Optimal transport (OT) theory focuses, among all maps $T:\mathbb{R}^d\rightarrow \mathbb{R}^d$ that can morph a probability measure onto another, on those that are the ``thriftiest'', i.e. such that the averaged cost $c(x, T(x))$ between $x$ and its image $T(x)$ be as small as possible. Many computational approaches have been proposed to estimate such Monge maps when $c$ is the $\ell_2^2$ distance, e.g., using entropic maps [Pooladian'22], or neural networks [Makkuva'20, Korotin'20]. We propose a new model for transport maps, built on a family of translation invariant costs $c(x, y):=h(x-y)$, where $h:=\tfrac{1}{2}\|\cdot\|_2^2+\tau$ and $\tau$ is a regularizer. We propose a generalization of the entropic map suitable for $h$, and highlight a surprising link tying it with the Bregman centroids of the divergence $D_h$ generated by $h$, and the proximal operator of $\tau$. We show that choosing a sparsity-inducing norm for $\tau$ results in maps that apply Occam's razor to transport, in the sense that the displacement vectors $\Delta(x):= T(x)-x$ they induce are sparse, with a sparsity pattern that varies depending on $x$. We showcase the ability of our method to estimate meaningful OT maps for high-dimensional single-cell transcription data, in the $34000$-$d$ space of gene counts for cells, without using dimensionality reduction, thus retaining the ability to interpret all displacements at the gene level.
翻译:最佳运输( OT) 理论在所有地图中 $T:\mathb{R ⁇ d\rightrow \ mathbb{R ⁇ d$, 可以将概率度量转换到另一个, 也就是“ lefitest ”, 也就是说, 平均成本( x, T(x) ) 美元与其图像 $T(x) 尽可能小。 许多计算方法都建议估算这样的蒙古地图, $C$ 是 $@ell_2x=2$的距离, 例如, 使用 entropic 地图[poolada' 22] 或 神经网络 [Makkuva'20, Korotin'20] 。 我们提出一个新的运输地图模型, 建于一个翻译成本( x) $( y) 的组合 :\\\\\\\\\\\\\\\\\\\\\\\\\\\\ xxxxxxxx 美元 。 许多计算方法 $, 其中, =remodemodeal=x max lialal=x 美元, lixxx 数据, 我们提出一个总数据。