Tuhin Sahai

Hearing the clusters of a graph: A distributed algorithm

We propose a novel distributed algorithm to cluster graphs. The algorithm recovers the solution obtained from spectral clustering without the need for expensive eigenvalue/vector computations. We prove that, by propagating waves through the graph, a local fast Fourier transform yields the local component of every eigenvector of the Laplacian matrix, thus providing clustering information. For large graphs, the proposed algorithm is orders of magnitude faster than random walk based approaches. We prove the equivalence of the proposed algorithm to spectral clustering and derive convergence rates. We demonstrate the benefit of using this decentralized clustering algorithm for community detection in social graphs, accelerating distributed estimation in sensor networks and efficient computation of distributed multi-agent search strategies.

Wave equation based algorithm for distributed eigenvector computation

We propose a novel distributed algorithm to compute eigenvectors and eigenvalues of the graph Laplacian matrix L. We prove that, by propagating waves through the graph, a local fast Fourier transform yields the local component of every eigenvector of L. For large graphs, the proposed algorithm is orders of magnitude faster than random walk based approaches. We prove the equivalence of the proposed algorithm to eigenvector computation and derive convergence rates. We also demonstrate its utility on a distributed estimation example.

Scalable approach to uncertainty quantification and robust design of interconnected dynamical systems

Development of robust dynamical systems and networks such as autonomous aircraft systems capable of accomplishing complex missions faces challenges due to the dynamically evolving uncertainties coming from model uncertainties, necessity to operate in a hostile cluttered urban environment, and the distributed and dynamic nature of the communication and computation resources. Model-based robust design is difficult because of the complexity of the hybrid dynamic models including continuous vehicle dynamics, the discrete models of computations and communications, and the size of the problem. We will overview recent advances in methodology and tools to model, analyze, and design robust autonomous aerospace systems operating in uncertain environment, with stress on efficient uncertainty quantification and robust design using the case studies of the mission including model-based target tracking and search, and trajectory planning in uncertain urban environment. To show that the methodology is generally applicable to uncertain dynamical systems, we will also show examples of application of the new methods to efficient uncertainty quantification of energy usage in buildings, and stability assessment of interconnected power networks.

Uncertainty quantification in hybrid dynamical systems

Uncertainty quantification (UQ) techniques are frequently used to ascertain output variability in systems with parametric uncertainty. Traditional algorithms for UQ are either system-agnostic and slow (such as Monte Carlo) or fast with stringent assumptions on smoothness (such as polynomial chaos and Quasi-Monte Carlo). In this work, we develop a fast UQ approach for hybrid dynamical systems by extending the polynomial chaos methodology to these systems. To capture discontinuities, we use a wavelet-based Wiener-Haar expansion. We develop a boundary layer approach to propagate uncertainty through separable reset conditions. The above methods are demonstrated on example problems.

Polynomial chaos based uncertainty quantification in Hamiltonian and chaotic systems

Polynomial chaos is a powerful technique for propagating uncertainty through ordinary and partial differential equations. Random variables are expanded in terms of orthogonal polynomials and differential equations are derived for the coefficients. Here we study the structure and dynamics of these differential equations when the original system has Hamiltonian structure or displays chaotic dynamics. In particular, we prove that the differential equations for the expansion coefficients in generalized polynomial chaos expansions of Hamiltonian systems retain the Hamiltonian structure relative to the ensemble average Hamiltonian. Additionally, using the forced Duffing oscillator as an example, we demonstrate that when the original dynamical system displays chaotic dynamics, the resulting dynamical system from polynomial chaos also displays chaotic dynamics, limiting its applicability.

On Learning Sparse Boolean Formulae for Explaining AI Decisions

In this paper, we consider the problem of learning Boolean formulae from examples obtained by actively querying an oracle that can label these examplesz as either positive or negative. This problem has received attention in both machine learning as well as formal methods communities, and it has been shown to have exponential worst-case complexity in the general case as well as for many restrictions. In this paper, we focus on learning sparse Boolean formulae which depend on only a small (but unknown) subset of the overall vocabulary of atomic propositions. We propose an efficient algorithm to learn these sparse Boolean formulae with a given confidence. This assumption of sparsity is motivated by the problem of mining explanations for decisions made by artificially intelligent (AI) algorithms, where the explanation of individual decisions may depend on a small but unknown subset of all the inputs to the algorithm. We demonstrate the use of our algorithm in automatically generating explanations of these decisions. These explanations will make intelligent systems more understandable and accountable to human users, facilitate easier audits and provide diagnostic information in the case of failure. The proposed approach treats the AI algorithm as a black-box oracle; hence, it is broadly applicable and agnostic to the specific AI algorithm. We illustrate the practical effectiveness of our approach on a diverse set of case studies.