A number of non-convex optimization problems can be convexified by “lifting” strategies. These methods yield convex formulations at the cost of substantially increased dimensionality. PhaseMax is a new type of convex relaxation that does not require lifting; it solves problems in their original low-dimensional parameter space.
Classical machine learning methods, include stochastic gradient descent (aka backprop), work great on one machine, but don’t scale well to the cloud or cluster setting. We propose a variety of algorithmic frameworks for scaling machine learning across many workers.
FASTA (Fast Adaptive Shrinkage/ Thresholding Algorithm) is an efficient, easy-to-use implementation of the Forward-Backward Splitting (FBS) method (also known as the proximal gradient method) for regularized optimization problems. Many variations on FBS are available in FASTA, including the popular accelerated variant FISTA (Beck and Teboulle ’09), the adaptive stepsize rule SpaRSA
PDHG is a powerful splitting method that can solve a wide range of constrained and non-differentiable optimization problems. Unlike the popular ADMM method, the PDHG approach usually does not require expensive minimization sub-steps. We provide adaptive stepsize selection rules that automate the solver, while increasing its speed and robustness.
The Split Bregman method is a technique for solving a variety of L1-regularized optimization problems, and is particularly effective for problems involving total-variation regularization. Split Bregman is one of the fastest solvers for Total-Variation denoising, image reconstruction from Fourier coefficients, convex image segmentation, and many other problems.
The Perfusion Imaging Toolkit (PIT) is a comprehensive set of tools for MR-based perfusion imaging. In addition to several different perfusion calculation tools, the software smoothly integrates file formatting, image denoising, registration, segmentation, mean curve extraction, and many other image processing tasks. PIT has the capability to generate perfusion measurements from regions of interest, as well as to generate pixel-by-pixel perfusion maps.