# Diego Klabjan (Northwestern University, Chicago): Scale Invariant Power Iteration under a Euclidean Ball Constraint

Vir: Seminar za numerično analizo

Several machine learning and statistical models can be formulated as maximization of a scale invariant function under a Euclidian ball constraint, for example, principle component analysis, Gaussian mixture models, non-negative matrix factorization. We generalize the power iteration algorithm to this setting and analyze convergence properties of the algorithm. Our main result states that if the starting point is close to a local maximum, then the iterates of the algorithm converge to this local maximum. Furthermore, the convergence rate is linear and thus equal to the rate of power iteration. Numerically we benchmark the algorithm against state-of-the-art methods for each individual problem class to find out that the algorithm either outperforms benchmark algorithms or is competitive to them. We conclude that our algorithm offers a robust and simple unified methodology for the aforementioned problems with a very attractive convergence rate.

Diego Klabjan is a professor at Northwestern University, Department of Industrial Engineering and Management Sciences. He is also Founding Director, Master of Science in Analytics and Director, Center for Deep Learning. After obtaining his doctorate from the School of Industrial and Systems Engineering of the Georgia Institute of Technology in 1999 in Algorithms, Combinatorics, and Optimization, in the same year he joined the University of Illinois at Urbana-Champaign. In 2007 he became an associate professor at Northwestern and in 2012 he was promoted to a full professor. His research is focused on machine learning, deep learning and analytics (modeling, methodologies, theoretical results) with concentration in finance, insurance, sports, and bioinformatics. Professor Klabjan has led projects with large companies such as Intel, Baxter, Allstate, AbbVie, FedEx Express, General Motors, United Continental, and many others, and he is also assisting numerous start-ups with their analytics needs. He is also a founder of Opex Analytics LLC.