Simple approximative algorithms for free-support Wasserstein barycenters
研究了离散测度Wasserstein重心的简单近似框架，仅需N-1或N(N-1)/2次标准最优传输计算，误差上界分别为N和2，数值实验显示实际误差通常只有几个百分点。
Abstract Computing Wasserstein barycenters of discrete measures has recently attracted considerable attention due to its wide variety of applications in data science. In general, this problem is NP-hard, calling for practical approximative algorithms. In this paper, we analyze a well-known simple framework for approximating Wasserstein- $${\varvec{p}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:math> barycenters, where we mainly consider the most common case $${\varvec{p}}={\varvec{2}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mrow></mml:math> and $${\varvec{p}}={\varvec{1}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:mrow></mml:math> , which is not as well discussed. The framework produces sparse support solutions and shows good numerical results in the free-support setting. Depending on the desired level of accuracy, this requires only $${\varvec{N}}-{\varvec{1}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mo>-</mml:mo><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:mrow></mml:math> or $${\varvec{N(N}}-{\varvec{1)/2 }}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mi>N</mml:mi><mml:mo>(</mml:mo><mml:mi>N</mml:mi></mml:mrow><mml:mo>-</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>)</mml:mo><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mrow></mml:math> standard two-marginal optimal transport (OT) computations between the $${\varvec{N}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:math> input measures, respectively, which is fast, memory-efficient and easy to implement using any OT solver as a black box. What is more, these methods yield a relative error of at most $${\varvec{N}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:math> and $${\varvec{2}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:math> , respectively, for both $${\varvec{p}}={\varvec{ 1, 2}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mrow></mml:math> . We show that these bounds are practically sharp. In light of the hardness of the problem, it is not surprising that such guarantees cannot be close to optimality in general. Nevertheless, these error bounds usually turn out to be drastically lower for a given particular problem in practice and can be evaluated with almost no computational overhead, in particular without knowledge of the optimal solution. In our numerical experiments, this guaranteed errors of at most a few percent.