Demixing problems in many areas such as hyperspectral imaging and
differential optical absorption spectroscopy (DOAS) often require finding
sparse nonnegative linear combinations of dictionary elements that match
observed data. We show how aspects of these problems, such as misalignment of
DOAS references and uncertainty in hyperspectral endmembers, can be modeled by
expanding the dictionary with grouped elements and imposing a structured
sparsity assumption that the combinations within each group should be sparse or
even 1-sparse. If the dictionary is highly coherent, it is difficult to obtain
good solutions using convex or greedy methods, such as non-negative least
squares (NNLS) or orthogonal matching pursuit. We use penalties related to the
Hoyer measure, which is the ratio of the $l_1$ and $l_2$ norms, as sparsity
penalties to be added to the objective in NNLS-type models. For solving the
resulting nonconvex models, we propose a scaled gradient projection algorithm
that requires solving a sequence of strongly convex quadratic programs. We
discuss its close connections to convex splitting methods and difference of
convex programming. We also present promising numerical results for example
DOAS analysis and hyperspectral demixing problems.

利用结构稀疏性假设将稀疏非负线性组合与组合间的稀疏性拓展到字典组合元素的分组上，采用 Hoyer 惩罚与缩放梯度投影算法解决了字典高度相干限制带来的问题，解决了 DOAS 分析和高光谱混合问题。

一种用于非负最小二乘问题中寻找结构稀疏解的方法及其应用

A Method for Finding Structured Sparse Solutions to Non-negative Least  Squares Problems with Applications

Super-resolution methods form high-resolution images from low-resolution
images. In this paper, we develop a new Bayesian nonparametric model for
super-resolution. Our method uses a beta-Bernoulli process to learn a set of
recurring visual patterns, called dictionary elements, from the data. Because
it is nonparametric, the number of elements found is also determined from the
data. We test the results on both benchmark and natural images, comparing with
several other models from the research literature. We perform large-scale human
evaluation experiments to assess the visual quality of the results. In a first
implementation, we use Gibbs sampling to approximate the posterior. However,
this algorithm is not feasible for large-scale data. To circumvent this, we
then develop an online variational Bayes (VB) algorithm. This algorithm finds
high quality dictionaries in a fraction of the time needed by the Gibbs
sampler.

本文提出了一种基于贝叶斯非参数模型的超分辨率方法，该方法使用 Beta-Bernoulli 过程从数据中学习一组可重复的可视化模式（称为字典元素），并用 Gibbs 采样近似后验进行实现，但是该算法不适用于大规模数据，因此，我们发展了一种在线变分贝叶斯算法，该算法在时间上比 Gibbs 采样器所需时间的一小部分内找到高质量的字典。