Reconstructing the detailed geometric structure of a face from a given image
is a key to many computer vision and graphics applications, such as motion
capture and reenactment. The reconstruction task is challenging as human faces
vary extensively when considering expressions, poses, textures, and intrinsic
geometries. While many approaches tackle this complexity by using additional
data to reconstruct the face of a single subject, extracting facial surface
from a single image remains a difficult problem. As a result, single-image
based methods can usually provide only a rough estimate of the facial geometry.
In contrast, we propose to leverage the power of convolutional neural networks
to produce a highly detailed face reconstruction from a single image. For this
purpose, we introduce an end-to-end CNN framework which derives the shape in a
coarse-to-fine fashion. The proposed architecture is composed of two main
blocks, a network that recovers the coarse facial geometry (CoarseNet),
followed by a CNN that refines the facial features of that geometry (FineNet).
The proposed networks are connected by a novel layer which renders a depth
image given a mesh in 3D. Unlike object recognition and detection problems,
there are no suitable datasets for training CNNs to perform face geometry
reconstruction. Therefore, our training regime begins with a supervised phase,
based on synthetic images, followed by an unsupervised phase that uses only
unconstrained facial images. The accuracy and robustness of the proposed model
is demonstrated by both qualitative and quantitative evaluation tests.

本文介绍了一种使用卷积神经网络构建面部几何细节的方法，包括由粗到细的两个网络模块和多阶段的训练阶段，可以从单张图像中提取面部表面。

从单张图像中学习详细的人脸重建

Learning Detailed Face Reconstruction from a Single Image

Exploiting geometric structure to improve the asymptotic complexity of
discrete assignment problems is a well-studied subject. In contrast, the
practical advantages of using geometry for such problems have not been
explored. We implement geometric variants of the Hopcroft--Karp algorithm for
bottleneck matching (based on previous work by Efrat el al.) and of the auction
algorithm by Bertsekas for Wasserstein distance computation. Both
implementations use k-d trees to replace a linear scan with a geometric
proximity query. Our interest in this problem stems from the desire to compute
distances between persistence diagrams, a problem that comes up frequently in
topological data analysis. We show that our geometric matching algorithms lead
to a substantial performance gain, both in running time and in memory
consumption, over their purely combinatorial counterparts. Moreover, our
implementation significantly outperforms the only other implementation
available for comparing persistence diagrams.

该研究介绍了如何使用几何结构来改善离散分配问题，并且实施了几何匹配算法，展示其明显的性能优势，主要在拓扑数据分析中应用。

几何有助于比较持久性图

Geometry Helps to Compare Persistence Diagrams

Discrete optimal transport solvers do not scale well on dense large problems
since they do not explicitly exploit the geometric structure of the cost
function. In analogy to continuous optimal transport we provide a framework to
verify global optimality of a discrete transport plan locally. This allows
construction of an algorithm to solve large dense problems by considering a
sequence of sparse problems instead. The algorithm lends itself to being
combined with a hierarchical multi-scale scheme. Any existing discrete solver
can be used as internal black-box.Several cost functions, including the noisy
squared Euclidean distance, are explicitly detailed. We observe a significant
reduction of run-time and memory requirements.

提供了一个与连续的最优传输类比的框架，以便于在本地验证离散传输计划的全局最优性，从而构建一种通过考虑一系列稀疏问题来解决大规模密集问题的算法，进而可以与分层多尺度方案相结合，明显减少了运行时间和内存要求。

一种用于稠密最优传输的稀疏多尺度算法

A Sparse Multi-Scale Algorithm for Dense Optimal Transport

We study the problem of detecting the presence of an underlying
high-dimensional geometric structure in a random graph. Under the null
hypothesis, the observed graph is a realization of an Erd\H{o}s-R\'enyi random
graph $G(n,p)$. Under the alternative, the graph is generated from the
$G(n,p,d)$ model, where each vertex corresponds to a latent independent random
vector uniformly distributed on the sphere $\mathbb{S}^{d-1}$, and two vertices
are connected if the corresponding latent vectors are close enough. In the
dense regime (i.e., $p$ is a constant), we propose a near-optimal and
computationally efficient testing procedure based on a new quantity which we
call signed triangles. The proof of the detection lower bound is based on a new
bound on the total variation distance between a Wishart matrix and an
appropriately normalized GOE matrix. In the sparse regime, we make a conjecture
for the optimal detection boundary. We conclude the paper with some preliminary
steps on the problem of estimating the dimension in $G(n,p,d)$.

研究了如何检测随机图中的高维几何结构问题，提出了一种基于新量的近乎最优和高效的测试方法，并针对稠密和稀疏情况提出了检测界和估计维度的猜想。