The universal approximation property (UAP) of neural networks is a
fundamental characteristic of deep learning. It is widely recognized that a
composition of linear functions and non-linear functions, such as the rectified
linear unit (ReLU) activation function, can approximate continuous functions on
compact domains. In this paper, we extend this efficacy to the scenario of
dynamical systems with controls. We prove that the control family
$\mathcal{F}_1 = \mathcal{F}_0 \cup \{ \text{ReLU}(\cdot)\} $ is enough to
generate flow maps that can uniformly approximate diffeomorphisms of
$\mathbb{R}^d$ on any compact domain, where $\mathcal{F}_0 = \{x \mapsto Ax+b:
A\in \mathbb{R}^{d\times d}, b \in \mathbb{R}^d\}$ is the set of linear maps
and the dimension $d\ge2$. Since $\mathcal{F}_1$ contains only one nonlinear
function and $\mathcal{F}_0$ does not hold the UAP, we call $\mathcal{F}_1$ a
minimal control family for UAP. Based on this, some sufficient conditions, such
as the affine invariance, on the control family are established and discussed.
Our result reveals an underlying connection between the approximation power of
neural networks and control systems.

通过证明参数矩阵、线性向量和 ReLU 激活函数这个控制组合能够统一逼近定义域在任意紧致区域的微分同胚，我们揭示了神经网络的逼近能力与控制系统之间的关联。

一种用于普适逼近的最简控制动态系统家族

A Minimal Control Family of Dynamical Syetem for Universal Approximation

Spatio-temporal coherency is a major challenge in synthesizing high quality
videos, particularly in synthesizing human videos that contain rich global and
local deformations. To resolve this challenge, previous approaches have
resorted to different features in the generation process aimed at representing
appearance and motion. However, in the absence of strict mechanisms to
guarantee such disentanglement, a separation of motion from appearance has
remained challenging, resulting in spatial distortions and temporal jittering
that break the spatio-temporal coherency. Motivated by this, we here propose
LEO, a novel framework for human video synthesis, placing emphasis on
spatio-temporal coherency. Our key idea is to represent motion as a sequence of
flow maps in the generation process, which inherently isolate motion from
appearance. We implement this idea via a flow-based image animator and a Latent
Motion Diffusion Model (LMDM). The former bridges a space of motion codes with
the space of flow maps, and synthesizes video frames in a warp-and-inpaint
manner. LMDM learns to capture motion prior in the training data by
synthesizing sequences of motion codes. Extensive quantitative and qualitative
analysis suggests that LEO significantly improves coherent synthesis of human
videos over previous methods on the datasets TaichiHD, FaceForensics and
CelebV-HQ. In addition, the effective disentanglement of appearance and motion
in LEO allows for two additional tasks, namely infinite-length human video
synthesis, as well as content-preserving video editing.

提出了一种名为 LEO 的新颖方法，通过将运动表示为一系列流场图，在生成过程中固有地分离运动和外貌，从而强调时空一致性，并取得了比以前方法更好的人类视频合成效果。

LEO：人体视频合成的生成式潜空间图像动画器

LEO: Generative Latent Image Animator for Human Video Synthesis

Generating robust and reliable correspondences across images is a fundamental
task for a diversity of applications. To capture context at both global and
local granularity, we propose ASpanFormer, a Transformer-based detector-free
matcher that is built on hierarchical attention structure, adopting a novel
attention operation which is capable of adjusting attention span in a
self-adaptive manner. To achieve this goal, first, flow maps are regressed in
each cross attention phase to locate the center of search region. Next, a
sampling grid is generated around the center, whose size, instead of being
empirically configured as fixed, is adaptively computed from a pixel
uncertainty estimated along with the flow map. Finally, attention is computed
across two images within derived regions, referred to as attention span. By
these means, we are able to not only maintain long-range dependencies, but also
enable fine-grained attention among pixels of high relevance that compensates
essential locality and piece-wise smoothness in matching tasks.
State-of-the-art accuracy on a wide range of evaluation benchmarks validates
the strong matching capability of our method.

提出了一种基于 Transformer 的 ASpanFormer 检测器自由匹配器，采用分层注意力结构，能够在全局和本地粒度上捕捉上下文，并具有自适应调整注意跨度的能力，以此来实现在匹配任务中对高相关性像素的细粒度关注。