最优平移不变空间及其Parseval生成元
Optimal Shift Invariant Spaces and Their Parseval Generators
摘要：
给定一组函数F ={f_1，...，f_m}，研究了求具有n个生成元{phi_1，.，φ_n}，它是F的函数的"最接近"，在这个意义上，V最小化从数据F到V的最小平方距离，在所有移位不变空间的集合上，可以由n个或更少的生成器生成。Eckart-Young定理使用奇异值分解来提供有限维中的相关问题的解决方案。我们将研究中的问题转化为一个不可数的有限维问题，每个问题都可以使用扩展的Eckart-Young定理来解决。我们证明了有限维的解决方案可以修补在一起，并转换，以获得最佳的移位不变的空间解决方案的原始问题，我们产生了一个Parseval框架的最佳空间。一个典型的应用是寻找描述给定类别的信号或图像的平移不变空间模型的问题（例如，胸部X射线的类别），从对m个信号或图像f_1，. f_m，可以是理论样本，也可以是实验数据。
英文摘要：
Given a set of functions F={f_1,...,f_m} of L2(Rd), we study the problem of finding the shift-invariant space V with n generators {phi_1,...,phi_n} that is ``closest'' to the functions of F in the sense that V minimize the least square distance from the data F to V over the set of all shift-invariant spaces that can be generated by n or less generators. The Eckart-Young Theorem uses the singular value decomposition to provide a solution to a related problem in finite dimension. We transform the problem under study into an uncountable set of finite dimensional problems each of which can be solved using an extension of the Eckart-Young Theorem. We prove that the finite dimensional solutions can be patched together and transformed to obtain the optimal shift-invariant space solution to the original problem, and we produce a Parseval frame for the optimal space. A typical application is the problem of finding a shift-invariant space model that describes a given class of signals or images (e.g., the class of chest X-Rays), from the observation of a set of m signals or images f_1,...f_m, which may be theoretical samples, or experimental data.
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