ℓâ‚€-norm Minimization for Basis Selection
ℓâ‚€-norm Minimization for Basis Selection
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发表时间：
2004-12
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通讯作者：
D. Wipf;B. Rao
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作者：
D. Wipf;B. Rao
在许多应用领域中，在给定过完备基向量字典的情况下，寻找信号的稀疏或最小l0范数表示是一个重要问题。不幸的是，所需的优化问题通常很棘手，因为随着候选基向量数量的增加，局部最小值的数量会组合增加。这一缺陷促使大多数研究人员转而尽量减少替代措施，如l1范数，导致更容易处理的计算方法。这个过程的缺点是，我们现在引入了最终目标和目标函数之间的不匹配。在本文中，我们展示了一种基于稀疏贝叶斯学习的方法，在减少麻烦的局部极小值的同时最小化l0范数。此外，我们推导出必要的条件，局部极小值发生通过这种方法和经验证明，通常有很多更少的一般问题的兴趣。
Finding the sparsest, or minimum l0-norm, representation of a signal given an overcomplete dictionary of basis vectors is an important problem in many application domains. Unfortunately, the required optimization problem is often intractable because there is a combinatorial increase in the number of local minima as the number of candidate basis vectors increases. This deficiency has prompted most researchers to instead minimize surrogate measures, such as the l1-norm, that lead to more tractable computational methods. The downside of this procedure is that we have now introduced a mismatch between our ultimate goal and our objective function. In this paper, we demonstrate a sparse Bayesian learning-based method of minimizing the l0-norm while reducing the number of troublesome local minima. Moreover, we derive necessary conditions for local minima to occur via this approach and empirically demonstrate that there are typically many fewer for general problems of interest.