Joint Learning of Linear Time-Invariant Dynamical Systems

The problem of identifying the transition matrices in linear time-invariant dynamical systems (LTIDS) has been studied extensively in the literature (Lai and Wei, 1983; Kailath et al., 2000; Buchmann and Chan, 2007). Recent works establish finite-time rates for accurately learning the dynamics in different online and offline settings (Faradonbeh et al., 2018; Simchowitz et al., 2018; Sarkar and Rakhlin, 2019). The existing results are established assuming that the goal is to identify the transition matrix of a single dynamical system. However, in many areas where LTIDS models (as in (1) below) are used, such as macroeconomics (Stock and Watson, 2016), functional genomics (Fujita et al., 2007), and neuroimaging (Seth et al., 2015), one observes multiple dynamical systems and needs to estimate the transition matrices for all of them jointly. Further, the underlying dynamical systems share commonalities, but also exhibit heterogeneity. For example, (Skripnikov and Michailidis, 2019a) analyze economic indicators of US states whose local economies share a strong manufacturing base. Moreover, in time course genetics experiments, one is interested in understanding the dynamics and drivers of gene expressions across related animal or cell line populations (Basu et al., 2015), while in neuroimaging, one has access to data from multiple subjects that suffer from the same disease (Skripnikov and Michailidis, 2019b). In all these settings, there are remarkable similarities in the dynamics of the systems, but some degree of heterogeneity is also present. Hence, it becomes natural to pursue a joint learning strategy of the systems'