Compressive Spectral Clustering

Data mining has become one of the fastest growing research topics in mathematics and computer science. Data such as high dimensional signals, magnetic resonance images, and hyperspectral images can be costly to acquire or it could be unobtainable to make even simple direct comparisons. Compressed sensing is a technique that addresses this issue. It is used for exact recovery of sparse signals using fewer measurements than the ambient dimension. Compressed sensing provides a bound on the error derived from making these few measurements of a signal. Our goal is to take advantage of these compressed sensing techniques to perform spectral clustering using much fewer measurements than the ambient dimension. The goal of clustering is to partition objects into groups such that objects within the same group are similar. Standard clustering such as k-means requires the space in which the objects are represented to be linearly separable. Spectral clustering methods allow for a wider range of underlying geometries, making them more flexible. Classification is the procedure of assigning labels to objects such that objects’ labels within the same cluster will match previously labeled objects from a training set. Classification is traditionally a type of supervised learning problem that tries to learn a function from the data in order to predict the output of an unknown input from known input and output pairs. Clustering is an unsupervised learning problem where one is only given the unlabeled data and the goal is to learn the underlying structure. Spectral clustering uses local distance between data points, (e.g. the Euclidian distance, d(xi, xj) = ‖xi − xj‖2) to construct a graph G = (V , E). A traditional choice of edge weights uses the Gaussian kernel,