TubeFM: Likelihood-Isometric Flow Matching for Noisy Inverse Problems

Posterior samplers for noisy inverse problems must represent uncertainty in measured as well as unmeasured directions. TubeFM learns conditional flows in an affine chart comprising a fiber coordinate and a noise-standardized measurement residual. We use this chart to connect posterior approximation, velocity regression, and numerical integration. An exact Wasserstein decomposition quantifies the error imposed by hard measurement constraints; a flow-stability bound transfers velocity error to image space; and explicit prior assumptions yield noise-uniform posterior curvature bounds. Gaussian analysis separates this target geometry from the dynamics of the chosen probability path. With the source, path, and error criterion transformed together, an affine change of coordinates preserves the mapped Runge–Kutta trajectory and evaluation count in exact arithmetic; changing the source can alter both. Experiments on 135 Gaussian problems, real handwritten images, procedural Fourier imaging, and an exploratory fastMRI pilot test these distinctions. Three-seed learned-flow comparisons show that a source-matched image model can outperform TubeFM in reconstruction, while hard projection removes measured-coordinate uncertainty. The MRI pilot reproduces this uncertainty distinction with nearly unchanged PSNR on two validation volumes. We also quantify the finite-ensemble effect in raw percentile coverage. Together, the analysis and controls identify what likelihood scaling contributes and separate it from source selection, posterior calibration, and solver efficiency.