Abstract
In the past decades, relevant sparse representation models and their corresponding dictionary learning algorithms have been explored extensively as they could be applied in various fields. However, most of them are focusing the linear model and nonlinear one is still less touched, although there are plenty of nonlinear scenarios in real applications. To further address this kind of the unmet challenge, in this work we mainly focus the following two works (i) propose a kernel transformation based method directly transforming the nonlinear analysis problem into a linear one, which is exactly the standard sparse analysis form but implies all nonlinear information of the original problem; (ii) present a nonlinear dictionary learning algorithm by leveraging the kernel trick and the KSVD-like manner, which has its root in analysis sparse model rather than synthesis model. Then, the proposed methods are employed to address the classification problem. Benchmark experimental results on three well-known datasets show that the proposed algorithm in (ii) outperforms some related linear algorithms and other existing nonlinear dictionary learning algorithms. Moreover, when the data is interfered by noise or some pixels are missing in the data, the algorithm is also effective, which proves its theoretical advantages owing to analysis sparse model's merits of the equality of all atoms and the much smaller dimensionality for signal representation to some extent. And the classification accuracy of the proposed method in (i) is slightly lower than that of (ii), but better than that of other state of art methods.
Highlights
S PARSE and redundant signal representation is a research hotspot in recent years
KERNEL SPARSE REPRESENTATION TRANSFORMATION METHOD we will propose the first work – a kernel transformation method (KTM) that can directly transform a nonlinear problem based on an analysis model into a linear one, but which implies all nonlinear information of the original problem
We find that the analysis kernel KSVD algorithm can achieve the 96.8% classification accuracy, which shows that our method is competitive among all compared methods
Summary
S PARSE and redundant signal representation is a research hotspot in recent years. Many scholars have developed strong interests in many domains [1], [2]. B. DICTIONARY LEARNING ALGORITHM FOR ANALYSIS SPARSE MODEL The basic optimization problem to solve the analysis sparse model is as follows min x,Ω y−x 2+. Note that when both the synthesis sparse model and the analysis sparse model depart, the analysis model becomes more interesting and powerful This case of analysis dictionary training is a challenging problem [8].In the past few years, model (5), as one kind of linear case, has been mainly studied [8], [30], [31]. In this work, we creatively combine the nonlinear kernel approach with the analysis sparse model, and propose two effective methods. The first one is about the kernel transformation method, which transforms the nonlinear analysis model into the linear form, and is solved directly by the linear dictionary learning algorithm.
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