Learning a nonparametric system of ordinary differential equations (ODEs)
from $n$ trajectory snapshots in a $d$-dimensional state space requires
learning $d$ functions of $d$ variables. Explicit formulations scale
quadratically in $d$ unless additional knowledge about system properties, such
as sparsity and symmetries, is available. In this work, we propose a linear
approach to learning using the implicit formulation provided by vector-valued
Reproducing Kernel Hilbert Spaces. By rewriting the ODEs in a weaker integral
form, which we subsequently minimize, we derive our learning algorithm. The
minimization problem's solution for the vector field relies on multivariate
occupation kernel functions associated with the solution trajectories. We
validate our approach through experiments on highly nonlinear simulated and
real data, where $d$ may exceed 100. We further demonstrate the versatility of
the proposed method by learning a nonparametric first order quasilinear partial
differential equation.

提出了一种使用向量值再生内积核空间提供的隐式公式的线性方法来学习非参数 ODE 系统的学习算法，其解决方案依赖于与解决方案轨迹相关的多元占用核函数，这在高度非线性的模拟和实际数据上进行了验证，并通过学习非参数一阶拟线性偏微分方程的方法证明了其多样性。

通过多变量占据核函数学习高维非参数微分方程

Learning High-Dimensional Nonparametric Differential Equations via  Multivariate Occupation Kernel Functions

Deep neural networks have dramatically advanced the state of the art for many
areas of machine learning. Recently they have been shown to have a remarkable
ability to generate highly complex visual artifacts such as images and text
rather than simply recognize them.
In this work we use neural networks to effectively invert low-dimensional
face embeddings while producing realistically looking consistent images. Our
contribution is twofold, first we show that a gradient ascent style approaches
can be used to reproduce consistent images, with a help of a guiding image.
Second, we demonstrate that we can train a separate neural network to
effectively solve the minimization problem in one pass, and generate images in
real-time. We then evaluate the loss imposed by using a neural network instead
of the gradient descent by comparing the final values of the minimized loss
function.

这篇论文利用深度神经网络逆向低维人脸嵌入并生成高质量真实的图像，探索了两种方法，通过引导图像使用梯度上升方法生成一致性良好的图像和通过训练单独的神经网络一次性解决最小化问题来实时生成图像，并对神经网络与梯度下降进行对比实验。

使用卷积神经网络反演面部嵌入

Inverting face embeddings with convolutional neural networks

In this paper we study an algorithm for solving a minimization problem
composed of a differentiable (possibly non-convex) and a convex (possibly
non-differentiable) function. The algorithm iPiano combines forward-backward
splitting with an inertial force. It can be seen as a non-smooth split version
of the Heavy-ball method from Polyak. A rigorous analysis of the algorithm for
the proposed class of problems yields global convergence of the function values
and the arguments. This makes the algorithm robust for usage on non-convex
problems. The convergence result is obtained based on the \KL inequality. This
is a very weak restriction, which was used to prove convergence for several
other gradient methods. First, an abstract convergence theorem for a generic
algorithm is proved, and, then iPiano is shown to satisfy the requirements of
this theorem. Furthermore, a convergence rate is established for the general
problem class. We demonstrate iPiano on computer vision problems: image
denoising with learned priors and diffusion based image compression.

本文研究了一种算法，用于解决由可微分（可能非凸）函数和凸（可能非可微分）函数组成的最小化问题。该算法 iPIANO 结合了前向 - 后向拆分和惯性力，是 Polyak 的不光滑分裂版本。对于所提出的问题类别，对算法进行了严格分析，证明了函数值和参数的全局收敛性，即使是非凸问题也具有强鲁棒性。