Browsing by Subject "Graphical models"
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Item 3D reconstruction from a single image(University of Minnesota. Institute for Mathematics and Its Applications, 2008-08) Rother, Diego; Sapiro, GuillermoItem Dynamic learning and resource management under uncertainties for smart grid and cognitive radio networks(2014-05) Yahyasoltani, NasinThe importance of timely applications and decisions in dynamic environments, has led to the integration of intelligent networks to increase efficiency and end-user satisfaction in various application domains including telecommunication and power grid networks. Contemporary intelligent networks require advanced statistical signal processing and optimization tools to learn, infer and control their operation. This integration poses new challenges and has witnessed the emergence of novel resource management and learning techniques to cope with dynamics. In addition, in order to have implementable resource management algorithms, it is crucial to model the underlying sources of uncertainty in the optimization framework. This thesis develops algorithms for resource allocation under channel uncertainty in cognitive radio (CR) communication networks and contributes to demand coordination under uncertainty in power networks.Demand coordination through real-time pricing is addressed first by capitalizing on the uncertainty involved in the consumption behavior of consumers. Prerequisite to the demand coordination task is learning the uncertainty present in power consumption data. The dependency of consumers' consumption behavior on the announced prices and their neighbors' behavior, is modeled through graphical models. In particular, the electric vehicle (EV) consumers are considered and the adopted model also captures dynamics of EV consumers' time-varying charging decisions. Leveraging the online convex optimization (OCO) framework, an online algorithm for tracking the model is devised. With minimal assumptions on the structure of the temporal dynamics, and while accounting for the possibly adversarial consumption behavior of consumers, the proposed online algorithm provides performance guarantees. The probability distributions obtained through the tracking algorithm are then deployed as input to stochastic economic profit maximization for real-time price setting.Learning in the presence of missing data is a pervasive problem in statistical data analysis. Next, attention is turned to tracking the dynamic charging behavior of EV consumers, when at each time slot some of the consumers' consumption decisions are possibly missing. The problem amounts to online classification with missing labels. An online algorithm is proposed to wed real-time estimation of the missing data with learning of complete data in the OCO framework.As regards CR networks, this thesis introduces novel resource allocation algorithms for orthogonal frequency-division multiple access (OFDMA) CR under channel uncertainty where the unique approaches can be fitted to a class of large-scale robust mixed-integer problems. Due to the lack of cooperation of the licensed system, CRs must resort to less efficient channel estimation techniques thus incurring an inevitable channel estimation error. It is shown that CR interference constraints under channel uncertainty can be cast as chance constraints. On the other hand, instead of just modeling the user rates by logarithmic functions of transmit-powers, justified under ideal Gaussian coding, practical finite-alphabet constellations are adopted which leads to an optimization objective of a weighted sum of mutual information. When multiple users are present, due to the combinatorial search for optimal subcarrier assignment, the problem is non-convex and hard to solve, as the optimization variables are coupled across all subcarriers. To circumvent the resulting computational hurdle, tight and conservative approximations of the chance constraint are introduced to break the coupling and enforce separability per subcarrier. The separableproblem across subcarriers opens the door to the dual decomposition approach, which leads to a near-optimal and computationally efficient solution.Item Grouping penalties and its applications to high-dimensional models(2014-06) Zhu, YunzhangPart I: In high-dimensional regression, grouping pursuit and feature selection have their own merits while complementing each other in battling the curse of dimensionality. To seek parsimonious model, we perform simultaneous grouping pursuit and feature selection over an arbitrary undirected graph with each node corresponding to one predictor. When the corresponding nodes are reachable from each other over the graph,regression coefficients can be grouped, whose absolute values are the same or close. This is motivated from gene network analysis, where genes tend to work in groups according to their biological functionalities. Through a nonconvex penalty, we develop a computational strategy and analyze the proposed method. Theoretical analysis indicates that the proposed method reconstructs the oracle estimator, that is, the unbiased least squares estimator given the true grouping, leading to consistent reconstruction of grouping structures and informative features, as well as to optimal parameter estimation. Simulation studies suggest that the method combines the benefit of grouping pursuit with that of feature selection, and compares favorably against its competitors in selection accuracy and predictive performance. An application to eQTL data is used to illustrate the methodology, where a network is incorporated into analysis through an undirected graph.Part II: Gaussian graphical models are useful to analyze and visualize conditional dependence relationships between interacting units. Motivated from network analysis under different experimental conditions, such as gene networks for disparate cancer subtypes, we model structural changes over multiple networks with possible heterogeneities. In particular, we estimate multiple precision matrices describing dependencies among interacting units through maximum penalized likelihood. Of particular interest are homogeneous groups of similar entries across and zero-entries of these matrices, referred to as clustering and sparseness structures, respectively. A non-convex method is proposed to seek a sparse representation for each matrix and identify clusters of the entries across the matrices. Computationally, we develop an efficient method on the basis of difference convex programming, the augmented Lagrangian method and the block-wise coordinate descent method, which is scalable to hundreds of graphs of thousands nodes through a simple necessary and sufficient partition rule, which divides nodes into smaller disjoint subproblems excluding zero-coefficients nodes for arbitrary graphs with convex relaxation. Theoretically, a finite-sample error bound is derived for the proposed method to reconstruct the clustering and sparseness structures. This leads to consistent reconstruction of these two structures simultaneously, permitting the number of unknown parameters to be exponential in the sample size, and yielding the optimal performance of the oracle estimator as if the true structures were given a priori. Simulation studies suggest that the method enjoys the benefit of pursuing these two disparate kinds of structures, and compares favorably against its convex counterpart in the accuracy of structure pursuit and parameter estimation.Item Network Structure Identification using Corrupt Data-Streams(2021-08) Subramanian, Venkat RamMany complex systems lend themselves to effective modeling described by a network of dynamically interacting agents. Such modeling is prevalent in many application domains that include climate science, neuroscience, internet-of-things, power grids, and econometrics. The evolution of these systems is governed by the interdependencies and interactions between the agents that can contain feedback loops. Identification of the presence or absence of influence pathways among the agents is of primary importance that enables subsequent analytics in networked systems such as identifying central agents and clusters, devising control strategies in distributed systems, and resource allocation. In most application domains, the nature of the relationships and interdependencies cannot be easily modeled using first principles. Furthermore, in many such systems, it is not possible to deliberately affect the system, and thus passive or noninvasive methods are required. The existing methods of network identification do not account for the common ways through which data gets corrupted. In real-world systems, sensor readings can be inaccurate, clocks can get out of sync, and messages can get lost in transmission over a wireless network. The focus of this research is to incorporate realistic modeling assumptions on data streams and characterize the effects of data corruption on network identification using passive means. We show that identifying the structure of networked systems using corrupt measurements results in the inference of spurious links. The effects of data corruption on network reconstruction are characterized with provable guarantees on the quality of construction with respect to the generative models considered. A wide range of generative models that underlie the data streams are considered that include static interactions (Markov random fields), linear time-invariant dynamical systems, and nonlinear dynamical models. We examine both causal and non-causal inference methods. In both cases, we provide an exact characterization of the location of spurious links. Our results show that the spurious links are localized to the neighborhood of the corrupted node. All our solution methodologies utilize only the time-series observations without any knowledge of the system parameters. Our precise characterization of the erroneous links is further exploited when the network has special structural properties. There are several physical systems, especially flow-driven systems like power grids, heat transfer networks, and fluid flow networks, where every dynamic coupling between the agents/nodes is bi-directional. In such systems, identifying unidirectional links in reconstruction lead to the conclusion that such links arise from data corruption. We utilize our precise characterization of spurious links to detect and localize all corrupt nodes in the network. It is imperative that learning the exact network representation of such systems without spurious links is needed for performing accurate state estimation, control, and optimization. To this, we developed methods to remove all spurious links and identify the exact structure of bi-directed networks despite of data corruption.