OT-Sufficient Bottleneck Flow Matching

1. Integration point: replace a standard supervised regression head, or add this objective to an encoder-decoder model. For each minibatch of covariates X and targets Y, compute a low-dimensional representation Z=r_phi(X) with dimension d much smaller than the input dimension. A conditional flow network v_theta receives (t,Y_t,Z) and predicts the transport velocity. At inference, draw Y_0 from a reference distribution such as N(0,I), then solve dY/dt=v_theta(t,Y,Z) using Euler or Heun steps. 2. Pseudocode: sample {(x_j,y_j)}; calculate z_j=r_phi(x_j); draw reference points y0_i; calculate C_ij=||y_j-y0_i||^2 and R_ij=||z_j-z_i||^2/(mean pairwise z-distance+delta); run Sinkhorn on exp(-(C+lambda R)/epsilon) to obtain P; sample or weight pairs (i,j) according to P; draw t uniformly from [0,1]; set yt=(1-t)y0_i+t y_j and u=y_j-y0_i; minimize ||v_theta(t,yt,z_j)-u||^2 and backpropagate through the encoder, optionally stopping gradients through P for stability. 3. The interpolation, quadratic cost, and Sinkhorn coupling come from the mathematical construction; epsilon, lambda, beta, bottleneck dimension d, and integration steps are estimated or selected empirically. 4. First experiment: construct a two-dimensional synthetic X with one true sufficient coordinate, an independent nuisance coordinate, and a bimodal or heteroscedastic Y conditional on the sufficient coordinate. Compare against a linear SDR bottleneck and an MLP trained with MSE at equal bottleneck size. Evaluate held-out conditional sliced Wasserstein distance, distributional log likelihood, mode coverage, and sensitivity to nuisance perturbations. Success means lower conditional Wasserstein error, preserved bimodality, and unchanged predictions when nuisance coordinates are randomized, without sacrificing training speed by more than the Sinkhorn overhead.