匹配追踪参数化的几何研究
A Geometrical Study of Matching Pursuit Parametrization
AI总结：
本文通过将连续参数化字典视为信号空间中的嵌入流形，利用微分几何工具，证明了离散字典达到最小密度准则时离散匹配追踪与弱化连续匹配追踪的收敛等价性，并引入几何梯度上升优化使弱化因子更接近1。
摘要：
本文研究了用于信号匹配追踪分解的字典参数化离散化的影响。我们的方法将连续参数化字典视为信号空间中的嵌入流形，从而可以应用微分（黎曼）几何工具。本文的主要贡献有两方面。首先，我们证明如果离散字典达到最小密度准则，那么相应的离散MP（dMP）在收敛性方面等价于一个弱化的假设连续MP。有趣的是，相应的弱化因子取决于离散字典的密度度量。其次，我们表明在原子dMP选择中插入简单的几何梯度上升优化可以保持先前的比较，但弱化因子至少比没有优化时更接近单位1两倍。最后，我们通过数值实验证实了我们对信号和图像在字典参数化规则离散化上分解的理论预测。
英文摘要：
This paper studies the effect of discretizing the parametrization of a dictionary used for Matching Pursuit decompositions of signals. Our approach relies on viewing the continuously parametrized dictionary as an embedded manifold in the signal space on which the tools of differential (Riemannian) geometry can be applied. The main contribution of this paper is twofold. First, we prove that if a discrete dictionary reaches a minimal density criterion, then the corresponding discrete MP (dMP) is equivalent in terms of convergence to a weakened hypothetical continuous MP. Interestingly, the corresponding weakness factor depends on a density measure of the discrete dictionary. Second, we show that the insertion of a simple geometric gradient ascent optimization on the atom dMP selection maintains the previous comparison but with a weakness factor at least two times closer to unity than without optimization. Finally, we present numerical experiments confirming our theoretical predictions for decomposition of signals and images on regular discretizations of dictionary parametrizations.
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