Rényi projections onto transportation polytopes: tail-sensitive couplings and a generalised Sinkhorn algorithm for robust risk modelling

We study projections of a strictly positive reference matrix A onto the transportation polytope Π(a,b) by minimising D_α(P‖A), α>1. The limit α↓1 recovers the Kullback–Leibler/Schrödinger–Sinkhorn projection, whereas α>1 penalises large likelihood ratios P_{ij}/A_{ij} more strongly. We prove existence and uniqueness, dual attainment, and the KKT representation P_{α,ij}=A_{ij}(u_i+v_j)_+^{1/(α−1)}, yielding a threshold support with a connected active graph. After gauge fixing, a generalised Sinkhorn-type alternating scheme converges globally and, under strict complementarity, locally linearly. On intervals where strict complementarity persists, the value and optimiser are C¹ in α; an explicit hypergradient enables calibration of α against a differentiable validation loss.

In risk modelling, a,b may represent separately calibrated loss marginals and A a historical or expert dependence prior. In a joint equity–credit or multi-asset loss grid, simultaneous threshold exceedances describe systemic-stress scenarios. If a KL-based fit underrepresents observed co-exceedances, α can be selected out of sample using a joint-exceedance score or a smooth expected-shortfall surrogate. Thus α is a likelihood-ratio sensitivity parameter, while the resulting change in tail risk remains data- and constraint-dependent rather than universally monotone.

The numerical study is a finite-dimensional proof of concept, including a resolved 2×3 transportation problem; no large-scale benchmark is claimed. For dense n×m inputs, a sweep requires O(Knm) arithmetic and O(nm) storage if each scalar row or column root is solved in at most K iterations. The contribution beyond entropic transport is the integration of structural theory, convergent computation, and differentiable calibration of α, linking constrained coupling to distributionally robust risk modelling.