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In this work, we study the problem of learning partially observed lineardynamical systems from a single sample trajectory. A major practical challengein the existing system identification methods is the undesirable dependency oftheir required sample size on the system dimension: roughly speaking, theypresume and rely on sample sizes that scale linearly with respect to the systemdimension. Evidently, in high-dimensional regime where the system dimension islarge, it may be costly, if not impossible, to collect as many samples from theunknown system. In this paper, we will remedy this undesirable dependency onthe system dimension by introducing an $\ell_1$-regularized estimation methodthat can accurately estimate the Markov parameters of the system, provided thatthe number of samples scale logarithmically with the system dimension. Ourresult significantly improves the sample complexity of learning partiallyobserved linear dynamical systems: it shows that the Markov parameters of thesystem can be learned in the high-dimensional setting, where the number ofsamples is significantly smaller than the system dimension. Traditionally, the$\ell_1$-regularized estimators have been used to promote sparsity in theestimated parameters. By resorting to the notion of "weak sparsity", we showthat, irrespective of the true sparsity of the system, a similar regularizedestimator can be used to reduce the sample complexity of learning partiallyobserved linear systems, provided that the true system is inherently stable.

Learning Partially Observed Linear Dynamical Systems from Logarithmic Number of Samples