arXiv subjects

Somayeh Sojoudi

Publications and source records attributed to Somayeh Sojoudi.

85 records · Page 5Linked to original sources

arXiv2019

Efficient Learning of Distributed Linear-Quadratic Controllers

In this work, we propose a robust approach to design distributed controllers for unknown-but-sparse linear and time-invariant systems. By leveraging modern techniques in distributed controller synthesis and structured linear inverse problems as applied to system identification, we show that near-optimal distributed controllers can be learned with sub-linear sample complexity and computed with near-linear time complexity, both measured with respect to the dimension of the system. In particular, we provide sharp end-to-end guarantees on the stability and the performance of the designed distributed controller and prove that for sparse systems, the number of samples needed to guarantee robust and near optimal performance of the designed controller can be significantly smaller than the dimension of the system. Finally, we show that the proposed optimization problem can be solved to global optimality with near-linear time complexity by iteratively solving a series of small quadratic programs.

math.OC↗

arXiv2019

Exact Guarantees on the Absence of Spurious Local Minima for Non-negative Rank-1 Robust Principal Component Analysis

This work is concerned with the non-negative rank-1 robust principal component analysis (RPCA), where the goal is to recover the dominant non-negative principal components of a data matrix precisely, where a number of measurements could be grossly corrupted with sparse and arbitrary large noise. Most of the known techniques for solving the RPCA rely on convex relaxation methods by lifting the problem to a higher dimension, which significantly increase the number of variables. As an alternative, the well-known Burer-Monteiro approach can be used to cast the RPCA as a non-convex and non-smooth $\ell_1$ optimization problem with a significantly smaller number of variables. In this work, we show that the low-dimensional formulation of the symmetric and asymmetric positive rank-1 RPCA based on the Burer-Monteiro approach has benign landscape, i.e., 1) it does not have any spurious local solution, 2) has a unique global solution, and 3) its unique global solution coincides with the true components. An implication of this result is that simple local search algorithms are guaranteed to achieve a zero global optimality gap when directly applied to the low-dimensional formulation. Furthermore, we provide strong deterministic and probabilistic guarantees for the exact recovery of the true principal components. In particular, it is shown that a constant fraction of the measurements could be grossly corrupted and yet they would not create any spurious local solution.

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arXiv2019

Boundary Defense against Cyber Threat for Power System Operation

The operation of power grids is becoming increasingly data-centric. While the abundance of data could improve the efficiency of the system, it poses major reliability challenges. In particular, state estimation aims to learn the behavior of the network from data but an undetected attack on this problem could lead to a large-scale blackout. Nevertheless, understanding vulnerability of state estimation against cyber attacks has been hindered by the lack of tools studying the topological and data-analytic aspects of the network. Algorithmic robustness is of critical need to extract reliable information from abundant but untrusted grid data. We propose a robust state estimation framework that leverages network sparsity and data abundance. For a large-scale power grid, we quantify, analyze, and visualize the regions of the network prone to cyber attacks. We also propose an optimization-based graphical boundary defense mechanism to identify the border of the geographical area whose data has been manipulated. The proposed method does not allow a local attack to have a global effect on the data analysis of the entire network, which enhances the situational awareness of the grid especially in the face of adversity. The developed mathematical framework reveals key geometric and algebraic factors that can affect algorithmic robustness and is used to study the vulnerability of the U.S. power grid in this paper.

eess.SP↗

arXiv2019

Graphical Lasso and Thresholding: Equivalence and Closed-form Solutions

Graphical Lasso (GL) is a popular method for learning the structure of an undirected graphical model, which is based on an $l_1$ regularization technique. The objective of this paper is to compare the computationally-heavy GL technique with a numerically-cheap heuristic method that is based on simply thresholding the sample covariance matrix. To this end, two notions of sign-consistent and inverse-consistent matrices are developed, and then it is shown that the thresholding and GL methods are equivalent if: (i) the thresholded sample covariance matrix is both sign-consistent and inverse-consistent, and (ii) the gap between the largest thresholded and the smallest un-thresholded entries of the sample covariance matrix is not too small. By building upon this result, it is proved that the GL method---as a conic optimization problem---has an explicit closed-form solution if the thresholded sample covariance matrix has an acyclic structure. This result is then generalized to arbitrary sparse support graphs, where a formula is found to obtain an approximate solution of GL. Furthermore, it is shown that the approximation error of the derived explicit formula decreases exponentially fast with respect to the length of the minimum-length cycle of the sparsity graph. The developed results are demonstrated on synthetic data, functional MRI data, traffic flows for transportation networks, and massive randomly generated data sets. We show that the proposed method can obtain an accurate approximation of the GL for instances with the sizes as large as $80,000\times 80,000$ (more than 3.2 billion variables) in less than 30 minutes on a standard laptop computer running MATLAB, while other state-of-the-art methods do not converge within 4 hours.

stat.ML↗

arXiv2019

Sharp Restricted Isometry Bounds for the Inexistence of Spurious Local Minima in Nonconvex Matrix Recovery

Nonconvex matrix recovery is known to contain no spurious local minima under a restricted isometry property (RIP) with a sufficiently small RIP constant $δ$. If $δ$ is too large, however, then counterexamples containing spurious local minima are known to exist. In this paper, we introduce a proof technique that is capable of establishing sharp thresholds on $δ$ to guarantee the inexistence of spurious local minima. Using the technique, we prove that in the case of a rank-1 ground truth, an RIP constant of $δ<1/2$ is both necessary and sufficient for exact recovery from any arbitrary initial point (such as a random point). We also prove a local recovery result: given an initial point $x_{0}$ satisfying $f(x_{0})\le(1-δ)^{2}f(0)$, any descent algorithm that converges to second-order optimality guarantees exact recovery.

cs.LG↗

arXiv2019

Learning Sparse Dynamical Systems from a Single Sample Trajectory

This paper addresses the problem of identifying sparse linear time-invariant (LTI) systems from a single sample trajectory generated by the system dynamics. We introduce a Lasso-like estimator for the parameters of the system, taking into account their sparse nature. Assuming that the system is stable, or that it is equipped with an initial stabilizing controller, we provide sharp finite-time guarantees on the accurate recovery of both the sparsity structure and the parameter values of the system. In particular, we show that the proposed estimator can correctly identify the sparsity pattern of the system matrices with high probability, provided that the length of the sample trajectory exceeds a threshold. Furthermore, we show that this threshold scales polynomially in the number of nonzero elements in the system matrices, but logarithmically in the system dimensions --- this improves on existing sample complexity bounds for the sparse system identification problem. We further extend these results to obtain sharp bounds on the $\ell_{\infty}$-norm of the estimation error and show how different properties of the system---such as its stability level and \textit{mutual incoherency}---affect this bound. Finally, an extensive case study on power systems is presented to illustrate the performance of the proposed estimation method.

eess.SY↗

arXiv2018

Transient Stability Analysis of Power Systems via Occupation Measures

We propose the application of occupation measure theory to the classical problem of transient stability analysis for power systems. This enables the computation of certified inner and outer approximations for the region of attraction of a nominal operating point. In order to determine whether a post-disturbance point requires corrective actions to ensure stability, one would then simply need to check the sign of a polynomial evaluated at that point. Thus, computationally expensive dynamical simulations are only required for post-disturbance points in the region between the inner and outer approximations. We focus on the nonlinear swing equations but voltage dynamics could also be included. The proposed approach is formulated as a hierarchy of semidefinite programs stemming from an infinite-dimensional linear program in a measure space, with a natural dual sum-of-squares perspective. On the theoretical side, this paper lays the groundwork for exploiting the oscillatory structure of power systems by using Hermitian (instead of real) sums-of-squares and connects the proposed approach to recent results from algebraic geometry.

math.OC↗

arXiv2018

How Much Restricted Isometry is Needed In Nonconvex Matrix Recovery?

When the linear measurements of an instance of low-rank matrix recovery satisfy a restricted isometry property (RIP)---i.e. they are approximately norm-preserving---the problem is known to contain no spurious local minima, so exact recovery is guaranteed. In this paper, we show that moderate RIP is not enough to eliminate spurious local minima, so existing results can only hold for near-perfect RIP. In fact, counterexamples are ubiquitous: we prove that every x is the spurious local minimum of a rank-1 instance of matrix recovery that satisfies RIP. One specific counterexample has RIP constant $δ=1/2$, but causes randomly initialized stochastic gradient descent (SGD) to fail 12% of the time. SGD is frequently able to avoid and escape spurious local minima, but this empirical result shows that it can occasionally be defeated by their existence. Hence, while exact recovery guarantees will likely require a proof of no spurious local minima, arguments based solely on norm preservation will only be applicable to a narrow set of nearly-isotropic instances.

cs.LG↗

arXiv2018

A Fog Robotic System for Dynamic Visual Servoing

Cloud Robotics is a paradigm where distributed robots are connected to cloud services via networks to access unlimited computation power, at the cost of network communication. However, due to limitations such as network latency and variability, it is difficult to control dynamic, human compliant service robots directly from the cloud. In this work, by leveraging asynchronous protocol with a heartbeat signal, we combine cloud robotics with a smart edge device to build a Fog Robotic system. We use the system to enable robust teleoperation of a dynamic self-balancing robot from the cloud. We first use the system to pick up boxes from static locations, a task commonly performed in warehouse logistics. To make cloud teleoperation more efficient, we deploy image based visual servoing (IBVS) to perform box pickups automatically. Visual feedbacks, including apriltag recognition and tracking, are performed in the cloud to emulate a Fog Robotic object recognition system for IBVS. We demonstrate the feasibility of real-time dynamic automation system using this cloud-edge hybrid, which opens up possibilities of deploying dynamic robotic control with deep-learning recognition systems in Fog Robotics. Finally, we show that Fog Robotics enables the self-balancing service robot to pick up a box automatically from a person under unstructured environments.

cs.RO↗

arXiv2018

Sample Complexity of Sparse System Identification Problem

In this paper, we study the system identification problem for sparse linear time-invariant systems. We propose a sparsity promoting block-regularized estimator to identify the dynamics of the system with only a limited number of input-state data samples. We characterize the properties of this estimator under high-dimensional scaling, where the growth rate of the system dimension is comparable to or even faster than that of the number of available sample trajectories. In particular, using contemporary results on high-dimensional statistics, we show that the proposed estimator results in a small element-wise error, provided that the number of sample trajectories is above a threshold. This threshold depends polynomially on the size of each block and the number of nonzero elements at different rows of input and state matrices, but only logarithmically on the system dimension. A by-product of this result is that the number of sample trajectories required for sparse system identification is significantly smaller than the dimension of the system. Furthermore, we show that, unlike the recently celebrated least-squares estimators for system identification problems, the method developed in this work is capable of \textit{exact recovery} of the underlying sparsity structure of the system with the aforementioned number of data samples. Extensive case studies on synthetically generated systems, physical mass-spring networks, and multi-agent systems are offered to demonstrate the effectiveness of the proposed method.

eess.SY↗

arXiv2018

Large-Scale Sparse Inverse Covariance Estimation via Thresholding and Max-Det Matrix Completion

The sparse inverse covariance estimation problem is commonly solved using an $\ell_{1}$-regularized Gaussian maximum likelihood estimator known as "graphical lasso", but its computational cost becomes prohibitive for large data sets. A recent line of results showed--under mild assumptions--that the graphical lasso estimator can be retrieved by soft-thresholding the sample covariance matrix and solving a maximum determinant matrix completion (MDMC) problem. This paper proves an extension of this result, and describes a Newton-CG algorithm to efficiently solve the MDMC problem. Assuming that the thresholded sample covariance matrix is sparse with a sparse Cholesky factorization, we prove that the algorithm converges to an $ε$-accurate solution in $O(n\log(1/ε))$ time and $O(n)$ memory. The algorithm is highly efficient in practice: we solve the associated MDMC problems with as many as 200,000 variables to 7-9 digits of accuracy in less than an hour on a standard laptop computer running MATLAB.

stat.ML↗

arXiv2017

Sparse Inverse Covariance Estimation for Chordal Structures

In this paper, we consider the Graphical Lasso (GL), a popular optimization problem for learning the sparse representations of high-dimensional datasets, which is well-known to be computationally expensive for large-scale problems. Recently, we have shown that the sparsity pattern of the optimal solution of GL is equivalent to the one obtained from simply thresholding the sample covariance matrix, for sparse graphs under different conditions. We have also derived a closed-form solution that is optimal when the thresholded sample covariance matrix has an acyclic structure. As a major generalization of the previous result, in this paper we derive a closed-form solution for the GL for graphs with chordal structures. We show that the GL and thresholding equivalence conditions can significantly be simplified and are expected to hold for high-dimensional problems if the thresholded sample covariance matrix has a chordal structure. We then show that the GL and thresholding equivalence is enough to reduce the GL to a maximum determinant matrix completion problem and drive a recursive closed-form solution for the GL when the thresholded sample covariance matrix has a chordal structure. For large-scale problems with up to 450 million variables, the proposed method can solve the GL problem in less than 2 minutes, while the state-of-the-art methods converge in more than 2 hours.

stat.ML↗

arXiv2017

Conic Optimization Theory: Convexification Techniques and Numerical Algorithms

Optimization is at the core of control theory and appears in several areas of this field, such as optimal control, distributed control, system identification, robust control, state estimation, model predictive control and dynamic programming. The recent advances in various topics of modern optimization have also been revamping the area of machine learning. Motivated by the crucial role of optimization theory in the design, analysis, control and operation of real-world systems, this tutorial paper offers a detailed overview of some major advances in this area, namely conic optimization and its emerging applications. First, we discuss the importance of conic optimization in different areas. Then, we explain seminal results on the design of hierarchies of convex relaxations for a wide range of nonconvex problems. Finally, we study different numerical algorithms for large-scale conic optimization problems.

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