On A Nonlinear Generalization of Sparse Coding and Dictionary Learning
On A Nonlinear Generalization of Sparse Coding and Dictionary Learning
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发表时间：
2013-06
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通讯作者：
J. Ho;Yuchen Xie;B. Vemuri
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作者：
J. Ho;Yuchen Xie;B. Vemuri
现有的字典学习算法都是基于数据是欧氏向量空间中的向量的假设，并且字典是使用欧氏向量空间结构及其欧氏L2度量从训练数据中学习的。然而，在许多应用中，特征和数据通常源自不支持全局线性（向量空间）结构的黎曼流形。此外，现有字典学习算法的外在观点变得不适合建模和纳入流形的内在几何形状，这对应用程序来说是潜在的重要和关键。本文提出了一种新的框架，稀疏编码和字典学习的数据在黎曼流形上，它表明，现有的稀疏编码和字典学习方法可以被认为是特殊的（欧几里德）的情况下，这里提出的更一般的框架。我们表明，字典和稀疏编码可以有效地计算几个重要类的黎曼流形，我们验证了所提出的方法，使用两个著名的分类问题，在计算机视觉和医学成像分析。
Existing dictionary learning algorithms are based on the assumption that the data are vectors in an Euclidean vector space ℝ d , and the dictionary is learned from the training data using the vector space structure of ℝ d and its Euclidean L2-metric. However, in many applications, features and data often originated from a Riemannian manifold that does not support a global linear (vector space) structure. Furthermore, the extrinsic viewpoint of existing dictionary learning algorithms becomes inappropriate for modeling and incorporating the intrinsic geometry of the manifold that is potentially important and critical to the application. This paper proposes a novel framework for sparse coding and dictionary learning for data on a Riemannian manifold, and it shows that the existing sparse coding and dictionary learning methods can be considered as special (Euclidean) cases of the more general framework proposed here. We show that both the dictionary and sparse coding can be effectively computed for several important classes of Riemannian manifolds, and we validate the proposed method using two well-known classification problems in computer vision and medical imaging analysis.