Open Access Publications from the University of California

## Numerical Optimization Methods for Image Processing and Machine Learning

• Author(s): Woodworth, Joseph Thomas
First, we are given a discrete sample of event locations, and we wish to produce a probability density that models the relative probability of events occurring in a spatial domain. Standard density estimation techniques do not incorporate priors informed by spatial data. Such methods can result in assigning significant positive probability to locations where events cannot realistically occur. In particular, when modeling residential burglaries, standard density estimation can predict residential burglaries occurring where there are no residences. Incorporating the spatial data can inform the valid region for the density. When modeling very few events, additional priors can help to correctly fill in the gaps. Learning and enforcing correlation between spatial data and event data can yield better estimates from fewer events. We propose a non-local version of Maximum Penalized Likelihood Estimation based on the $H^1$ Sobolev seminorm regularizer that computes non-local weights from spatial data to obtain more spatially accurate density estimates. We evaluate this method in application to a residential burglary data set from San Fernando Valley with the non-local weights informed by housing data or a satellite image.
Second, we analyze a method for compressed sensing. The $\ell^0$ minimization of compressed sensing is often relaxed to $\ell^1$, which yields easy computation using the shrinkage mapping known as soft thresholding, and can be shown to recover the original solution under certain hypotheses. Recent work has derived a general class of shrinkages and associated nonconvex penalties that better approximate the original $\ell^0$ penalty and empirically can recover the original solution from fewer measurements. We specifically examine $p$-shrinkage and firm thresholding. We prove that given data and a measurement matrix from a broad class of matrices, one can choose parameters for these classes of shrinkages to guarantee exact recovery of the sparsest solution. We further prove convergence of the algorithm iterative $p$-shrinkage (IPS) for solving one such relaxed problem.