LAUSR

In this article, we present a novel approach to reconstruct the topology of linear dynamical systems with latent nodes. The network is allowed to have directed loops and bi-directed edges. We show that the imaginary part of the inverse power spectral density matrix (IPSDM), realized from the time-series data and shown to be skew symmetric, can unveil the connectivity structure. Necessary and sufficient conditions are provided for the unique decomposition of a given skew symmetric into sum of a sparse skew symmetric and a low rank skew symmetric matrices. An optimization based algorithm is developed to decompose the imaginary part of IPSDM to yield the sparse matrix $\mathbf{S}$ and the low-rank matrix $\mathbf{L}$. $\mathbf{S}$ embeds information about the topology of a subgraph restricted to the observed nodes and $\mathbf{L}$ provides information about the topology between the observed nodes and the hidden nodes. Algorithms are provided to reconstruct the topology of the network between the observed nodes using $\mathbf{S}$ and the links related to latent nodes using $\mathbf{L}$. Moreover, for finite number of data samples, we provide concentration bounds on the entry-wise distance between the true IPSDM and the estimated IPSDM.