Ian Hsieh, Soumya Snigdha Kundu, Tom Vercauteren +1cs.LG
Entropic optimal transport (EOT) has been shown to offer a computationally tractable approximation to exact optimal transport. However, the standard Sinkhorn-Knopp algorithm has two main limitations. First, given discrete measures with $N$ points, each iteration requires $O(N^2)$ operations, which restricts its use on large-scale datasets (e.g. $N\geq10^4$). Second, it uses the independent coupling as a reference measure for regularisation. This assigns mass to high-cost transport edges at moderate regularisation strengths. We propose SinkSLOT, which addresses both limitations by putting forth the expected sliced lifted transport plan as a natural way to sparsify the Gibbs kernel with a non-independent prior coupling. We prove that: 1) SinkSLOT converges; 2) with $L$ slices, each resulting sparse Sinkhorn iteration costs $O(LN)$; and 3) the resulting objective is a divergence requiring no debiasing. Experiments on synthetic benchmarks show that SinkSLOT delivers substantial speedups over state-of-the-art dense and sparse EOT methods. We also demonstrate the applicability of the proposed divergence in a gradient flow experiment. The code is publicly available at https://github.com/cai4cai/SinkSLOT.
This paper uses a minimum divergence framework to introduce a new way of calculating model weights that can be used to average probabilistic predictions from statistical and machine learning models. The method is general and can be applied regardless of whether the models under consideration are fit to data using frequentist, Bayesian, or some other fitting method. The proposed method is motivated in two different ways and is shown empirically to perform better than or on a par with standard model averaging methods, including model stacking and model averaging that relies on Akaike-style negative exponentiated model weighting, especially when the sample size is small. Our theoretical analysis explains why the method has a small-sample advantage.
We propose a new regularized optimal transport (OT) formulation, termed sliced-regularized optimal transport (SROT). Unlike entropic OT (EOT), which regularizes the transport plan toward an independent coupling, SROT regularizes it toward a smoothened sliced OT (SOT) plan. To the best of our knowledge, SROT is the first approach to leverage a version of SOT plan as a reference to improve classical OT. We provide a formal definition of SROT, derive its dual formulation, and provide a post-Bayesian interpretation of SROT. We then develop a Sinkhorn-style algorithm for efficient computation, retaining the same scalability advantages as EOT. By incorporating a scalable SOT plan as a prior, SROT yields more accurate approximations of the exact OT plan than EOT under the same level of regularization. Moreover, the resulting transport plan improves upon the reference SOT plan itself. We further introduce the corresponding OT divergence induced by SROT, named SROT divergence, and analyze its topological and computational properties. Finally, we validate our approach through experiments on synthetic datasets and color transfer tasks, demonstrating that SROT is better than both EOT and SOT in approximating exact OT. Additional experiments on gradient flows further highlight the advantages of SROT divergence.