最优稀疏框架
Optimally Sparse Frames
AI总结：
针对大维信号处理中框架分解计算复杂的问题，引入稀疏框架概念，并利用Spectral Tetris算法构造出相对于标准单位向量基最优稀疏的紧框架。
摘要：
框架已成为一种用于分析或传输信号时获得冗余但稳定分解的手段，同时也促进了稀疏展开。然而，当信号维数较大时，计算信号的框架测量通常需要大量的加法和乘法，这使得在计算预算有限的应用中框架分解变得棘手。为了解决这个问题，本文聚焦于有限维希尔伯特空间中的框架，并将此类框架的稀疏性作为一种新范式引入。在我们的术语中，稀疏框架是指其元素在标准正交基中具有稀疏表示的框架，从而能够实现低复杂度的框架分解。为了引入最优性的精确含义，我们以扩展每个框架向量时所需的标准正交基向量的数量之和作为稀疏性度量。然后，我们分析了最近引入的用于构造单位范数紧框架的Spectral Tetris算法，并证明了该算法生成的紧框架实际上相对于标准单位向量基是最优稀疏的。最后，我们证明，即使将Spectral Tetris推广到构造与给定框架算子相关的单位范数框架，也能产生最优稀疏框架。
英文摘要：
Frames have established themselves as a means to derive redundant, yet stable decompositions of a signal for analysis or transmission, while also promoting sparse expansions. However, when the signal dimension is large, the computation of the frame measurements of a signal typically requires a large number of additions and multiplications, and this makes a frame decomposition intractable in applications with limited computing budget. To address this problem, in this paper, we focus on frames in finite-dimensional Hilbert spaces and introduce sparsity for such frames as a new paradigm. In our terminology, a sparse frame is a frame whose elements have a sparse representation in an orthonormal basis, thereby enabling low-complexity frame decompositions. To introduce a precise meaning of optimality, we take the sum of the numbers of vectors needed of this orthonormal basis when expanding each frame vector as sparsity measure. We then analyze the recently introduced algorithm Spectral Tetris for construction of unit norm tight frames and prove that the tight frames generated by this algorithm are in fact optimally sparse with respect to the standard unit vector basis. Finally, we show that even the generalization of Spectral Tetris for the construction of unit norm frames associated with a given frame operator produces optimally sparse frames.
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