论文标题
不平衡的Sobolev下降
Unbalanced Sobolev Descent
论文作者
论文摘要
我们引入了不平衡的Sobolev下降（USD），这是一种用于将高维源分布传输到不一定具有相同质量的目标分布的粒子下降算法。我们定义了分布之间的Sobolev-fisher差异，并表明它与对流反应传输方程以及分布之间的Wasserstein-Fisher-Rao指标有关。 USD沿Sobolev-Fisher差异的证人功能的梯度流动粒子（对流步骤），并相对于此证人功能（反应步骤）重新质量颗粒的质量。可以将反应步骤视为颗粒的出生死亡过程，其生长速率与证人功能成正比。当Sobolev-fisher证人功能在繁殖的内核希尔伯特空间（RKHS）中估计时，在轻度假设下，我们表明，USD在最大平均差异（MMD）意义上以渐近（在无限粒子的极限（在无限粒子的极限上）收敛到目标分布。然后，我们提供了两种方法，以通过神经网络估算Sobolev-Fisher证人，从而产生了两种神经USD算法。第一个在重量上以镜下降的形式实现了反应步骤，而第二个则通过颗粒的出生死亡过程实现了反应步骤。我们在合成示例上显示，与以前的粒子下降算法相比，USD传输具有或不得更快的质量保护的分布，并最终证明了其用于分子生物学分析的使用，其中我们的方法自然适合基于单细胞RNA测序谱。代码可在https://github.com/ibm/usd上找到。
We introduce Unbalanced Sobolev Descent (USD), a particle descent algorithm for transporting a high dimensional source distribution to a target distribution that does not necessarily have the same mass. We define the Sobolev-Fisher discrepancy between distributions and show that it relates to advection-reaction transport equations and the Wasserstein-Fisher-Rao metric between distributions. USD transports particles along gradient flows of the witness function of the Sobolev-Fisher discrepancy (advection step) and reweighs the mass of particles with respect to this witness function (reaction step). The reaction step can be thought of as a birth-death process of the particles with rate of growth proportional to the witness function. When the Sobolev-Fisher witness function is estimated in a Reproducing Kernel Hilbert Space (RKHS), under mild assumptions we show that USD converges asymptotically (in the limit of infinite particles) to the target distribution in the Maximum Mean Discrepancy (MMD) sense. We then give two methods to estimate the Sobolev-Fisher witness with neural networks, resulting in two Neural USD algorithms. The first one implements the reaction step with mirror descent on the weights, while the second implements it through a birth-death process of particles. We show on synthetic examples that USD transports distributions with or without conservation of mass faster than previous particle descent algorithms, and finally demonstrate its use for molecular biology analyses where our method is naturally suited to match developmental stages of populations of differentiating cells based on their single-cell RNA sequencing profile. Code is available at https://github.com/ibm/usd .
