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arxiv: 2606.17420 · v1 · pith:KXGENVP5new · submitted 2026-06-16 · 📡 eess.IV · cs.AI· q-bio.QM

Feynman Kac Reweighted Schr\"odinger Bridge Matching for Surface-Based Tau PET Harmonization

Jianwei Zhang , Xinyu Nie , Jiaxin Yue , Yonggang Shi This is my paper

Pith reviewed 2026-06-26 23:03 UTC · model grok-4.3

classification 📡 eess.IV cs.AIq-bio.QM
keywords tau PET harmonizationSchrödinger bridge matchingFeynman Kac reweightingsurface-based neuroimagingAlzheimer's diseasemulti-site imagingSUVR mapsAPOE subgroup alignment
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The pith

FKRSBM aligns tau PET images across sites by reweighting Schrödinger bridges to separate scanner effects from biological subgroups.

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

The paper proposes FKRSBM to harmonize tau PET SUVR maps from different scanners and radiotracers. It replaces the usual noise-prior route with a direct entropy-regularized optimal transport between source and target distributions. A Feynman Kac reweighting supplies subgroup-aware endpoint proposals that are realized through stratified importance sampling, leaving the bridge solver and network unchanged. Applied to spherical convolutions on cortical meshes, the method is tested by mapping PI-2620 data into the AV-1451 domain. It reports tighter distributional match, fewer sign flips in tau positivity, better APOE alignment, and higher downstream classification accuracy than ComBat, CycleGAN, diffusion baselines, and plain DSBM.

Core claim

FKRSBM learns a direct stochastic transport process between source and target distributions via entropy-regularized optimal transport. It incorporates a subgroup-aware endpoint proposal derived from a Feynman Kac reweighting of the reference bridge measure, implemented 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 of tau PET SUVR maps.

What carries the argument

subgroup-aware endpoint proposal derived from Feynman Kac reweighting of the reference bridge measure, implemented through stratified importance sampling

If this is right

  • achieves superior distributional alignment compared with ComBat, CycleGAN, a diffusion method, and unregularized DSBM
  • reduces tau-positivity sign mismatch
  • strengthens APOE subgroup alignment
  • improves downstream disease classification performance

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach may generalize to harmonizing other surface-based modalities such as cortical thickness or functional connectivity maps.
  • If the reweighting truly isolates biology, the same sampling procedure could be applied to longitudinal within-subject data to reduce scanner drift.
  • Clinical trials using tau PET endpoints across centers could adopt the method to increase statistical power without requiring matched subgroup compositions.

Load-bearing premise

The subgroup-aware endpoint proposal derived from Feynman Kac reweighting of the reference bridge measure successfully separates site-induced shifts from biological variation such as tau-positivity status without introducing new biases.

What would settle it

A new multi-site tau PET dataset in which FKRSBM-harmonized maps still exhibit site-specific differences in tau-positivity rates or yield no gain in disease classification accuracy relative to the unharmonized data.

Figures

Figures reproduced from arXiv: 2606.17420 by Jianwei Zhang, Jiaxin Yue, Xinyu Nie, Yonggang Shi.

Figure 1
Figure 1. Figure 1: Distributional and spatial differences between tau PET trac￾ers. 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 < 0.001), with AV-1451 yielding systematically higher SUVR values. Right: Cohort￾averaged SUVR projected onto the lateral cortical surface reveals tracer￾specif… view at source ↗
Figure 2
Figure 2. Figure 2: Overview of the proposed Feynman–Kac Reweighted Schrodinger Bridge (FKRSBM) for cross-tracer tau PET harmonization. ¨ Top: Schematic of the bridge between source (X0, PI-2620) and target (X1, 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) rew… view at source ↗
Figure 3
Figure 3. 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. 0.0 0.5 1.0 1.5 2.0 2.5 ROI mean Source ComBat CycleGAN DF DSBM FKRSBM(Ours) Target 0.0 0.5 1.0 1.5 2.0 2.5… view at source ↗
Figure 4
Figure 4. 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). TABLE III TAU POSITIVITY SIGN MISMATCH ANALYSIS AFTER HARMONIZATION. POS→NEG AND… view at source ↗
Figure 5
Figure 5. Figure 5: Subgroup alignment by APOE ε4 status. Each column corresponds to a harmonization method; the two rows show APOE ε4 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… view at source ↗
read the original 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 Schr\"oodinger 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 Schr\"oodinger Bridge Matching (DSBM), FKRSBM achieves superior distributional alignment, reduced tau-positivity sign mismatch, stronger APOE subgroup alignment, and improved downstream disease classification performance.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes the Feynman Kac Reweighted Schrödinger Bridge Matching (FKRSBM) model for surface-based tau PET harmonization. It learns a direct entropy-regularized stochastic transport between source (HABS-HD PI-2620) and target (ADNI AV-1451) distributions via Schrödinger Bridge Matching, augmented by a subgroup-aware endpoint proposal obtained from Feynman Kac reweighting of the reference bridge measure and realized through stratified importance sampling at the data level. The approach uses a spherical convolutional network on cortical meshes for vertex-level mapping and is reported to outperform ComBat, CycleGAN, a diffusion baseline (DF), and unregularized DSBM on distributional alignment, tau-positivity sign consistency, APOE subgroup preservation, and downstream disease classification.

Significance. If the central performance claims are supported by the data, the work would offer a practical, architecture-preserving extension of bridge-matching methods that explicitly incorporates subgroup structure to mitigate conflation of site and biological effects in multi-tracer tau PET studies. The data-level implementation of the reweighting is a design choice that could facilitate adoption.

major comments (2)
  1. [Abstract] Abstract: the claim that the subgroup-aware endpoint proposal 'separates site-induced shifts from biological variation such as tau-positivity status' is load-bearing for the entire contribution. The description states that this is achieved 'entirely through stratified importance sampling at the data level' with 'no changes to the underlying bridge-matching solver or network architecture.' It is therefore unclear whether the reweighting modifies the full path measure of the learned bridge or only the endpoint marginals; if the latter, the stochastic transport may still mix site and subgroup effects, undermining the separation guarantee.
  2. [Abstract] Abstract (evaluation paragraph): the reported superiority on 'distributional alignment, reduced tau-positivity sign mismatch, stronger APOE subgroup alignment, and improved downstream disease classification' is presented without reference to specific quantitative tables, statistical tests, or error bars. Because these metrics directly support the claim that FKRSBM avoids new biases, the absence of the supporting numerical evidence in the provided manuscript text prevents verification that the improvements are not driven by post-hoc choices or subgroup imbalance.
minor comments (1)
  1. [Abstract] The abstract refers to 'Feynman Kac Reweighted Schrödinger Bridge Matching' and 'Diffusion Schrödinger Bridge Matching (DSBM)' without an explicit equation or pseudocode block defining the reweighting operator or the importance-sampling weights; a short methods subsection with these definitions would improve reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their insightful comments. We address each major comment below with clarifications and indicate where revisions will strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that the subgroup-aware endpoint proposal 'separates site-induced shifts from biological variation such as tau-positivity status' is load-bearing for the entire contribution. The description states that this is achieved 'entirely through stratified importance sampling at the data level' with 'no changes to the underlying bridge-matching solver or network architecture.' It is therefore unclear whether the reweighting modifies the full path measure of the learned bridge or only the endpoint marginals; if the latter, the stochastic transport may still mix site and subgroup effects, undermining the separation guarantee.

    Authors: The Feynman Kac reweighting adjusts the reference bridge measure to produce subgroup-aware endpoint marginals. The Schrödinger Bridge is then matched directly between these reweighted marginals, so the learned path measure incorporates the subgroup structure throughout the transport rather than only at the endpoints. The data-level stratified importance sampling realizes this reweighting without altering the solver or network. We will revise the abstract to state explicitly that the reweighting influences the full bridge measure via the adjusted marginals. revision: yes

  2. Referee: [Abstract] Abstract (evaluation paragraph): the reported superiority on 'distributional alignment, reduced tau-positivity sign mismatch, stronger APOE subgroup alignment, and improved downstream disease classification' is presented without reference to specific quantitative tables, statistical tests, or error bars. Because these metrics directly support the claim that FKRSBM avoids new biases, the absence of the supporting numerical evidence in the provided manuscript text prevents verification that the improvements are not driven by post-hoc choices or subgroup imbalance.

    Authors: The abstract summarizes key findings at a high level. Full quantitative results with error bars, statistical tests, and tables appear in the Results section. We will add brief parenthetical references (e.g., 'Table 2') in the abstract to direct readers to the supporting evidence. revision: yes

Circularity Check

0 steps flagged

No circularity: method introduces independent reweighting via stratified sampling; claims rest on external comparisons

full rationale

The derivation introduces FKRSBM as a direct stochastic transport with a subgroup-aware endpoint proposal implemented through stratified importance sampling at the data level, without altering the bridge-matching solver. Performance claims (distributional alignment, tau-positivity sign mismatch, APOE alignment, downstream classification) are evaluated against external baselines (ComBat, CycleGAN, DF, DSBM) on held-out HABS-HD to ADNI data. No equation or step reduces a claimed prediction to a fitted parameter or self-citation by construction; the reweighting is presented as an additive procedure whose correctness is assessed empirically rather than defined into the result. This is the common case of an independent methodological contribution.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only; no explicit free parameters, ad-hoc axioms, or invented entities are stated. Relies on standard optimal transport and diffusion concepts.

axioms (1)
  • standard math Entropy-regularized optimal transport yields a valid stochastic transport process between source and target distributions.
    Standard background assumption in Schrödinger bridge literature.

pith-pipeline@v0.9.1-grok · 5844 in / 1251 out tokens · 32433 ms · 2026-06-26T23:03:41.415407+00:00 · methodology

discussion (0)

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Reference graph

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