多尺度字典学习：非渐近界和鲁棒性
Multiscale Dictionary Learning: Non-Asymptotic Bounds and Robustness
摘要：
高维数据集被低维结构很好地近似。在过去的十年中，这种经验观察激发了对检测，测量和建模技术的研究，以利用这些低维内在结构，从而对高维统计，机器学习和信号处理产生了许多影响。流形学习（其中低维结构是流形）和字典学习（其中低维结构是来自有限字典的向量的稀疏线性组合的集合）是该领域中两个突出的理论和计算框架。尽管它们表面上的区别，最近推出的几何多分辨率分析（GMRA）提供了一个强大的，计算效率高，多尺度的程序，同时学习流形和字典。在这项工作中，我们证明了非渐近概率界的近似误差的GMRA为丰富的一类数据生成的统计模型，包括“嘈杂”的流形，从而建立理论的鲁棒性的程序和确认经验观察。特别是，如果一个数据集聚集在一个低维流形附近，我们的结果表明，GMRA的近似误差是完全独立的环境尺寸。因此，我们的工作建立GMRA作为一个可证明的快速算法的字典学习近似和稀疏性的保证。我们包括几个数值实验证实这些理论结果，我们的理论框架提供了新的工具，用于评估行为的流形学习和字典学习程序上的一大类有趣的模型。
英文摘要：
High-dimensional datasets are well-approximated by low-dimensional structures. Over the past decade, this empirical observation motivated the investigation of detection, measurement, and modeling techniques to exploit these low-dimensional intrinsic structures, yielding numerous implications for high-dimensional statistics, machine learning, and signal processing. Manifold learning (where the low-dimensional structure is a manifold) and dictionary learning (where the low-dimensional structure is the set of sparse linear combinations of vectors from a finite dictionary) are two prominent theoretical and computational frameworks in this area. Despite their ostensible distinction, the recently-introduced Geometric Multi-Resolution Analysis (GMRA) provides a robust, computationally efficient, multiscale procedure for simultaneously learning manifolds and dictionaries. In this work, we prove non-asymptotic probabilistic bounds on the approximation error of GMRA for a rich class of data-generating statistical models that includes "noisy" manifolds, thereby establishing the theoretical robustness of the procedure and confirming empirical observations. In particular, if a dataset aggregates near a low-dimensional manifold, our results show that the approximation error of the GMRA is completely independent of the ambient dimension. Our work therefore establishes GMRA as a provably fast algorithm for dictionary learning with approximation and sparsity guarantees. We include several numerical experiments confirming these theoretical results, and our theoretical framework provides new tools for assessing the behavior of manifold learning and dictionary learning procedures on a large class of interesting models.
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