Department of Computer Science | Institute of Theoretical Computer Science | CADMO
Theory of Combinatorial Algorithms
Prof. Emo Welzl and Prof. Bernd Gärtner
Department of Computer Science | Institute of Theoretical Computer Science | CADMO
Prof. Emo Welzl and Prof. Bernd Gärtner
| Mittagsseminar Talk Information |
Date and Time: Thursday, December 11, 2025, 12:15 pm
Duration: 30 minutes
Location: CAB G51
Speaker: Maja Gwozdz
We study Entropic Optimal Transport on weighted graphs, namely, a modified version of the Sinkhorn algorithm, where the transport kernel is given by a continuous-time random walk. While the Sinkhorn algorithm provides strong theoretical guarantees, we show that on certain graph families with a growing diameter, convergence can become arbitrarily slow. In order to overcome this disadvantage, for each diffusion time T > 0, we define a geometry-adapted stabilised kernel MT = AT(R) + η uTvT, where AT(R) is the heat kernel truncated at radius R ≍ √T in the graph distance, η > 0 is a stabilisation parameter, uT(x) is the inverse volume of the ball B(x,√T), and v is the reference vertex measure. By means of Gaussian heat-kernel upper bounds, we bound the projective diameter of MT and obtain a concrete contraction factor ρ(MT) in Hilbert's projective metric.
For finite graphs, this geometric control has three important algorithmic consequences. The equivalence w.r.t. residual error demonstrates that the true Hilbert distance to the fixed point is bounded below and above, and the constants depend only on ρ(MT), which provides rigorous stopping rules based on residuals. Secondly, we are able to describe kernel changes for multi-scale schedules in T by a relatively simple logarithmic oscillation, namely: Ek = osc log(Mk/Mk-1). This also leads to a bound on the drift of the corresponding fixed points and iteration counts (the number of Sinkhorn iterations at scale k) that guarantee precise Hilbert tolerances. Finally, under the assumption of a two-sided Gaussian heat-kernel bound, we obtain a bound on the approximation bias between transport plans computed with MT and with the raw heat kernel AT.
In the final part of the talk, we show that on countably infinite graphs, under relatively mild and natural assumptions, the plan induced by the fixed point of the stabilised Sinkhorn map is precisely the Kullback-Leibler I-projection of MT onto the marginal constraints. We also present empirical simulation results and briefly discuss the practical challenges of the algorithmic implementation.
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