393 research outputs found
Sparse Multivariate Factor Regression
We consider the problem of multivariate regression in a setting where the
relevant predictors could be shared among different responses. We propose an
algorithm which decomposes the coefficient matrix into the product of a long
matrix and a wide matrix, with an elastic net penalty on the former and an
penalty on the latter. The first matrix linearly transforms the
predictors to a set of latent factors, and the second one regresses the
responses on these factors. Our algorithm simultaneously performs dimension
reduction and coefficient estimation and automatically estimates the number of
latent factors from the data. Our formulation results in a non-convex
optimization problem, which despite its flexibility to impose effective
low-dimensional structure, is difficult, or even impossible, to solve exactly
in a reasonable time. We specify an optimization algorithm based on alternating
minimization with three different sets of updates to solve this non-convex
problem and provide theoretical results on its convergence and optimality.
Finally, we demonstrate the effectiveness of our algorithm via experiments on
simulated and real data
Online Joint Topology Identification and Signal Estimation with Inexact Proximal Online Gradient Descent
Identifying the topology that underlies a set of time series is useful for
tasks such as prediction, denoising, and data completion. Vector autoregressive
(VAR) model based topologies capture dependencies among time series, and are
often inferred from observed spatio-temporal data. When the data are affected
by noise and/or missing samples, the tasks of topology identification and
signal recovery (reconstruction) have to be performed jointly. Additional
challenges arise when i) the underlying topology is time-varying, ii) data
become available sequentially, and iii) no delay is tolerated. To overcome
these challenges, this paper proposes two online algorithms to estimate the VAR
model-based topologies. The proposed algorithms have constant complexity per
iteration, which makes them interesting for big data scenarios. They also enjoy
complementary merits in terms of complexity and performance. A performance
guarantee is derived for one of the algorithms in the form of a dynamic regret
bound. Numerical tests are also presented, showcasing the ability of the
proposed algorithms to track the time-varying topologies with missing data in
an online fashion.Comment: 14 pages including supplementary material, 2 figures, submitted to
IEEE Transactions on Signal Processin
Online Machine Learning for Graph Topology Identification from Multiple Time Series
High dimensional time series data are observed in many complex systems. In networked data, some of the time series are influenced by other time series. Identifying these relations encoded in a graph structure or topology among the time series is of paramount interest in certain applications since the identified structure can provide insights about the underlying system and can assist in inference tasks. In practice, the underlying topology is usually sparse, that is, not all the participating time series in influence each other. The goal of this dissertation pertains to study the problem of sparse topology identification under various settings.
Topology identification from time series is a challenging task. The first major challenge in topology identification is that the assumption of static topology does not hold always in practice since most of the practical systems are evolving with time. For instance, in econometrics, social networks, etc., the relations among the time series can change over time. Identifying the topologies of such dynamic networks is a major challenge.
The second major challenge is that in most practical scenarios, the data is not available at once - it is coming in a streaming fashion. Hence, batch approaches are either not applicable or they become computationally expensive since a batch algorithm is needed to be run when a new datum becomes available.
The third challenge is that the multi-dimensional time series data can contain missing values due faulty sensors, privacy and security reasons, or due to saving energy.
We address the aforementioned challenges in this dissertation by proposing online/-batch algorithms to solve the problem of time-varying topology identification. A model based on vector autoregressive (VAR) process is adopted initially. The parameters of the VAR model reveal the topology of the underlying network. First, two online algorithms are proposed for the case of streaming data. Next, using the same VAR model, two online algorithms under the framework of online optimization are presented to track the time-varying topologies. To evaluate the performance of propose online algorithms, we show that both the proposed algorithms incur a sublinear static regret. To characterize the performance theoretically in time-varying scenarios, a bound on the dynamic regret for one of the proposed algorithms (TIRSO) is derived. Next, using a structural equation model (SEM) for topology identification, an online algorithm for tracking time-varying topologies is proposed, and a bound on the dynamic regret is also derived for the proposed algorithm. Moreover, using a non-stationary VAR model, an algorithm for dynamic topology identification and breakpoint detection is also proposed, where the notion of local structural breakpoint is introduced to accommodate the concept of breakpoint where instead of the whole topology, only a few edges vary. Finally, the problem of tracking VAR-based time-varying topologies with missing data is investigated. Online algorithms are proposed where the joint signal and topology estimation is carried out. Dynamic regret analysis is also presented for the proposed algorithm. For all the previously mentioned works, simulation tests about the proposed algorithms are also presented and discussed in this dissertation. The numerical results of the proposed algorithms corroborate with the theoretical analysis presented in this dissertation.publishedVersio
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