization with a penalty Φ encoding covariate structure, preventing the transport from conflating site effects with biological variation. We show that the resulting source-anchored conditional reweighting can be implemented as standard bridge matching under a Feynman–Kac reweighted endpoint proposal obtained via Radon–Nikodým reweighting, with an interpretable parameter λ controlling the strength of the constraint. Crucially, the Feynman–Kac reweighting operates entirely at the data-sampling level (via reweighted importance sampling), requiring no changes to the DSBM solver or network architecture. This modular design means that any future improvement to the SB solvers can be incorporated without modifying the endpoint constraint. Operationally, FKRSBM trains a bridge-matching model by drawing Brownian-bridge samples from endpoint pairs whose target endpoint is selected by a covariate-aware Feynman–Kac reweighted proposal, then harmonizes a source surface by integrating the learned stochastic dynamics into the target tracer domain. We evaluate the method on unpaired tau PET SUVR maps, harmonizing HABS-HD PI-2620 data to the ADNI AV-1451 domain. Across image-level and region-level evaluations, FKRSBM achieves the strongest overall distributional alignment, lowest residual site separability, and best tau-subgroup consistency among ComBat [23], CycleGAN [10], DF [24], and unregularized DSBM [20] baselines, while preserving subject-level cortical patterns. These results indicate that Feynman–Kac reweighting of the reference bridge measure improves SB harmonization not only theoretically, but also in practical cross-tracer tau PET experiments. Figure 1: Distributional and spatial differences between tau PET tracers. Left: Bar plots of scan-level mean left hemisphere cortical SUVR for PI-2620 (HABS-HD, blue) and AV-1451 (ADNI, orange) show a significant shift in central tendency (Mann–Whitney U, p<0.001p<0.001), with AV-1451 yielding systematically higher SUVR values. Right: Cohort-averaged SUVR projected onto the lateral cortical surface reveals tracer-specific spatial signal patterns, motivating the need for harmonization across tracers. Figure 2: Overview of the proposed Feynman–Kac Reweighted Schrödinger Bridge (FKRSBM) for cross-tracer tau PET harmonization. Top: Schematic of the bridge between source (X0X_0, PI-2620) and target (X1X_1, AV-1451) tau distributions. The DSBM (dashed purple) transports mass between marginals without regard to biological subgroup, producing crossing trajectories that mix subjects across subgroups. FKRSBM (solid teal) reweights the endpoint coupling so that trajectories preferentially connect subjects sharing the same biological subgroup (tau positive vs. negative), yielding straighter, subgroup-consistent paths while preserving the path-space dynamics. Middle (Training stage): At each iteration, a time t and a Brownian-bridge sample XtX_t are drawn from a Feynman–Kac reweighted endpoint coupling, and a hierarchical surface network operating across icosahedral resolutions (ico 3–6) predicts the drift vθ(t,Xt)v_θ(t,X_t), regressed against the analytic bridge drift. Bottom (Inference stage): A source SUVR map is harmonized by integrating the learned SDE from t=0t=0 to t=1t=1, producing a trajectory of intermediate cortical maps that terminates in the target tracer space. 2 Method 2.1 Problem Formulation Let π0 _0 and π1 _1 denote the source and target data distributions, respectively, with samples represented as x∈ℝdx ^d. In the unpaired harmonization setting, no subject-level correspondence is available across domains. The goal is to map samples from π0 _0 toward the target distribution π1 _1 while reducing acquisition-related variability and preserving biologically meaningful variation. We formulate this task within the Schrödinger Bridge framework. This section first reviews the standard Schrödinger Bridge problem and bridge matching, then introduces a covariate-informed endpoint regularization, establishes its equivalence to bridge matching under a Feynman–Kac reweighted reference bridge measure, and derives the resulting training and inference procedures. 2.2 Background: Schrödinger Bridge and Bridge Matching 2.2.1 The Schrödinger Bridge Problem Let Ω denote the space of continuous trajectories xtt∈[0,1]\x_t\_t∈[0,1] with xt∈ℝdx_t ^d, and let ϵW_ε be a reference Wiener measure with diffusion scale ϵε. The Schrödinger Bridge (SB) problem seeks a path measure Q∗Q that interpolates between π0 _0 and π1 _1 while remaining as close as possible to ϵW_ε in KL divergence: Q∗=argminQ∈ℙ(Ω)Q0=π0,Q1=π1KL(Q∥ϵ).Q = *argmin_ subarraycQ ( )\\ Q_0= _0,\;Q_1= _1 subarrayKL(Q\,\|\,W_ε). (1) The solution Q∗Q can be represented as an Itô diffusion sharing the same diffusion coefficient as the reference process: dxt=b∗(t,xt)dt+ϵdWt,dx_t=b (t,x_t)\,dt+ ε\,dW_t, (2) where WtW_t is standard Brownian motion and b∗(t,x)b (t,x) is the optimal drift. Intuitively, Eq. (1) selects, among all stochastic processes connecting π0 _0 to π1 _1, the one that deviates minimally from pure Brownian motion. 2.2.2 Structure of the Optimal Solution When the marginals are point masses π0=δx0 _0= _x_0 and π1=δx1 _1= _x_1, the solution reduces to the Brownian bridge connecting x0x_0 to x1x_1. For general non-degenerate marginals, Q∗Q is a mixture of Brownian bridges: the joint endpoint distribution (x0,x1)(x_0,x_1) follows an optimal entropic coupling γ∗∈Π(π0,π1)γ ∈ ( _0, _1), and conditional on (x0,x1)∼γ∗(x_0,x_1) γ , the intermediate trajectory xtt∈(0,1)\x_t\_t∈(0,1) follows a Brownian bridge from x0x_0 to x1x_1. This decomposition separates the problem into two parts: (i) learning the coupling γ∗γ between marginals, and (i) the conditional path dynamics, which are analytically tractable as Brownian bridges. The optimal drift takes the form b∗(t,x)=ϵ∇xlog⁡φ(t,x),b (t,x)=ε _x (t,x), (3) where φ(t,x) (t,x) is a space–time harmonic function associated with π1 _1. This corresponds to a Doob h-transform of the Wiener process. In practice, φ(t,x) (t,x) is unavailable in closed form for complex, high-dimensional distributions. 2.2.3 Bridge Matching Rather than estimating the Schrödinger potential φ explicitly, bridge matching[25] learns the transport dynamics by regressing a parameterized drift vθ(t,x)v_θ(t,x) onto the conditional drift of Brownian bridges connecting samples from the two endpoint marginals. Specifically, for endpoint pairs (X0,X1)(X_0,X_1) and an intermediate bridge sample XtX_t, the model is trained at randomly sampled times t∈(0,1)t∈(0,1) by minimizing ℒ(θ)=[‖vθ(t,Xt)−vBB(t,Xt;X0,X1)‖22],L(θ)=E [ \|v_θ(t,X_t)-v_B(t,X_t;X_0,X_1) \|_2^2 ], (4) where the Brownian bridge drift conditioned on endpoints (X0,X1)(X_0,X_1) is given by vBB(t,x;X0,X1)=X1−x1−t.v_B(t,x;X_0,X_1)= X_1-x1-t. (5) The intermediate state Xt=(1−t)X0+tX1+ϵt(1−t)ZX_t=(1-t)X_0+tX_1+ ε t(1-t)Z with Z∼(0,I)Z (0,I) is sampled from the corresponding conditional Brownian bridge. The Diffusion Schrödinger Bridge Matching (DSBM) framework [20, 25] operationalizes this idea by alternating between forward and backward bridge constructions and fitting the drift via regression. 2.3 Endpoint-Regularized Schrödinger Bridge The DSBM-style bridge matching loss in Eq. (4) is commonly estimated by sampling endpoint pairs independently from the product measure π0⊗π1 _0 _1. In the context of multi-site neuroimaging, this means the model treats all cross-site pairings as equally plausible, for example, a tau-negative subject from site A is just as likely to be paired with a tau-positive subject from site B as with a tau-negative subject from site B. When sites differ in the composition of tau-positive and tau-negative subjects, such agnostic pairing can bias the learned transport, conflating site effects with biological variation. To address this, we introduce an endpoint penalty Φ:ℝd×ℝd→ℝ :R^d×R^d that encodes prior knowledge about which pairings are biologically meaningful. We then define an endpoint-regularized SB objective that augments the standard KL minimization with a penalty on the expected value of Φ under the learned path measure. We first define a reference path measure R as an independent mixture of Brownian bridges. Let x0→x1W^x_0→ x_1 denote the Brownian bridge path measure connecting x0x_0 to x1x_1 with diffusion scale inherited from ϵW_ε. The reference measure is: R(⋅)=∫ℝd×ℝdx0→x1(⋅)π0(dx0)π1(dx1).R(·)=\! _R^d×R^d\!\!W^x_0→ x_1(·)\, _0(dx_0)\, _1(dx_1). (6) Under R, the endpoints are drawn independently: (X0,X1)∼π0⊗π1(X_0,X_1) _0 _1. This is the implicit reference used by product-pairing DSBM[25] training. For a penalty strength λ≥0λ≥ 0, we seek a path measure that stays close to the reference R while penalizing biologically implausible endpoint pairings: Pλ⋆∈argminP≪RP0=π0KL(P∥R)+λP[Φ(X0,X1)],P _λ∈ *argmin_ subarraycP R\\ P_0= _0 subarray \KL(P\,\|\,R)+λ\,E_P[ (X_0,X_1)] \, (7) where P≪RP R denotes that P is absolutely continuous with respect to R, and the constraint P0=π0P_0= _0 fixes the source marginal. Here, λ is the weight that adjusts the penalty level, and Φ(X0,X1) (X_0,X_1) scores how implausible a pairing (X0,X1)(X_0,X_1) is: the KL term keeps P close to the independent bridge mixture, while the penalty term discourages endpoint pairs with high Φ . As λ increases, the solution increasingly favors path measures whose endpoint pairs have low penalty Φ —i.e., biologically consistent pairings. The key question is then how to solve Eq. (7) in practice. We show next that this regularized objective admits an implementation as standard bridge matching under a Feynman–Kac reweighted reference bridge measure. 2.4 Feynman–Kac Reweighting of the Reference Bridge Measure The key observation is that the source-anchored endpoint penalty in Eq. (7) can be absorbed into the reference path measure through a Feynman–Kac-type exponential reweighting. Classical Feynman–Kac theory describes how a reference stochastic process can be modified by multiplying its path measure by an exponential potential functional [26]. In our setting, the potential is not a running cost accumulated along the path; instead, it is an endpoint potential Φ(X0,X1) (X_0,X_1) that encodes biological compatibility between a source and target sample. This endpoint-potential reweighting changes the endpoint coupling used to construct the bridge mixture, while leaving the conditional Brownian bridge law between any fixed endpoint pair unchanged. Define the source-anchored Feynman–Kac reweighted reference RλR_λ via the Radon–Nikodým derivative[27]: dRλdR(ω)=exp⁡(−λΦ(X0(ω),X1(ω)))Zλ(X0(ω)), dR_λdR(ω)= \! (\!-λ (X_0(ω),X_1(ω)) )Z_λ(X_0(ω)), (8) where the source-conditional normalizing constant is Zλ(x0):=X1∼π1[exp⁡(−λΦ(x0,X1))].Z_λ(x_0):=E_X_1 _1\! [ \! (\!-λ (x_0,X_1) ) ]. (9) The Feynman–Kac factor reweights trajectories in R according to the compatibility of their endpoints: paths connecting well-matched pairs with low Φ receive higher weight, whereas paths connecting mismatched pairs are downweighted. The normalization is conditional on X0X_0, so the source marginal remains π0 _0 while the target endpoint is sampled from a biased conditional proposal. Crucially, the potential depends only on the endpoints (X0,X1)(X_0,X_1) and not on intermediate states along the path. Therefore, the reweighting modifies the endpoint law but does not modify the Brownian bridge dynamics conditioned on each endpoint pair. We formalize this through three propositions. Proposition 1 (Endpoint regularization equals Feynman–Kac reweighting) For any P≪RP R with P0=π0P_0= _0, KL(P∥Rλ) (P\,\|\,R_λ) =KL(P∥R)+λP[Φ(X0,X1)] =KL(P\,\|\,R)+λ\,E_P[ (X_0,X_1)] +X0∼π0[log⁡Zλ(X0)]. +E_X_0 _0[ Z_λ(X_0)]. (10) Consequently, minimizing the regularized objective in Eq. (7) is equivalent to minimizing KL(P∥Rλ)KL(P\,\|\,R_λ). Proof 2.1. By definition of KL divergence and the Radon–Nikodým chain rule, KL(P∥Rλ) (P\,\|\,R_λ) =P[log⁡dPdRλ] =E_P\!\! [ dPdR_λ ] =P[log⁡dPdR+log⁡dRdRλ]. =E_P\!\! [ dPdR+ dRdR_λ ]. (11) From Eq. (8), the inverse Radon–Nikodým derivative is dRdRλ(ω)=Zλ(X0(ω))exp⁡(λΦ(X0(ω),X1(ω))). dRdR_λ(ω)=Z_λ(X_0(ω)) \! (λ (X_0(ω),X_1(ω)) ). (12) Substituting and expanding: KL(P∥Rλ) (P\,\|\,R_λ) =KL(P∥R) =KL(P\,\|\,R) +λP[Φ(X0,X1)]+P[log⁡Zλ(X0)]. +λ\,E_P[ (X_0,X_1)]+E_P[ Z_λ(X_0)]. (13) Since P0=π0P_0= _0, the final term equals X0∼π0[log⁡Zλ(X0)]E_X_0 _0[ Z_λ(X_0)] and is independent of the conditional endpoint proposal. Minimizing KL(P∥Rλ)KL(P\,\|\,R_λ) over P is equivalent to minimizing Eq. (7). Proposition 2.2 (Endpoint proposal under the Feynman–Kac reweighted reference). The endpoint distribution under RλR_λ has source marginal π0 _0 and target conditional r1|0λ(x1∣x0)=e−λΦ(x0,x1)π1(x1)Zλ(x0).r^λ_1|0(x_1 x_0)= e^-λ (x_0,x_1)\, _1(x_1)Z_λ(x_0). (14) Proof 2.3. Let A∈ℬ(ℝd×ℝd)A (R^d×R^d) be a measurable endpoint set. Applying Eq. (8): Rλ((X0,X1)∈A) R_λ\! ((X_0,X_1)∈ A ) =R[(X0,X1)∈AdRλdR] =E_R\!\! [1_\(X_0,X_1)∈ A\ dR_λdR ] =R[(X0,X1)∈Ae−λΦ(X0,X1)Zλ(X0)]. =E_R\!\! [1_\(X_0,X_1)∈ A\\, e^-λ (X_0,X_1)Z_λ(X_0) ]. (15) Since (X0,X1)∼π0⊗π1(X_0,X_1) _0 _1 under R and the denominator depends only on x0x_0: Rλ((X0,X1)∈A) R_λ\! ((X_0,X_1)∈ A ) =∫Ae−λΦ(x0,x1)Zλ(x0)π0(dx0)π1(dx1), =\! _A\! e^-λ (x_0,x_1)Z_λ(x_0)\, _0(dx_0)\, _1(dx_1), (16) establishing Eq. (14). Proposition 2.2 shows that the Feynman–Kac reweighting modifies the conditional proposal for endpoints: rather than drawing target endpoints independently of the source sample, the reweighted reference favors pairings with low penalty Φ . For instance, when Φ penalizes tau-positivity mismatch, the Feynman–Kac reweighted proposal preferentially pairs tau-positive subjects with tau-positive subjects and tau-negative subjects with tau-negative subjects across sites. Proposition 2.4 (Conditional bridge dynamics are unchanged). For any endpoints (x0,x1)(x_0,x_1), Rλ(⋅∣X0=x0,X1=x1)=x0→x1(⋅).R_λ(· X_0\!=\!x_0,X_1\!=\!x_1)=W^x_0→ x_1(·). (17) Proof 2.5. Let B⊆ΩB be measurable in the path-space sigma-algebra. By Bayes’ rule: Rλ(B∣X0=x0,X1=x1) R_λ(B X_0\!=\!x_0,X_1\!=\!x_1) =R[B 1X0=x0,X1=x1dRλdR]R[X0=x0,X1=x1dRλdR]. = E_R\! [1_B\,1_\X_0=x_0,X_1=x_1\\, dR_λdR ]E_R\! [1_\X_0=x_0,X_1=x_1\\, dR_λdR ]. (18) Since dRλdR∝exp⁡(−λΦ(X0,X1)) dR_λdR (-λ (X_0,X_1)) depends only on (X0,X1)(X_0,X_1), it is constant when conditioning on (X0,X1)=(x0,x1)(X_0,X_1)=(x_0,x_1) and cancels: Rλ(B∣X0=x0,X1=x1) R_λ(B X_0\!=\!x_0,X_1\!=\!x_1) =R(B∣X0=x0,X1=x1). =R(B X_0\!=\!x_0,X_1\!=\!x_1). (19) By definition of R in Eq. (6), R(⋅∣X0=x0,X1=x1)=x0→x1(⋅)R(· X_0\!=\!x_0,X_1\!=\!x_1)=W^x_0→ x_1(·). Proposition 2.4 is critical for practical implementation: the Feynman–Kac reweighting changes which endpoint pairs are favored but does not alter the Brownian bridge dynamics between any given pair. This means the bridge matching regression target vBBv_B remains the same—only the sampling distribution over endpoint pairs changes. 2.5 Feynman–Kac Reweighted DSBM: Objective and Algorithm The preceding identities justify implementing the source-anchored endpoint penalty by performing standard bridge matching under the conditional endpoint proposal induced by the Feynman–Kac reweighted reference RλR_λ. The reweighted objective requires sampling from RλR_λ. One way to view the reweighting is to sample from the unreweighted product measure (X0,X1)∼π0⊗π1(X_0,X_1) _0 _1 and correct via importance weighting. Let w~=exp⁡(−λΦ(X0,X1)) w= (-λ (X_0,X_1)) and the resulting Feynman–Kac reweighted DSBM objective ℒλ(θ)L_λ(θ) is given by: X0∼π0[X1∼π1[w~(X0,X1)t,Xt|X0,X1[‖vθ−vBB‖22]]X1∼π1[w~(X0,X1)]].E_X_0 _0\! [ E_X_1 _1\! [ w(X_0,X_1)\,E_t,X_t|X_0,X_1\! [\|v_θ-v_B\|_2^2 ] ]E_X_1 _1[ w(X_0,X_1)] ]. (20) However, this weighted estimator is inefficient when λ is large: label-mismatched pairs receive very small weights but still require a full forward and backward pass. Our implementation therefore moves the importance weights into the sampling distribution. For each source sample x0x_0, we sample the target endpoint from the empirical conditional proposal qλ(x1∣x0)=exp⁡[−λΦ(x0,x1)]π^1(x1)∑x1′∈1exp⁡[−λΦ(x0,x1′)]π^1(x1′),q_λ(x_1 x_0)= [-λ (x_0,x_1)]\, π_1(x_1) _x _1 _1 [-λ (x_0,x _1)]\, π_1(x _1), (21) where 1D_1 denotes the target training set and π^1 π_1 is its empirical measure. The bridge-matching loss is then evaluated without an explicit per-sample weight, because the endpoint pairs are already drawn from the Feynman–Kac reweighted proposal. This is the sampling-based form of importance sampling: computational effort is concentrated on pairs with non-negligible reweighting weight rather than spent on pairs whose gradients would be nearly zero. 2.5.1 Endpoint Penalty Design The endpoint penalty Φ(x0,x1) (x_0,x_1) encodes prior knowledge about which cross-distribution pairings are semantically meaningful. Because the penalty enters only through the importance weights w~=exp⁡(−λΦ) w= (-λ ) and does not receive gradients, its design is flexible and can incorporate arbitrary covariate information. For the tau PET harmonization task, where sites differ in the proportion of tau-positive and tau-negative subjects, we define Φ as a tau-positivity mismatch penalty: Φ(x0,x1)=[g(x0)≠g(x1)], (x_0,x_1)=1[g(x_0)≠ g(x_1)], (22) where g(⋅)∈τ+,τ−g(·)∈\τ^+,τ^-\ denotes the tau positivity label associated with each sample, determined by thresholding the regional SUVR values using the standard criterion described in Section 3.1. Under this penalty, same-group pairs receive weight w~=1 w=1 while cross-group pairs receive weight w~=e−λ w=e^-λ, which decreases with λ. This encourages the transport to align each tau-positivity subgroup with its target counterpart rather than with the target marginal as a whole. More generally, Φ can incorporate multiple covariates as a weighted sum of mismatch terms, or encode feature-space distances between samples. The modular design allows practitioners to specify domain-appropriate constraints without modifying the underlying transport algorithm. 2.5.2 Feynman–Kac Reweighted Sampling for Efficient Training While the weighted view in Eq. (20) is useful conceptually, it can be computationally wasteful in practice due to near zero gradient pass due to small weight from large lambda. The training objective estimated by the implemented sampler is ℒλ(θ)=X0∼π^0X1∼qλ(⋅∣X0)[t,Xt|X0,X1[‖vθ−vBB‖22]].L_λ(θ)=E_X_0 π_0E_X_1 q_λ(· X_0)\! [E_t,X_t|X_0,X_1\! [\|v_θ-v_B\|_2^2 ] ]. (23) This is the configuration used for the reported FKRSBM experiments. Following the DSBM framework [20], training proceeds in two stages. In the pretraining stage, endpoint pairs are drawn from the reweighted endpoint proposal (Eq. (21)) and the model learns an initial drift by regressing toward Brownian bridge velocities. In the finetuning stage, the model generates self-consistent endpoints by solving the learned SDE forward and backward from samples drawn by the same endpoint proposal, and refines the drift using the resulting bridge-matching regression targets. An exponential moving average (EMA) of the parameters is maintained throughout for stable trajectory generation. Algorithm 1 summarizes the complete procedure. 2.6 Inference via SDE Sampling At inference time, given a source sample x0∼π0x_0 _0, we generate the harmonized output by simulating the learned stochastic differential equation from t=0t=0 to t=1t=1. Because the Schrödinger Bridge solution is inherently a diffusion process (Eq. (2)), we retain the stochastic dynamics at inference and solve: dxt=vθ(t,xt)dt+ϵdWt,x0∼π0,t∈[0,1].dx_t=v_θ(t,x_t)\,dt+ ε\,dW_t, x_0 _0, t∈[0,1]. (24) The harmonized sample is x^1=xt=1 x_1=x_t=1, the solution evaluated at t=1t=1. We solve Eq. (24) numerically using the Euler–Maruyama method with N uniformly spaced steps. At each step k=0,1,…,N−1k=0,1,…,N-1 with step size h=1/Nh=1/N: xtk+1=xtk+h⋅vθ(tk,xtk)+ϵhξk,ξk∼(0,Id),x_t_k+1=x_t_k+h· v_θ(t_k,x_t_k)+ ε h\; _k, _k (0,I_d), (25) where tk=k/Nt_k=k/N. In all experiments, we use N=100N=100 steps, which provides a stable tradeoff between integration accuracy and computational cost. Input: Source and target datasets 0,1D_0,D_1; tau-positivity labels g; diffusion scale ϵε; reweighting strength λ; pretrain steps NpreN_pre; finetune steps NfineN_fine; batch size B; EMA decay γ. Initialize forward and backward drifts vθf,vθbv_θ^f,v_θ^b; θEMA←θ _EMA←θ; Optionally oversample minority tau-positivity groups in 0D_0 and 1D_1; // Stage 1: Pretrain with Feynman--Kac reweighted endpoint sampling for n=1,…,Npren=1,…,N_pre do Draw source batch X01:BX_0^1:B from 0D_0; For each X0iX_0^i, sample X1iX_1^i from qλ(⋅∣X0i)q_λ(· X_0^i) in Eq. (21); t∼Unif([0,1])⊗Bt ([0,1]) B; Z∼(0,Id)⊗BZ (0,I_d) B; Xt←(1−t)X0+tX1+ϵt(1−t)ZX_t←(1-t)X_0+tX_1+ ε t(1-t)Z; Update vθfv_θ^f and vθbv_θ^b using the unweighted bridge-matching MSE to the analytic forward and backward Brownian-bridge drifts; θEMA←γθEMA+(1−γ)θ _EMA←γ\, _EMA+(1\!-\!γ)\,θ; end for // Stage 2: Finetune with self-consistent dynamics for n=1,…,Nfinen=1,…,N_fine do Draw reweighted endpoint batch (X01:B,X11:B)(X_0^1:B,X_1^1:B) as above; X^1← X_1← forward SDE from X0X_0 using vθEMAfv^f_ _EMA (Eq. (24)); X^0← X_0← backward SDE from X1X_1 using vθEMAbv^b_ _EMA; t∼Unif([0,1])⊗Bt ([0,1]) B; Z∼(0,Id)⊗BZ (0,I_d) B; Xt(a)←Interpt(X^0,X1,Z)X_t^(a) _t( X_0,X_1,Z); Xt(b)←Interpt(X0,X^1,Z)X_t^(b) _t(X_0, X_1,Z); Update vθfv_θ^f on bridges (X^0,X1)( X_0,X_1) and vθbv_θ^b on bridges (X0,X^1)(X_0, X_1) using unweighted MSE; θEMA←γθEMA+(1−γ)θ _EMA←γ\, _EMA+(1\!-\!γ)\,θ; end for // Inference Given x0∼π0x_0 _0, simulate SDE (Eq. (24)) with vθEMAv_ _EMA from t=0t\!=\!0 to t=1t\!=\!1 to obtain harmonized x^1 x_1; Output: Parameters (θ,θEMA)(θ, _EMA) Algorithm 1 Feynman–Kac Reweighted Schrödinger Bridge Matching 2.7 Implementation The backbone network is a conditional surface U-Net (CUNet) composed of spherical convolution layers [22] arranged in an encoder–decoder architecture on the sixth-order icosahedral mesh. We use channel widths [32,64,128,256][32,64,128,256], latent dimension 256, group normalization, ReLU activations in convolutional blocks, and sinusoidal time embeddings injected into residual blocks. No auxiliary clinical condition is provided to the CUNet; the endpoint reweighting is implemented entirely through the tau-positivity-aware sampler. Before input to the network, data goes through a log transform to suppress abnormally high SUVR values. For noise term ϵε and reweighting strength λ, our model uses ϵ=0.01ε=0.01 and λ=4λ=4 picked from grid search results. The model is trained at batch size 16 in pretraining for 20,000 iterations and batch size 4 in finetuning for 10,000 iterations. Both inference and finetune use 100 sampling steps, SDE inference, EMA decay 0.999, and post-translation surface smoothing with σ=0.05σ=0.05. The learning rates used in pretraining and finetuning are 10−410^-4 and 10−510^-5, respectively. The best model is selected based on validation-set performance metrics as in Table 2. All experiments were conducted on an NVIDIA RTX A6000 GPU with 48 GB of memory. 3 Results 3.1 Dataset and Preprocessing For the tau PET harmonization experiments, we utilized two public datasets: the Health and Aging Brain Study Health Disparities (HABS-HD) [28] and the Alzheimer’s Disease Neuroimaging Initiative (ADNI) [29]. Each raw tau PET acquisition consisted of six consecutive 5-minute frames (30 minutes total), beginning approximately 75 minutes after tracer injection. All frames are coregistered to the first frame and averaged to create the average PET image. Then the PET image is processed through PETSurfer [30] along with the temporally closest T1w image processed by the FreeSurfer pipeline [31]. The standard uptake value ratio (SUVR) was calculated by dividing the raw tau signal by the mean signal from the inferior cerebellum. The SUVR values are resampled to the cortical surface obtained from T1 using mri_vol2surf command in FreeSurfer. Because tau PET data suffers from off-target binding where signals from extracerebral tissue such as the meninges contaminate the cortical signal, we apply artifact removal [24] to the SUVR feature map. The feature map is then resampled to a 6th-order icosahedron surface through the mri_surf2surf command. The tau positivity is computed by thresholding the mean cortical SUVR by the method in [32]. The statistics of datasets are shown in Table 1. Table 1: Subject-level demographics, scan counts, and data splits for the HABS-HD and ADNI tau PET datasets used in the SUVR harmonization experiment. CN: Cognitively Normal; MCI: Mild Cognitive Impairment; AD: Alzheimer’s Disease. Splits follow a 6/2/2 ratio on the subject level. Attribute HABS-HD ADNI N (Subjects) 1797 826 Number of Scans 2459 1480 Tracer Types PI-2620 AV-1451 Age (Mean ± SD, years) 66.2 ± 8.7 74.3 ± 8.1 Sex (M / F) 1105 / 692 417 / 409 Diagnosis (CN / MCI / AD) 1323 / 385 / 89 460 / 283 / 83 Train / Val / Test 1078 / 359 / 360 496 / 165 / 165 3.2 Tau PET SUVR Harmonization Across Tracers We designate HABS-HD as the source domain and ADNI as the target domain. This choice is motivated by ADNI’s standardized diagnostic protocols and by the widespread use of AV-1451 (Flortaucipir) as a tau PET tracer in Alzheimer’s disease research. The objective is to harmonize HABS-HD PI-2620 SUVR maps into the ADNI AV-1451 target domain. Both datasets are split into train/validation/test sets separately, as shown in Table 1. The model is trained on the training set and validated on the validation set for model selection. All reported metrics are computed on the held-out test set. We compare our model with ComBat [23], surface-based CycleGAN [10], surface DDPM-based harmonization (DF) [24], and the unregularized Diffusion Schrödinger Bridge matching (DSBM). 3.2.1 Harmonization Evaluation Table 2: Quantitative comparison of harmonization methods. Metrics include Wasserstein Distance (↓), PCC (↑), absolute Somers’ D of site classification (↓), and Wasserstein Distance within tau-positive and tau-negative subgroups (↓). Somers’ D = 2⋅·AUC−-1; values near 0 indicate site-indistinguishable harmonization. Bold indicates best. Left Hemisphere Right Hemisphere Method WD↓ PCC↑ |D|↓|D| WD τpos↓ _pos WD τneg↓ _neg WD↓ PCC↑ |D|↓|D| WD τpos↓ _pos WD τneg↓ _neg ComBat 0.0619 0.9517 0.3314 0.2296 0.0493 0.0357 0.9605 0.1824 0.1076 0.0327 CycleGAN 0.0601 0.9468 0.0700 0.3628 0.1578 0.0631 0.9547 0.6140 0.1430 0.0942 DF 0.1254 0.9244 0.7600 0.4279 0.2069 0.1946 0.9414 0.3960 0.3705 0.3685 DSBM 0.0496 0.9359 0.1694 0.1187 0.0627 0.0285 0.9484 0.1520 0.0935 0.0449 FKRSBM 0.0079 0.9519 0.0696 0.0486 0.0043 0.0100 0.9503 0.0504 0.0376 0.0072 The quantitative results are shown in Table 2. We evaluate image-level harmonization using Pearson Correlation Coefficient (PCC) between each source image and its harmonized output to assess pattern preservation and Wasserstein Distance (WD) between the harmonized and target distributions to assess distributional alignment in the unpaired setting. The left and right hemispheres are evaluated separately. Residual site separability is quantified using the absolute Somers’ D of a site classifier (SVM on ROI mean features) (|D|=|2⋅AUC−1||D|=|2·AUC-1|), where values closer to zero indicate stronger site-effect removal. We also report WD within tau-positive and tau-negative subgroups to evaluate whether harmonization aligns distributions at the subgroup level. Table 2 shows that our model provides the best overall distributional alignment, the lowest residual site separability, and the strongest tau-subgroup alignment among the evaluated methods in both hemispheres, while preserving source spatial patterns at a PCC comparable to competing harmonization methods. For qualitative evaluation, we plotted SUVR maps of example subjects in Figure 3. The three rows show representative subjects with different tau-burden patterns, including high temporal tau pathology and lower-SUVR cases with more subtle temporal elevation. Due to the distributional difference between AV-1451 (target) and PI-2620 (source), SUVR values are expected to increase after harmonization. FKRSBM and DSBM generally preserve the topography of pathology, whereas CycleGAN and DF show visible artifacts or loss of detail in some examples. Figure 3: Harmonization visualization for three representative HABS-HD subjects and harmonized outputs produced by ComBat, CycleGAN, DF, DSBM and FKRSBM. The rows illustrate examples with different tau-burden patterns, including clear temporal tau pathology and lower-SUVR cases with subtler temporal elevation. Figure 4: Temporal Region-level harmonization comparison across Desikan–Killiany ROIs. Each group of bars shows the mean SUVR for a single ROI in temporal lobe, with error bars denoting standard deviation. The three rows correspond to the overall test set (top), the tau-negative subgroup (middle), and the tau-positive subgroup (bottom). Figure 4 presents the results as grouped bar plots. Across the reported ROIs and subgroups, our model shows the closest alignment to the target in both mean and variance. ComBat corrects the global mean shift but, owing to its location–scale assumption, leaves residual biases where the tracer-induced shift is nonlinear or region-dependent. CycleGAN aligns ROI means while substantially compressing variance, consistent with mode-collapse-like reduction; this is most pronounced in the tau-positive subgroup (bottom row), where the higher dynamic range exposes the limited expressiveness of the cycle-consistency constraint. DF yields shifted means and inflated variance in several ROIs, attenuating fine regional structure. The DSBM improves overall mean alignment but shows systematic under- or over-correction in the stratified rows, particularly for tau-positive subjects, where product endpoint sampling conflates tracer effects with tau-burden subgroups during transport. In contrast, our model uses the Feynman–Kac reweighting to steer transport toward tau-subgroup-consistent pairings, yielding harmonized per-ROI means and standard deviations that more closely track the target across all three strata. 3.2.2 Tau Positivity Preservation Table 3: Tau positivity sign mismatch analysis after harmonization. pos→ and neg→ indicate the direction of status flips. Left Hemisphere Right Hemisphere Method n_total n_mismatch mismatch_pct pos→ neg→ n_total n_mismatch mismatch_pct pos→ neg→ ComBat 503 33 6.6% 0 33 503 26 5.2% 0 26 CycleGAN 503 36 7.2% 36 0 503 34 6.8% 34 0 DF 503 35 7.0% 35 0 503 38 7.6% 38 0 DSBM 503 88 17.5% 18 70 503 67 13.3% 16 51 FKRSBM 503 13 2.6% 9 4 503 15 3.0% 4 11 Table 4: Downstream diagnostic separability on the combined test set. Best result per column in bold. Method CN vs AD CN vs MCI vs AD F1 Acc. Spec. Sens. F1 Acc. Spec. Sens. Unharmonized 0.7404±0.02750.7404±0.0275 0.7657±0.01860.7657±0.0186 0.8625±0.01230.8625±0.0123 0.6871±0.03980.6871±0.0398 0.5314±0.03010.5314±0.0301 0.5410±0.02760.5410±0.0276 0.7739±0.01290.7739±0.0129 0.5474±0.02670.5474±0.0267 ComBat 0.7426±0.01150.7426±0.0115 0.7690±0.01210.7690±0.0121 0.8649±0.03330.8649±0.0333 0.6846±0.01740.6846±0.0174 0.5122±0.03330.5122±0.0333 0.5220±0.03250.5220±0.0325 0.7629±0.01560.7629±0.0156 0.5258±0.03150.5258±0.0315 CycleGAN 0.7373±0.03320.7373±0.0332 0.7659±0.02830.7659±0.0283 0.8743±0.02420.8743±0.0242 0.6793±0.03670.6793±0.0367 0.5175±0.03820.5175±0.0382 0.5278±0.03830.5278±0.0383 0.7675±0.01670.7675±0.0167 0.5366±0.03600.5366±0.0360 DF 0.7360±0.04390.7360±0.0439 0.7698±0.03020.7698±0.0302 0.8877±0.02540.8877 0.0254 0.6692±0.06320.6692±0.0632 0.4914±0.04220.4914±0.0422 0.4953±0.04560.4953±0.0456 0.7497±0.02250.7497±0.0225 0.5009±0.04390.5009±0.0439 DSBM 0.7286±0.04560.7286±0.0456 0.7541±0.03700.7541±0.0370 0.8400±0.03490.8400±0.0349 0.6839±0.05240.6839±0.0524 0.5095±0.04270.5095±0.0427 0.5212±0.04680.5212±0.0468 0.7628±0.02130.7628±0.0213 0.5244±0.04360.5244±0.0436 FKRSBM 0.8019±0.03730.8019 0.0373 0.8169±0.03040.8169 0.0304 0.8832±0.02310.8832±0.0231 0.7621±0.04770.7621 0.0477 0.5794±0.06390.5794 0.0639 0.5862±0.06020.5862 0.0602 0.7952±0.02890.7952 0.0289 0.5888±0.05960.5888 0.0596 Figure 5: Subgroup alignment by APOE ε 4 status. Each column corresponds to a harmonization method; the two rows show APOE ε 4 non-carriers (top) and carriers (bottom). Within each panel, boxplots compare the mean tau SUVR of the harmonized HABS-HD data (colored) with the target ADNI data (blue) for the left and right hemispheres. The Kolmogorov–Smirnov (KS) statistic between the harmonized and target distributions is reported at the top of each panel; lower values indicate better distributional alignment. A clinically meaningful harmonization should align the tau positivity status of individual subjects. Each HABS-HD hemisphere is assigned a binary tau label (positive or negative) using the standard definition described in Section 3.1. After harmonizing the HABS-HD SUVR maps into the ADNI domain, we recompute tau positivity using the ADNI-derived cutoff and compare the post-harmonization label with the original pre-harmonization label. A status flip—tau-positive becoming negative or vice versa—indicates that the harmonization has altered the biological signal sufficiently to change a subject’s clinical categorization. Table 3 reports the results for 503 test hemispheres. FKRSBM has the lowest mismatch rate in this comparison: 2.6% (13/503) in the left hemisphere and 3.0% (15/503) in the right hemisphere. The direction of flips is also balanced, with roughly comparable numbers of pos→ and neg→ errors, suggesting no systematic bias in either direction. In contrast, the DSBM produces the highest mismatch rate (17.5% left, 13.3% right), with a strong asymmetry: the majority of flips are neg→ , consistent with product endpoint sampling in DSBM-style training transporting some tau-negative subjects toward higher-SUVR regions of the target distribution. This is the failure mode that the Feynman–Kac reweighting is designed to reduce. ComBat exhibits moderate mismatch rates (6.6% left, 5.2% right) with all flips in the neg→ direction, consistent with an upward shift that pushes borderline-negative subjects above the positivity threshold. CycleGAN shows a comparable mismatch rate (7.2% left, 6.8% right) but with flips exclusively in the pos→ direction, consistent with the variance compression observed in the ROI-level analysis. The low and balanced mismatch rate of FKRSBM suggests that Feynman–Kac reweighted transport better preserves tau-positivity subgroup structure during harmonization. 3.2.3 Downstream Clinical Assessment To assess whether harmonization aligns clinically relevant biological stratification beyond the tau-positivity label, we perform a diagnostic classification experiment and a subgroup analysis based on APOE ε 4 carrier status, a major genetic risk factor for Alzheimer’s disease. APOE ε 4 carrier status is not used during training. Figure 5 shows boxplots of mean tau SUVR for APOE ε 4 non-carriers (top row) and carriers (bottom row) across all methods, with the target ADNI distribution shown in blue. Among all methods, FKRSBM achieves the lowest KS statistics in all panels except right-hemisphere carriers, where ComBat is slightly better, indicating that the harmonized distributions more closely match the target within each genetic subgroup. Importantly, FKRSBM preserves the expected biological ordering: APOE ε 4 carriers consistently show higher mean SUVR than non-carriers after harmonization, mirroring the pattern observed in the target ADNI data. This supports the interpretation that the Feynman–Kac reweighted transport reduces tracer-induced site effects without collapsing the biologically meaningful separation between genetic risk groups. We further evaluate whether the harmonized data improves pooled-domain diagnostic separability. For each hemisphere, the surface SUVR map is reduced to a feature vector of Desikan–Killiany ROI means. A support vector machine (SVM) classifier is trained on the combined (harmonized HABS-HD ∪ ADNI) test set for two tasks: CN vs. AD and the three-class CN vs. MCI vs. AD problem. This analysis is intended as a downstream statistical-power probe: if harmonization reduces site- or tracer-related variability while retaining disease-related signal, pooled data should support better diagnostic separation. Because the diagnostic groups are imbalanced, we subsample each class to match the size of the smallest class. To ensure robustness, we repeat this random subsampling procedure across 5 independent random seeds and report the mean and standard deviation of F1 score, accuracy, specificity, and sensitivity from 3-fold cross-validation. The folds are split at the subject level to prevent data leakage from subjects with multiple scans. Table 4 presents the results. FKRSBM yields the highest F1 score across two tasks. The superior performance metrics indicate that our model can better support downstream clinical tasks. 4 Conclusion We presented Feynman–Kac Reweighted Schrödinger Bridge Matching (FKRSBM), a surface-based harmonization framework that combines Schrödinger Bridge matching with a tau-positivity-aware endpoint proposal. The Feynman–Kac reweighting is implemented at the sampling level, so it preserves the Brownian bridge regression target and the underlying DSBM architecture while concentrating training on biologically plausible source–target pairs. In cross-tracer tau PET harmonization from HABS-HD PI-2620 to ADNI AV-1451, FKRSBM improved distributional alignment, tau-positivity preservation, APOE subgroup alignment, and pooled-domain diagnostic separability relative to the evaluated baselines. These results suggest that endpoint-aware coupling is important when unpaired source and target cohorts differ in biological subgroup composition. Future work should extend the method to multi-site harmonization, continuous covariates, and other surface-based imaging modalities. REFERENCES References [1] J.-P. Fortin, D. Parker, B. Tunç, T. Watanabe, M. A. Elliott, K. Ruparel, D. R. Roalf, T. D. Satterthwaite, R. C. Gur, R. E. Gur, R. T. Schultz, R. Verma, and R. T. Shinohara, “Harmonization of multi-site diffusion tensor imaging data,” NeuroImage, vol. 161, p. 149–170, 2017. [2] J.-P. Fortin, N. Cullen, Y. I. Sheline, W. D. Taylor, I. Aselcioglu, P. A. Cook, P. Adams, C. Cooper, M. Fava, P. J. McGrath et al., “Harmonization of cortical thickness measurements across scanners and sites,” Neuroimage, vol. 167, p. 104–120, 2018. [3] W. E. Johnson, C. Li, and A. Rabinovic, “Adjusting batch effects in microarray expression data using empirical Bayes methods,” Biostatistics, vol. 8, no. 1, p. 118–127, 2007. [4] G. Varoquaux and V. Cheplygina, “Machine learning for medical imaging: methodological failures and recommendations for the future,” NPJ digital medicine, vol. 5, no. 1, p. 48, 2022. [5] Y. Ganin, E. Ustinova, H. Ajakan, P. Germain, H. Larochelle, F. Laviolette, M. Marchand, and V. Lempitsky, “Domain-adversarial training of neural networks,” in Journal of Machine Learning Research, vol. 17, no. 59, 2016, p. 1–35. [6] A. F. Marquand, I. Rezek, J. Buitelaar, and C. F. Beckmann, “Understanding heterogeneity in clinical cohorts using normative models: Beyond case-control studies,” Biological Psychiatry, vol. 80, no. 7, p. 552–561, 2016. [7] S. Kang, S.-W. Kim, J.-K. Seong, and A. D. N. Initiative, “Disentangling brain atrophy heterogeneity in Alzheimer’s disease: A deep self-supervised approach with interpretable latent space,” NeuroImage, vol. 297, p. 120737, 2024. [8] J.-Y. Zhu, T. Park, P. Isola, and A. A. Efros, “Unpaired image-to-image translation using cycle-consistent adversarial networks,” in Proceedings of the IEEE International Conference on Computer Vision (ICCV), 2017. [9] B. E. Dewey, C. Zhao, J. C. Reinhold, A. Carass, K. C. Fitzgerald, E. S. Sotirchos, S. Saidha, J. Oh, D. L. Pham, P. A. Calabresi et al., “Deepharmony: A deep learning approach to contrast harmonization across scanner changes,” Magnetic resonance imaging, vol. 64, p. 160–170, 2019. [10] C. Zhao, A. Carass, J. Lee, Y. He, and J. L. Prince, “Harmonization of multi-site diffusion mri data via a cycle-consistent adversarial network,” in International Conference on Medical Image Computing and Computer-Assisted Intervention. Springer, 2019, p. 159–167. [11] A. H. Jha, S. Anand, M. Singh, and V. Veeravasarapu, “Disentangling factors of variation with cycle-consistent variational auto-encoders,” arXiv preprint arXiv:1804.10469, 2018. [12] D. Moyer, G. Ver Steeg, C. M. Tax, and P. M. Thompson, “Scanner invariant representations for weakly supervised domain adaptation with adversarial learning,” in International Conference on Medical Image Computing and Computer-Assisted Intervention. Springer, 2020, p. 3–13. [13] L. Zuo, B. E. Dewey, Y. Liu, Y. He, S. D. Newsome, E. M. Mowry, S. M. Resnick, J. L. Prince, and A. Carass, “Information-based disentangled representation learning for unsupervised mr harmonization,” IEEE Transactions on Medical Imaging, vol. 40, no. 11, p. 3095–3107, 2021. [14] N. K. Dinsdale, M. Jenkinson, and A. I. Namburete, “Deep learning-based unlearning of dataset bias for mri harmonisation and confound removal,” NeuroImage, vol. 228, p. 117689, 2021. [15] F. Locatello, S. Bauer, M. Lucic, G. Raetsch, S. Gelly, B. Schölkopf, and O. Bachem, “Challenging common assumptions in the unsupervised learning of disentangled representations,” in International Conference on Machine Learning. PMLR, 2019, p. 4114–4124. [16] C. P. Burgess, I. Higgins, A. Pal, L. Matthey, N. Watters, G. Desjardins, and A. Lerchner, “Understanding disentangling in β-vae,” arXiv preprint arXiv:1804.03599, 2018. [17] C. Meng, Y. He, Y. Song, J. Song, J. Wu, J.-Y. Zhu, and S. Ermon, “Sdedit: Guided image synthesis and editing with stochastic differential equations,” in International Conference on Learning Representations, 2021. [18] V. De Bortoli, J. Thornton, J. Heng, and A. Doucet, “Diffusion schrödinger bridge with applications to score-based generative modeling,” in Advances in Neural Information Processing Systems (NeurIPS), 2021. [19] G.-H. Liu, A. Vahdat, D.-A. Huang, E. A. Theodorou, W. Nie, and A. Anandkumar, “I2sb: Image-to-image schrödinger bridge,” in International Conference on Machine Learning. PMLR, 2023, p. 21 228–21 248. [20] V. De Bortoli, I. Korshunova, A. Mnih, and A. Doucet, “Schrödinger bridge flow for unpaired data translation,” arXiv preprint arXiv:2409.09347, 2024. [21] J. Ma, Y. Tan, and R. Xu, “Schrödinger bridge for generative AI: Soft-constrained formulation and convergence analysis,” arXiv preprint arXiv:2510.11829, 2025. [22] F. Zhao, Z. Wu, L. Wang, W. Lin, J. H. Gilmore, S. Xia, D. Shen, and G. Li, “Spherical deformable u-net: Application to cortical surface parcellation and development prediction,” IEEE Transactions on Medical Imaging, vol. 40, no. 4, p. 1217–1228, 2021. [23] J.-P. Fortin, N. Cullen, Y. I. Sheline, W. D. Taylor, I. Aselcioglu, P. A. Cook, P. Adams, C. Cooper, M. Fava, P. J. McGrath, M. McInnis, M. L. Phillips, M. H. Trivedi, M. M. Weissman, and R. T. Shinohara, “Harmonization of cortical thickness measurements across scanners and sites,” NeuroImage, vol. 167, p. 104–120, 2018. [Online]. Available: https://w.sciencedirect.com/science/article/pii/S105381191730931X [24] J. Yue, J. Zhang, L. Zhong, Y. Shi, A. D. N. Initiative et al., “Tau pet harmonization via surface-based diffusion model,” in 2025 IEEE 22nd International Symposium on Biomedical Imaging (ISBI). IEEE, 2025, p. 1–5. [25] Y. Shi, V. De Bortoli, A. Campbell, and A. Doucet, “Diffusion schrödinger bridge matching,” Advances in Neural Information Processing Systems, vol. 36, p. 62 183–62 223, 2023. [26] P. Moral, Feynman-Kac formulae: genealogical and interacting particle systems with applications. Springer, 2004. [27] P. Billingsley, Probability and Measure, 3rd ed., ser. Wiley Series in Probability and Mathematical Statistics. New York: John Wiley & Sons, 1995. [28] S. E. O’Bryant et al., “The health & aging brain among latino elders (HABLE) study methods and participant characteristics,” Alzheimer’s & Dementia: Diagnosis, Assessment & Disease Monitoring, vol. 13, no. 1, p. e12202, 2021. [29] S. Mueller et al., “The Alzheimer’s disease neuroimaging initiative,” Neuroimaging Clinics of North America, vol. 15, no. 4, p. 869–877, 11 2005. [30] D. N. Greve, C. Svarer, P. M. Fisher, L. Feng, A. E. Hansen, W. Baare, B. Rosen, B. Fischl, and G. M. Knudsen, “Cortical surface-based analysis reduces bias and variance in kinetic modeling of brain pet data,” Neuroimage, vol. 92, p. 225–236, 2014. [31] B. Fischl, “Freesurfer,” Neuroimage, vol. 62, no. 2, p. 774–781, 2012. [32] C. R. Jack, D. A. Bennett, K. Blennow, M. C. Carrillo, B. Dunn, S. B. Haeberlein, D. M. Holtzman, W. Jagust, F. Jessen, J. Karlawish, E. Liu, J. L. Molinuevo, T. Montine, C. Phelps, K. P. Rankin, C. C. Rowe, P. Scheltens, E. Siemers, H. M. Snyder, and R. Sperling, “Nia-a research framework: Toward a biological definition of alzheimer’s disease,” Alzheimer’s & Dementia, vol. 14, no. 4, p. 535–562, 2018.

Instant research discovery