CIF: Small: Exploiting Sparsity for Dimensionality Reduction

利用统计或确定性信号描述符中存在的稀疏性的现有方法大多依赖于线性模型，并且可用的压缩采样(CS)实现是模拟的。另一方面，主要的应用领域，如信号压缩和复杂系统的降阶逼近，需要双线性模型。此外，日常生活中使用的相关技术，例如语音和图像压缩，是数字技术。因此，需要进行基础研究来解释双线性模型中的稀疏性，并将CS置于与数字压缩模块同等的地位，后者依赖于主成分分析(PCA)或典型相关分析(CCA)进行降维，并以率失真极限作为性能基准。本研究旨在开发识别稀疏性的降维、压缩和重建任务的算法和相应的性能限制。实现这些目标的关键是稀疏PCA、稀疏CCA和相关量化方案的最佳公式，以及用于与稀疏过完全基扩展(SOBE)进行比较的稀疏感知的率失真度量。我们的愿景是拥有可用的工具和价值系数，以利用稀疏性的“正确”形式应用于“正确”的应用领域。稀疏主元分析、稀疏CCA和SOBE方法的优化借鉴了稀疏意识回归、基寻踪、子空间跟踪和以L 1范数正则的最小化问题的坐标下降求解器的当代进展。研究议程利用这些工具来调查一些具有挑战性的方向。这包括：(D1)非理想链路上的稀疏性利用和功率感知压缩；(D2)稀疏性认知，用于具有记忆的(非)平稳过程的自适应压缩；(D3)隐藏源的稀疏性感知重建，以及稀疏系统的降维识别；以及(D4)稀疏性的确定性描述符和统计描述符之间的比较。

Existing approaches to exploit sparsity present in statistical or deterministic signal descriptors have mostly relied on linear models, and available compressive sampling (CS) implementations are analog. On the other hand, major application domains, such as signal compression and reduced-rank approximation of complex systems, entail bilinear models. In addition, the pertinent technology used in everyday life, for e.g., speech and image compression, is digital. Accordingly, the need arises for fundamental research to account for sparsity in bilinear models, as well as put CS on equal footing with digital compression modules, which rely on the `workhorses' for dimensionality reduction, namely principal component analysis (PCA) or canonical correlation analysis (CCA), and have their performance benchmarked by rate-distortion limits.This research aims at developing algorithms, and corresponding performance limits for sparsity-cognizant dimensionality reduction, compression, and reconstruction tasks. Key to achieving these goals are optimal formulations for sparse PCA, sparse CCA, and associated quantization schemes, along with sparsity-aware rate-distortion metrics for comparison with sparse overcomplete basis expansions (SOBE). The vision is to have available tools and figures of merit to exploit the `right' form of sparsity for the `right' application domain. Optimization of sparse PCA, sparse CCA, and SOBE approaches draws from contemporary advances in sparsity-aware regression, basis pursuit, subspace tracking, and coordinate-descent solvers of minimization problems regularized with the l_1 norm of the unknowns. The research agenda leverages these tools to investigate a number of challenging directions. These include: (d1) sparsity-exploiting and power-aware compression over non-ideal links; (d2) sparsity-cognizant, adaptive compression for (non-)stationary processes with memory; (d3) sparsity-aware reconstruction of hidden sources, and reduced-rank identification of sparse systems; and (d4) comparisons between deterministic and statistical descriptors of sparsity.