Video coding is a mathematical optimization problem of rate and distortion
essentially. To solve this complex optimization problem, two popular video
coding frameworks have been developed: block-based hybrid video coding and
end-to-end learned video coding. If we rethink video coding from the
perspective of optimization, we find that the existing two frameworks represent
two directions of optimization solutions. Block-based hybrid coding represents
the discrete optimization solution because those irrelevant coding modes are
discrete in mathematics. It searches for the best one among multiple starting
points (i.e. modes). However, the search is not efficient enough. On the other
hand, end-to-end learned coding represents the continuous optimization solution
because the gradient descent is based on a continuous function. It optimizes a
group of model parameters efficiently by the numerical algorithm. However,
limited by only one starting point, it is easy to fall into the local optimum.
To better solve the optimization problem, we propose to regard video coding as
a hybrid of the discrete and continuous optimization problem, and use both
search and numerical algorithm to solve it. Our idea is to provide multiple
discrete starting points in the global space and optimize the local optimum
around each point by numerical algorithm efficiently. Finally, we search for
the global optimum among those local optimums. Guided by the hybrid
optimization idea, we design a hybrid optimization video coding framework,
which is built on continuous deep networks entirely and also contains some
discrete modes. We conduct a comprehensive set of experiments. Compared to the
continuous optimization framework, our method outperforms pure learned video
coding methods. Meanwhile, compared to the discrete optimization framework, our
method achieves comparable performance to HEVC reference software HM16.10 in
PSNR.

提出用混合的离散和连续优化方法来解决视频编码问题，并设计了一个混合优化视频编码框架，实验结果表明相比于单纯连续或者离散优化方法，该方法具有更好的表现。

走向混合优化视频编码

Towards Hybrid-Optimization Video Coding

We consider the problem of conditional independence testing of $X$ and $Y$
given $Z$ where $X,Y$ and $Z$ are three real random variables and $Z$ is
continuous. We focus on two main cases - when $X$ and $Y$ are both discrete,
and when $X$ and $Y$ are both continuous. In view of recent results on
conditional independence testing (Shah and Peters, 2018), one cannot hope to
design non-trivial tests, which control the type I error for all absolutely
continuous conditionally independent distributions, while still ensuring power
against interesting alternatives. Consequently, we identify various, natural
smoothness assumptions on the conditional distributions of $X,Y|Z=z$ as $z$
varies in the support of $Z$, and study the hardness of conditional
independence testing under these smoothness assumptions. We derive matching
lower and upper bounds on the critical radius of separation between the null
and alternative hypotheses in the total variation metric. The tests we consider
are easily implementable and rely on binning the support of the continuous
variable $Z$. To complement these results, we provide a new proof of the
hardness result of Shah and Peters.

研究了在条件分布连续的情况下，基于条件独立性测试 $X$ 和 $Y$ 关于 $Z$，$X$ 和 $Y$ 均为离散或连续变量的问题。考虑到条件独立性测试的研究，对于所有绝对连续的条件分布都无法设计非平凡的测试以控制所有的一类错误并确保在有趣的替代情况下仍然具有功效，因此在条件分布的各个自然光滑性假设下研究了条件独立性测试的难度，并以总变差度量指标的临界分离半径为下界和上界。最后，提供了一种新的证明方法以证明 Shah 与 Peters 的结论。

极小极大化最优条件独立性测试

Minimax Optimal Conditional Independence Testing

There are competing schools of thought about the question of whether
spacetime is fundamentally either continuous or discrete. Here, we consider the
possibility that spacetime could be simultaneously continuous and discrete, in
the same mathematical way that information can be simultaneously continuous and
discrete. The equivalence of continuous and discrete information, which is of
key importance in information theory, is established by Shannon sampling
theory: of any bandlimited signal it suffices to record discrete samples to be
able to perfectly reconstruct it everywhere, if the samples are taken at a rate
of at least twice the bandlimit. It is known that physical fields on generic
curved spaces obey a sampling theorem if they possess an ultraviolet cutoff.
Most recently, methods of spectral geometry have been employed to show that
also the very shape of a curved space (i.e., of a Riemannian manifold) can be
discretely sampled and then reconstructed up to the cutoff scale. Here, we
develop these results further, and we here also consider the generalization to
curved spacetimes, i.e., to Lorentzian manifolds.

在这篇论文中，我们讨论了时空基本上是连续的还是离散的问题，并考虑到时空可能同时是连续和离散的，与信息同时连续和离散的数学方式相同，并且研究了采样定理在物理场上的应用，发现一些很有意思的结果。