Abstract
The sparse modeling is an evident manifestation capturing the parsimony principle just described, and sparse models are widespread in statistics, physics, information sciences, neuroscience, computational mathematics, and so on. In statistics the many applications of sparse modeling span regression, classification tasks, graphical model selection, sparse M-estimators and sparse dimensionality reduction. It is also particularly effective in many statistical and machine learning areas where the primary goal is to discover predictive patterns from data which would enhance our understanding and control of underlying physical, biological, and other natural processes, beyond just building accurate outcome black-box predictors. Common examples include selecting biomarkers in biological procedures, finding relevant brain activity locations which are predictive about brain states and processes based on fMRI data, and identifying network bottlenecks best explaining end-to-end performance. Moreover, the research and applications of efficient recovery of high-dimensional sparse signals from a relatively small number of observations, which is the main focus of compressed sensing or compressive sensing, have rapidly grown and became an extremely intense area of study beyond classical signal processing. Likewise interestingly, sparse modeling is directly related to various artificial vision tasks, such as image denoising, segmentation, restoration and superresolution, object or face detection and recognition in visual scenes, and action recognition. In this manuscript, we provide a brief introduction of the basic theory underlying sparse representation and compressive sensing, and then discuss some methods for recovering sparse solutions to optimization problems in effective way, together with some applications of sparse recovery in a machine learning problem known as sparse dictionary learning.
Abstract (translated)
稀疏建模是一种明显的表现,捕捉到我刚才描述的简洁性原则,稀疏模型在统计、物理、信息科学、神经科学、计算数学等领域广泛应用。在统计中,稀疏建模的许多应用涵盖了回归、分类任务、图形模型选择、稀疏高斯估计和稀疏维度减少。它还在许多统计和机器学习领域中特别有效,其主要目标是从数据中发现预测模式,这将增强我们对基础物理、生物和自然过程的理解和控制,超越了仅仅建立准确的黑盒预测器。常见的例子包括在生物学过程中选择生物标记物、基于FMRI数据的 Brain 活动位置找到相关的脑活动区域、并确定网络瓶颈的最佳解释,最有效地解释整体性能。此外,研究和应用从相对少量的观察数据中高效恢复高维稀疏信号的研究和应用,这是压缩感知或压缩感知的主要关注点,已经迅速增长并成为 classical 信号处理之外极为强烈的研究领域。类似地,稀疏建模直接与各种人工视觉任务相关,例如图像去噪、分割、恢复和超分辨率、视觉场景中的物体或面部检测和识别,以及动作识别。在本文中,我们简要介绍了稀疏表示和压缩感知的基础理论,然后讨论了如何有效地恢复优化问题的稀疏解决方案,以及在稀疏字典学习机器学习问题中的稀疏恢复应用。
URL
https://arxiv.org/abs/2308.13960
