BLISS: Global Blind Identification of Linear Systems with Sparse Inputs
This work addresses the fundamental problem of blind system identification for researchers in control theory and machine learning by bridging two historically separate fields, though it appears to be an incremental theoretical extension rather than a paradigm shift.
The paper tackles blind linear system identification by connecting it to sparse dictionary learning, showing that under sparse input assumptions, blind system identification becomes a generalization of complete dictionary learning. This connection enables global identifiability guarantees and empirical demonstration that alternating direction method of multipliers can globally recover ground-truth systems from single trajectories with sufficient samples and input sparsity.
Linear system identification and sparse dictionary learning can both be seen as structured matrix factorization problems. However, these two problems have historically been studied in isolation by the systems theory and machine learning communities. Although linear system identification enjoys a mature theory when inputs are known, blind linear system identification remains poorly understood beyond restrictive settings. In contrast, complete sparse dictionary learning has recently benefited from strong global identifiability results and scalable nonconvex algorithms. In this work, we bridge these two areas by showing that under a sparse input assumption, fully observed blind system identification becomes a generalization of complete dictionary learning. This connection allows us to develop global identifiability guarantees for blind system identification, by leveraging techniques from the complete dictionary learning literature. We further show empirically that a principled application of the alternating direction method of multipliers can globally recover the ground-truth system from a single trajectory, provided sufficient samples and input sparsity.