We propose to learn non-convex regularizers with a prescribed upper bound on
their weak-convexity modulus. Such regularizers give rise to variational
denoisers that minimize a convex energy. They rely on few parameters (less than
15,000) and offer a signal-processing interpretation as they mimic handcrafted
sparsity-promoting regularizers. Through numerical experiments, we show that
such denoisers outperform convex-regularization methods as well as the popular
BM3D denoiser. Additionally, the learned regularizer can be deployed to solve
inverse problems with iterative schemes that provably converge. For both CT and
MRI reconstruction, the regularizer generalizes well and offers an excellent
tradeoff between performance, number of parameters, guarantees, and
interpretability when compared to other data-driven approaches.

我们提出通过学习具有预设上界的非凸正则化器。这些正则化器产生了能够最小化一个凸能量的变分去噪器。它们依赖较少的参数（少于 15000 个），并且具有信号处理的解释性，因为它们模仿手工稀疏促进的正则化器。通过数值实验，我们证明了这样的去噪器在性能上优于凸正则化方法以及流行的 BM3D 去噪器。此外，学习到的正则化器可以用于求解具有可证收敛性的迭代方案的逆问题。对于 CT 和 MRI 重建，该正则化器具有很好的泛化性能，并在性能、参数数量、保证性和可解释性等方面与其他数据驱动方法相比具有出色的平衡。

学习弱凸规则化器以收敛图像重构算法

Learning Weakly Convex Regularizers for Convergent Image-Reconstruction  Algorithms

This work studies the denoising of piecewise smooth graph signals that
exhibit inhomogeneous levels of smoothness over a graph, where the value at
each node can be vector-valued. We extend the graph trend filtering framework
to denoising vector-valued graph signals with a family of non-convex
regularizers, which exhibit superior recovery performance over existing convex
regularizers. Using an oracle inequality, we establish the statistical error
rates of first-order stationary points of the proposed non-convex method for
generic graphs. Furthermore, we present an ADMM-based algorithm to solve the
proposed method and establish its convergence. Numerical experiments are
conducted on both synthetic and real-world data for denoising, support
recovery, event detection, and semi-supervised classification.

本研究研究了分段平滑图信号的去噪问题，使用非凸正则化器对值为向量的图信号进行了扩展，提高了恢复性能。提出了基于 ADMM 的算法来解决问题和证明了其收敛性。

非凸惩罚的矢量图趋势滤波

Vector-Valued Graph Trend Filtering with Non-Convex Penalties

Dense image matching is a fundamental low-level problem in Computer Vision,
which has received tremendous attention from both discrete and continuous
optimization communities. The goal of this paper is to combine the advantages
of discrete and continuous optimization in a coherent framework. We devise a
model based on energy minimization, to be optimized by both discrete and
continuous algorithms in a consistent way. In the discrete setting, we propose
a novel optimization algorithm that can be massively parallelized. In the
continuous setting we tackle the problem of non-convex regularizers by a
formulation based on differences of convex functions. The resulting hybrid
discrete-continuous algorithm can be efficiently accelerated by modern GPUs and
we demonstrate its real-time performance for the applications of dense stereo
matching and optical flow.

本文提出一种基于能量最小化的模型，以离散和连续算法的一致方式进行优化，在离散设置中，我们提出了一种新的可大规模并行化的优化算法，在连续设置中，我们处理了非凸正则化的问题，所得到的混合离散连续算法可以通过现代 GPU 高效加速，并展示了在密集立体匹配和光流应用中的实时性能。