Low-dimensional Embedding of Large-scale Infinite-dimensional Function Spaces with Application to Human Brain Connectome

Graph-based dimensionality reduction techniques assume that each datapoint can be written as a fixed width vector with a well-defined distance measure among datapoints; also, they typically assume that the number of instances is small enough to perform matrix inversion or pseudo-inversion. This paper considers dimensionality reduction on data using graph-based methods when two extreme circumstances arise: (1) the data is composed of functional objects instead of feature vectors, and (2) the data contains many objects so that standard methods become numerically and computationally intractable. In the case of functional data, even if a finite representation is available (for example a linked-list of points), converting that representation into a standard vector representation is required so that a meaningful distance measure can be defined. In this paper, we present a two-stage framework to embed a large-scale functional space into a low-dimensional vector space while preserving the original similarity in the reduced space. In particular, after converting the function space into a high-dimensional vector space in the first stage, we have developed a fast and robust approximation to the Laplacian-based dimensionality reduction methods in the second stage to deal with large number of datapoints.