Variational Autoencoders (VAEs) represent the given data in a low-dimensional
latent space, which is generally assumed to be Euclidean. This assumption
naturally leads to the common choice of a standard Gaussian prior over
continuous latent variables. Recent work has, however, shown that this prior
has a detrimental effect on model capacity, leading to subpar performance. We
propose that the Euclidean assumption lies at the heart of this failure mode.
To counter this, we assume a Riemannian structure over the latent space, which
constitutes a more principled geometric view of the latent codes, and replace
the standard Gaussian prior with a Riemannian Brownian motion prior. We propose
an efficient inference scheme that does not rely on the unknown normalizing
factor of this prior. Finally, we demonstrate that this prior significantly
increases model capacity using only one additional scalar parameter.

本文提出了在潜空间上采用 Riemannian 结构和 Riemannian Brownian motion 先验来代替传统高斯先验，从而提高 Variational Autoencoder 的模型能力。

带黎曼布朗运动先验的变分自编码器

Variational Autoencoders with Riemannian Brownian Motion Priors

Optimal transportation distances are valuable for comparing and analyzing
probability distributions, but larger-scale computational techniques for the
theoretically favorable quadratic case are limited to smooth domains or
regularized approximations. Motivated by fluid flow-based transportation on
$\mathbb{R}^n$, however, this paper introduces an alternative definition of
optimal transportation between distributions over graph vertices. This new
distance still satisfies the triangle inequality but has better scaling and a
connection to continuous theories of transportation. It is constructed by
adapting a Riemannian structure over probability distributions to the graph
case, providing transportation distances as shortest-paths in probability
space. After defining and analyzing theoretical properties of our new distance,
we provide a time discretization as well as experiments verifying its
effectiveness.

该论文提出了一种基于图顶点的概率分布最优传输方法，它不仅满足三角不等式，而且具有更好的可扩展性和与连续传输理论的联系，同时提供了一个适用于该方法的时间离散化方法并验证了其有效性。

连续图传输距离

Continuous-Flow Graph Transportation Distances

The paper addresses the problem of low-rank trace norm minimization. We
propose an algorithm that alternates between fixed-rank optimization and
rank-one updates. The fixed-rank optimization is characterized by an efficient
factorization that makes the trace norm differentiable in the search space and
the computation of duality gap numerically tractable. The search space is
nonlinear but is equipped with a particular Riemannian structure that leads to
efficient computations. We present a second-order trust-region algorithm with a
guaranteed quadratic rate of convergence. Overall, the proposed optimization
scheme converges super-linearly to the global solution while maintaining
complexity that is linear in the number of rows and columns of the matrix. To
compute a set of solutions efficiently for a grid of regularization parameters
we propose a predictor-corrector approach that outperforms the naive
warm-restart approach on the fixed-rank quotient manifold. The performance of
the proposed algorithm is illustrated on problems of low-rank matrix completion
and multivariate linear regression.

本文提出一种通过交替固定秩优化与秩一更新来解决低秩矩阵迹范数最小化问题的算法，针对具有理曼结构的非线性搜索空间，利用有效的因子分解实现了迹范数在搜索空间的可微以及对偶间隙的数值计算可行，提出了一个拥有保证二次收敛速度的二阶信任域算法，并以低秩矩阵完成和多元线性回归问题为例说明了该算法的性能。