Convex Relaxations for Robust Identification of Hybrid Models

Northeastern University Department of Electrical and Computer Engineering Doctor of Philosophy in Electrical Engineering Convex Relaxations for Robust Identification of Hybrid Models by Necmiye Ozay In order to extract useful information from a data set, it is necessary to understand the underlying model. This dissertation addresses two of the main challenges in identification of such models. First, the data is often generated by multiple, unknown number of sources; that is, the underlying model is usually a mixture model or hybrid model. This requires solving the identification and data association problems simultaneously. Second, the data is usually corrupted by noise necessitating robust identification schemes. Motivated by these challenges we consider the problem of robust identification of hybrid models that interpolate the data within a given noise bound. In particular, for static data we try to fit affine subspace arrangements or more generally a mixture of algebraic surfaces to the data; and for dynamic data we try to infer the underlying switched affine dynamical system. Clearly, as stated, these problems admit infinitely many solutions. For instance, one can always find a trivial hybrid model with as many submodels/subspaces as the number of data points (i.e. one submodel/subspace per data point). In order to regularize the problem, we define suitable a priori model sets and objective functions that seek ”simple” models. Although this leads to generically NP-Hard problems, we develop computationally efficient algorithms based on convex relaxations. Additionally, we discuss a related problem: robust model (in)validation for affine hybrid systems. Before a given system description, obtained either from first principles or an identification step, can be used to design controllers, it must be validated using additional experimental data. In this dissertation, we show that the invalidation problem for switched affine systems can be solved within a similar framework by exploiting a combination of elements from convex analysis and the classical theory of moments. Finally, the effectiveness of the proposed methods are illustrated using both simulations and several non-trivial applications in computer vision such as video and dynamic texture segmentation, two-view motion segmentation and human activity analysis. In all cases the proposed methods significantly outperform existing approaches both in terms of accuracy and resilience to noise. Acknowledgements First and foremost, I would like to thank my adviser, Professor Mario Sznaier for his support in all aspects of my graduate life. His guidance, encouragement and enthusiasm made this journey quite fun for me. I am especially thankful to him for giving me the independence to pursue my own research ideas but also being there with “crazy” ideas whenever I got stuck. His being a great teacher, technical expertise, attention to mathematical rigor and broad vision have been and will be a constant source of inspiration for me. I would like to thank Professor Octavia Camps who introduced me to the fields of computer vision and pattern recognition. Her expertise and suggestions have been very useful in finding interesting computer vision applications for hybrid system identification algorithms developed in this dissertation. I would also like to express my gratitude to Professor Constantino Lagoa of Penn State. My research has benefited a lot from discussions and interactions with him. I am very grateful to Professor Dana Brooks and Professor Gilead Tadmor for serving on my dissertation committee and for their insightful comments on my work. I would also like to thank Professor Brooks for letting me sit in his group meetings. I feel very fortunate to meet with great friends during my graduate studies. I had a very pleasant start to my studies in the U.S. thanks to two special friends, Roberto Lublinerman and Dimitris Zarpalas who have been a great source of support and encouragement. I would also like to thank my lab mates at the Robust Systems Laboratory at Northeastern for their help and support, and for all the fun times we spent together chatting about this and that (and sometimes about research). In particular, Mustafa Ayazoglu helped generating some of the simulation data and figures in this dissertation (he can even print videos on an eps file). My thanks also go to the CDSP gang and Ms. Joan Pratt for their friendship, the pizza parties and the wonderful times with the dragonboat team (go go go Huskies!!). I am also grateful to Sila Kurugol not only for being a very close friend but also for our collaboration on medical image segmentation. I want to take this opportunity to thank my professors and teachers back in Turkey for providing me with a solid education and background. I would also like to thank my closest friends, particularly Bahar, Dilek and Gugu, for being next to me whenever I need, despite the physical distance. Finally, my greatest appreciation goes to my family, especially to my parents Kafiye and Hami for their endless love and support, and for being on the other end of the webcam each and every day.