Many applications collect a large number of time series, for example, the financial data of companies quoted in a stock exchange, the health care data of all patients that visit the emergency room of a hospital, or the temperature sequences continuously measured by weather stations across the US. These data are often referred to as unstructured. A first task in its analytics is to derive a low dimensional representation, a graph or discrete manifold, that describes well the interrelations among the time series and their intrarelations across time. This paper presents a computationally tractable algorithm for estimating this graph that structures the data. The resulting graph is directed and weighted, possibly capturing causal relations, not just reciprocal correlations as in many existing approaches in the literature. A convergence analysis is carried out. The algorithm is demonstrated on random graph datasets and real network time series datasets, and its performance is compared to that of related methods. The adjacency matrices estimated with the new method are close to the true graph in the simulated data and consistent with prior physical knowledge in the real dataset tested.

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A semi-parametric, non-linear regression model in the presence of latent variables is introduced. These latent variables can correspond to unmodeled phenomena or unmeasured agents in a complex networked system. This new formulation allows joint estimation of certain non-linearities in the system, the direct interactions between measured variables, and the effects of unmodeled elements on the observed system. The particular form of the model adopted is justified, and learning is posed as a regularized empirical risk minimization. This leads to classes of structured convex optimization problems with a "sparse plus low-rank" flavor. Relations between the proposed model and several common model paradigms, such as those of Robust Principal Component Analysis (PCA) and Vector Autoregression (VAR), are established. Particularly in the VAR setting, the lowrank contributions can come from broad trends exhibited in the time series. Details of the algorithm for learning the model are presented. Experiments demonstrate the performance of the model and the estimation algorithm on simulated and real data.

We present a new method for simultaneously denoising and unwrapping phase in multi-frequency homodyne time-of-flight ranging for the formation of accurate depth maps despite low SNR of raw measurements. This is achieved with a new generalized approximate message passing (GAMP) algorithm for minimum mean-squared error estimation of the phase. A detailed, physically-accurate acquisition model is central in achieving high accuracy, and the use of the GAMP methodology allows low computational complexity despite dense dependencies and the nonlinearity and non-Gaussianity of the acquisition model. Numerical simulations demonstrate that our integrated approach performs better than separate unwrapping followed by denoising. This performance translates to lowering the optical power consumption of time-of-flight cameras for a fixed acquisition quality.

A class of models for describing sets of time series generated by interacting agents using directed, weighted graphs is introduced. A computationally tractable algorithm for estimating the graph adjacency matrix of this model from observed time series data is presented. The performance guarantees of this algorithm for prediction are outlined under several assumptions on the properties of the dynamics of the system of agents and on the true values of the parameters. These guarantees are tested empirically through simulation studies using several random graph models.