The Alternating Direction Method of Multipliers (ADMM) has gained significant
attention across a broad spectrum of machine learning applications.
Incorporating the over-relaxation technique shows potential for enhancing the
convergence rate of ADMM. However, determining optimal algorithmic parameters,
including both the associated penalty and relaxation parameters, often relies
on empirical approaches tailored to specific problem domains and contextual
scenarios. Incorrect parameter selection can significantly hinder ADMM's
convergence rate. To address this challenge, in this paper we first propose a
general approach to optimize the value of penalty parameter, followed by a
novel closed-form formula to compute the optimal relaxation parameter in the
context of linear quadratic problems (LQPs). We then experimentally validate
our parameter selection methods through random instantiations and diverse
imaging applications, encompassing diffeomorphic image registration, image
deblurring, and MRI reconstruction.

在这篇论文中，我们首先提出了一种通用方法来优化惩罚参数的值，然后在线性二次问题（LQPs）的背景下，提出了一种新的封闭形式公式来计算最佳松弛参数。通过随机实例化和多样化的图像应用实验证明了我们参数选择方法的有效性，其中包括差分同胚图像配准、图像去模糊和 MRI 重建。

线性二次问题的 ADMM 及过松弛 ADMM 参数的优化

Optimizing ADMM and Over-Relaxed ADMM Parameters for Linear Quadratic  Problems

We investigate a framework for binary image denoising via restricted
Boltzmann machines (RBMs) that introduces a denoising objective in quadratic
unconstrained binary optimization (QUBO) form and is well-suited for quantum
annealing. The denoising objective is attained by balancing the distribution
learned by a trained RBM with a penalty term for derivations from the noisy
image. We derive the statistically optimal choice of the penalty parameter
assuming the target distribution has been well-approximated, and further
suggest an empirically supported modification to make the method robust to that
idealistic assumption. We also show under additional assumptions that the
denoised images attained by our method are, in expectation, strictly closer to
the noise-free images than the noisy images are. While we frame the model as an
image denoising model, it can be applied to any binary data. As the QUBO
formulation is well-suited for implementation on quantum annealers, we test the
model on a D-Wave Advantage machine, and also test on data too large for
current quantum annealers by approximating QUBO solutions through classical
heuristics.

研究了一种基于受限玻尔兹曼机的二元图像去噪框架，其引入了一个二次无约束二元优化（QUBO）形式的去噪目标，非常适用于量子退火。

基于 QUBO 和受限玻尔兹曼机的量子退火适用的图像去噪框架

An Image-Denoising Framework Fit for Quantum Annealing via QUBO and  Restricted Boltzmann Machines

The pattern of zero entries in the inverse covariance matrix of a
multivariate normal distribution corresponds to conditional independence
restrictions between variables. Covariance selection aims at estimating those
structural zeros from data. We show that neighborhood selection with the Lasso
is a computationally attractive alternative to standard covariance selection
for sparse high-dimensional graphs. Neighborhood selection estimates the
conditional independence restrictions separately for each node in the graph and
is hence equivalent to variable selection for Gaussian linear models. We show
that the proposed neighborhood selection scheme is consistent for sparse
high-dimensional graphs. Consistency hinges on the choice of the penalty
parameter. The oracle value for optimal prediction does not lead to a
consistent neighborhood estimate. Controlling instead the probability of
falsely joining some distinct connectivity components of the graph, consistent
estimation for sparse graphs is achieved (with exponential rates), even when
the number of variables grows as the number of observations raised to an
arbitrary power.

该文介绍了如何使用 lasso 算法来进行高维稀疏图的协方差无关估计，实现了变量选择，并控制了图中误连接不同的连通分量的概率，最终实现了稀疏图的一致性估计。