Barycentric One-to-Many Alignment

1. Integration point: add this as a cross-modal alignment loss between source embeddings X={x_i} and target embeddings Y={y_j}, or between graph-node prototypes and instance embeddings. Compute embeddings with the existing encoders; do not alter the main task head initially. 2. Pseudocode: inputs are X, Y, source weights a, and target weights b. Initialize a nonnegative coupling P with row sums a and column sums b, for example by Sinkhorn on a cheap squared-distance cost. Repeat K times: compute z_i=(sum_j P_ij*y_j)/(a_i+1e-8); compute L=sum_{i,i'} a_i*a_i'*(dot(x_i,x_i')-dot(z_i,z_i'))^2; differentiate L through z and P; update encoder parameters and optionally update P by projected gradient or entropic Sinkhorn using the current barycentric objective. To avoid O(n^2) memory, estimate the pair sum with random source pairs or use the moment form ||S_mu||_F^2+||S_z||_F^2-2||M||_F^2, where S_mu=X^T diag(a) X, S_z=Z^T diag(a) Z, and M=X^T diag(a) Z. 3. The paper supplies the barycentric objective and its moment reduction; the coupling is estimated numerically, while Sinkhorn temperature, iteration count, and pair subsampling are empirical choices. Add a ridge term epsilon*sum_i a_i||z_i||^2 if barycenters become unstable. 4. First experiment: use CIFAR-100 or a small single-cell prototype-to-cell split, with one source prototype paired to 5-20 target instances. Compare pointwise cosine or OT alignment, ordinary GW, and this loss at equal encoder FLOPs. Measure retrieval accuracy, prototype-to-instance top-k recall, and sensitivity to duplicated or dispersed target instances. Success is improved retrieval under one-to-many spread without degrading one-to-one accuracy, plus lower variance across random target refinements.