Learning Low-Dimensional Nonlinear Structures from High-Dimensional Noisy Data: An Integral Operator Approach

With rapid technological advancements in data collection and processing, massive large-scale and high-dimensional data sets are widely available nowadays in diverse research fields such as astronomy, business analytics, human genetics and microbiology. A common feature of these data sets is that their statistical and geometric properties can be well understood via a meaningful low-rank representation of reduced dimensionality. Learning low-dimensional structures from these high-dimensional noisy data is one of the central topics in statistics and data science. Moreover, nonlinear structures have been found predominant and intrinsic in many real-world data sets, which may not be easily captured or preserved in commonly used linear or quasi-linear methods such as principal component analysis (PCA), singular value decomposition (SVD) [53] and multidimensional scaling (MDS) [18]. As a longstanding and well-recognized technique for analyzing data sets with possibly nonlinear structures, kernel methods have been shown effective in various applications ranging from clustering, data visualization to classification and prediction [75, 47, 59]. On the other hand, spectral methods [23], as a fundamental tool for dimension reduction, are oftentimes applied in combination with kernel methods to better capture the underlying low-dimensional nonlinear structure in the data. These approaches are commonly referred to as nonlinear dimension reduction techniques; see Section 1.1 below for a brief overview.