The quantization of a (probability) measure is replacing it by a sum of Dirac
masses that is close enough to it (in some metric space of probability
measures). Various methods exists to do so, but the situation of quantizing a
conditional law has been less explored. We propose a method, called DCMQ,
involving a Huber-energy kernel-based approach coupled with a deep neural
network architecture. The method is tested on several examples and obtains
promising results.

使用 Huber 能量核心方法与深度神经网络结构的 DCMQ 方法，在多个案例中取得了有前途的结果。

深度条件度量量化

Deep Conditional Measure Quantization

This paper introduces a numerical algorithm to compute the $L_2$ optimal
transport map between two measures $\mu$ and $\nu$, where $\mu$ derives from a
density $\rho$ defined as a piecewise linear function (supported by a
tetrahedral mesh), and where $\nu$ is a sum of Dirac masses.
I first give an elementary presentation of some known results on optimal
transport and then observe a relation with another problem (optimal sampling).
This relation gives simple arguments to study the objective functions that
characterize both problems.
I then propose a practical algorithm to compute the optimal transport map
between a piecewise linear density and a sum of Dirac masses in 3D. In this
semi-discrete setting, Aurenhammer et.al [\emph{8th Symposium on Computational
Geometry conf. proc.}, ACM (1992)] showed that the optimal transport map is
determined by the weights of a power diagram. The optimal weights are computed
by minimizing a convex objective function with a quasi-Newton method. To
evaluate the value and gradient of this objective function, I propose an
efficient and robust algorithm, that computes at each iteration the
intersection between a power diagram and the tetrahedral mesh that defines the
measure $\mu$.
The numerical algorithm is experimented and evaluated on several datasets,
with up to hundred thousands tetrahedra and one million Dirac masses.

本文介绍了一种计算两个测度之间 L2 最优传输图的数值算法，其中一个测度源自由分段线性函数（由四面体网格支持）定义的密度 rho，而另一个测度则是 Dirac 质量之和。作者提出了一个实用的算法来计算分段线性密度与 Dirac 质量总和之间的最优传输图。

$L_2$ 三维半离散最优输运的数值算法

A numerical algorithm for $L_2$ semi-discrete optimal transport in 3D

This paper studies sparse spikes deconvolution over the space of measures. We
focus our attention to the recovery properties of the support of the measure,
i.e. the location of the Dirac masses. For non-degenerate sums of Diracs, we
show that, when the signal-to-noise ratio is large enough, total variation
regularization (which is the natural extension of the L1 norm of vectors to the
setting of measures) recovers the exact same number of Diracs. We also show
that both the locations and the heights of these Diracs converge toward those
of the input measure when the noise drops to zero. The exact speed of
convergence is governed by a specific dual certificate, which can be computed
by solving a linear system. We draw connections between the support of the
recovered measure on a continuous domain and on a discretized grid. We show
that when the signal-to-noise level is large enough, the solution of the
discretized problem is supported on pairs of Diracs which are neighbors of the
Diracs of the input measure. This gives a precise description of the
convergence of the solution of the discretized problem toward the solution of
the continuous grid-free problem, as the grid size tends to zero.

本文研究测度空间中稀疏高斯脉冲信号的反卷积问题，提出总变差正则化方法用于测度恢复，当信噪比足够大时，可以精确恢复脉冲信号的支撑点位置。

稀疏脉冲解卷积的精确支持恢复

Exact Support Recovery for Sparse Spikes Deconvolution

Given a distribution $\rho$ on persistence diagrams and observations
$X_1,...X_n \stackrel{iid}{\sim} \rho$ we introduce an algorithm in this paper
that estimates a Fr\'echet mean from the set of diagrams $X_1,...X_n$. If the
underlying measure $\rho$ is a combination of Dirac masses $\rho = \frac{1}{m}
\sum_{i=1}^m \delta_{Z_i}$ then we prove the algorithm converges to a local
minimum and a law of large numbers result for a Fr\'echet mean computed by the
algorithm given observations drawn iid from $\rho$. We illustrate the
convergence of an empirical mean computed by the algorithm to a population mean
by simulations from Gaussian random fields.

该论文介绍了一种可以从任意分布的坚持图像计算出 Frechet mean 的算法，并证明了当分布是 Dirac masses 的组合时，算法收敛于局部最小值，并给出了一个取自该分布的观测序列的 Frechet mean 的大数定理结果。同时，作者还用高斯随机场的模拟表明算法计算出的经验 mean 会收敛到总体 mean。