There is a growing gap between the impressive results of deep image
generative models and classical algorithms that offer theoretical guarantees.
The former suffer from mode collapse or memorization issues, limiting their
application to scientific data. The latter require restrictive assumptions such
as log-concavity to escape the curse of dimensionality. We partially bridge
this gap by introducing conditionally strongly log-concave (CSLC) models, which
factorize the data distribution into a product of conditional probability
distributions that are strongly log-concave. This factorization is obtained
with orthogonal projectors adapted to the data distribution. It leads to
efficient parameter estimation and sampling algorithms, with theoretical
guarantees, although the data distribution is not globally log-concave. We show
that several challenging multiscale processes are conditionally log-concave
using wavelet packet orthogonal projectors. Numerical results are shown for
physical fields such as the $\varphi^4$ model and weak lensing convergence maps
with higher resolution than in previous works.

介绍了条件强对数凹模型的概念，该模型将数据分布因子化为强对数凹条件概率分布的乘积，并使用适应于数据分布的正交投影器来实现因子化，从而实现了有效的参数估计和抽样算法，并在物理字段等多个领域展示了该模型的实用性。

有条件强对数凹生成模型

Conditionally Strongly Log-Concave Generative Models

This article is concerned with the numerical solution of subspace
optimization problems, consisting of minimizing a smooth functional over the
set of orthogonal projectors of fixed rank. Such problems are encountered in
particular in electronic structure calculation (Hartree-Fock and Kohn-Sham
Density Functional Theory -DFT- models). We compare from a numerical analysis
perspective two simple representatives, the damped self-consistent field (SCF)
iterations and the gradient descent algorithm, of the two classes of methods
competing in the field: SCF and direct minimization methods. We derive
asymptotic rates of convergence for these algorithms and analyze their
dependence on the spectral gap and other properties of the problem. Our
theoretical results are complemented by numerical simulations on a variety of
examples, from toy models with tunable parameters to realistic Kohn-Sham
computations. We also provide an example of chaotic behavior of the simple SCF
iterations for a nonquadratic functional.

通过数值分析对自封闭场迭代和梯度下降算法两种代表方法进行了比较，研究了固定秩的正交投影器子空间优化问题的渐近收敛速率，并在理论结果和数值模拟示例上完善了该研究。

直接最小化与自洽迭代的收敛分析

Convergence analysis of direct minimization and self-consistent  iterations

Consider a dataset of vector-valued observations that consists of noisy
inliers, which are explained well by a low-dimensional subspace, along with
some number of outliers. This work describes a convex optimization problem,
called REAPER, that can reliably fit a low-dimensional model to this type of
data. This approach parameterizes linear subspaces using orthogonal projectors,
and it uses a relaxation of the set of orthogonal projectors to reach the
convex formulation. The paper provides an efficient algorithm for solving the
REAPER problem, and it documents numerical experiments which confirm that
REAPER can dependably find linear structure in synthetic and natural data. In
addition, when the inliers lie near a low-dimensional subspace, there is a
rigorous theory that describes when REAPER can approximate this subspace.

该论文提出了一种名为 REAPER 的凸优化问题来处理向量值观测数据中的低维子空间和离群值，利用正交投影子空间参数化线性子空间，并使用正交投影子空间的松弛达到凸优化的形式，并提供了 REAPER 的有效算法和数值实验结果的文献。