  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.
