Electroencephalographic (EEG) monitoring of neural activity is widely used for sleep disorder diagnostics and research. The standard of care is to manually classify 30-second epochs of EEG time-domain traces into 5 discrete sleep stages. Unfortunately, this scoring process is subjective and time-consuming, and the defined stages do not capture the heterogeneous landscape of healthy and clinical neural dynamics. This motivates the search for a data-driven and principled way to identify the number and composition of salient, reoccurring brain states present during sleep. To this end, we propose a Hierarchical Dirichlet Process Hidden Markov Model (HDP-HMM), combined with wide-sense stationary (WSS) time series spectral estimation to construct a generative model for personalized subject sleep states. In addition, we employ multitaper spectral estimation to further reduce the large variance of the spectral estimates inherent to finite-length EEG measurements. By applying our method to both simulated and human sleep data, we arrive at three main results: 1) a Bayesian nonparametric automated algorithm that recovers general temporal dynamics of sleep, 2) identification of subject-specific "microstates" within canonical sleep stages, and 3) discovery of stage-dependent sub-oscillations with shared spectral signatures across subjects.

We consider the problem of recovering a sequence of vectors, $(x_k)_{k=0}^K$, for which the increments $x_k-x_{k-1}$ are $S_k$-sparse (with $S_k$ typically smaller than $S_1$), based on linear measurements $(y_k = A_k x_k + e_k)_{k=1}^K$, where $A_k$ and $e_k$ denote the measurement matrix and noise, respectively. Assuming each $A_k$ obeys the restricted isometry property (RIP) of a certain order---depending only on $S_k$---we show that in the absence of noise a convex program, which minimizes the weighted sum of the $\ell_1$-norm of successive differences subject to the linear measurement constraints, recovers the sequence $(x_k)_{k=1}^K$ \emph{exactly}. This is an interesting result because this convex program is equivalent to a standard compressive sensing problem with a highly-structured aggregate measurement matrix which does not satisfy the RIP requirements in the standard sense, and yet we can achieve exact recovery. In the presence of bounded noise, we propose a quadratically-constrained convex program for recovery and derive bounds on the reconstruction error of the sequence. We supplement our theoretical analysis with simulations and an application to real video data. These further support the validity of the proposed approach for acquisition and recovery of signals with time-varying sparsity.