Optimizing Sparse Adaptive Representations of Signals Using Energy-Based Algorithm Enhancements

稀疏分解算法根据构成词典的称为原子的过完备的有限支持函数集自适应地扩展信号。这些非线性算法的目标是找到一种同时稀疏、高效和健壮的表示，以及信息丰富和可塑性的表示。研究人员正在对各种信号的稀疏分解的能量含量进行建模，以优化性能。这项工作将导致更好的稀疏分解的自适应原子选择策略，以及更符合一类信号的内在结构的词典。这将导致一种确定稀疏表示的哪些项实际上属于信号以及哪些是分解的伪像的方法。这项研究对依赖于波形表示的应用具有重要意义，例如地质和生物医学数据分析、内容检索、源分离、音乐声音转换等。稀疏表示为标准的正交化扩展提供了一种有吸引力的替代方案。稀疏表示的一些算法的一个特性是创建的术语在物理上没有意义，但反映了算法的贪婪。研究人员将这种现象称为暗能量，因为这些项在信号重建中被取消了。尽管之前针对这些虚假术语的研究将它们视为一种麻烦，但有证据表明，暗能量包含了有关该信号及其与词典一致性的有用信息。它还可以为选择原子或学习更好的原子提供更好的策略。研究人员正在探索暗能量的性质及其对原始信号、词典和用于产生分解的策略的意义。

Sparse decomposition algorithms adaptively expand a signal in terms of an over-complete set of finite-support functions called atoms that comprise a dictionary. These nonlinear algorithms aim to find a representation that is at once sparse, efficient, and robust, as well as informative and malleable. The investigators are modeling the energy content of sparse decompositions for a variety of signals in order to optimize performance. This work will lead to better adaptive atom selection strategies for sparse decompositions, as well as dictionaries that are more coherent to the intrinsic structures of a class of signals. It will result in a way to determine which terms of a sparse representation actually belong to the signal and which are artifacts of the decomposition. This research has implications for applications that rely on representations of waveforms such as geological and biomedical data analyses, content retrieval, source separation, music sound transformation, etc.Sparse representations provide an attractive alternative to standard orthonormal expan-sions. One property of some algorithms for sparse representations is the creation of terms that are not physically meaningful, but reflect instead the greediness of the algorithm. The investigators refer to this phenomenon as dark energy because these terms are cancelled in the signal reconstruction. Although previous work addressing these spurious terms has viewed them as a nuisance, there is evidence suggesting that dark energy embodies useful information about the signal and its coherency with the dictionary. It can also provide a better strategy for choosing atoms or learning better ones. The investigators are exploring the nature of dark energy and its significance for the original signal, the dictionary, and the strategies used to generate decompositions.