论文标题
通过通用凸函数正规的最佳传输问题：几何和算法方法
Optimal transport problems regularized by generic convex functions: A geometric and algorithmic approach
论文作者
论文摘要
In order to circumvent the difficulties in solving numerically the discrete optimal transport problem, in which one minimizes the linear target function $P\mapsto\langle C,P\rangle:=\sum_{i,j}C_{ij}P_{ij}$, Cuturi introduced a variant of the problem in which the target function is altered by a convex one $φ（p）= \ langle c，p \ rangle-λ\ mathcal {h}（p）$，其中$ \ mathcal {h} $是香农熵，$λ$是一个正常数。我们在这里将其公式化为$φ（p）= \ langle c，p \ rangle+λf（p）$的目标函数，其中$ f $是通用的严格凸出光滑函数。我们还提出了一种迭代方法来找到数值解决方案，并澄清提出的方法在$ f（p）= \ frac {1} {2} {2} {2} \ | p \ |^2 $时特别有效。
In order to circumvent the difficulties in solving numerically the discrete optimal transport problem, in which one minimizes the linear target function $P\mapsto\langle C,P\rangle:=\sum_{i,j}C_{ij}P_{ij}$, Cuturi introduced a variant of the problem in which the target function is altered by a convex one $Φ(P)=\langle C,P\rangle-λ\mathcal{H}(P)$, where $\mathcal{H}$ is the Shannon entropy and $λ$ is a positive constant. We herein generalize their formulation to a target function of the form $Φ(P)=\langle C,P\rangle+λf(P)$, where $f$ is a generic strictly convex smooth function. We also propose an iterative method for finding a numerical solution, and clarify that the proposed method is particularly efficient when $f(P)=\frac{1}{2}\|P\|^2$.
