Venkat Chandrasekaran

Anshuman Pradhan, Kyra H. Adams, Venkat Chandrasekaran et al.

Modeling groundwater levels continuously across California's Central Valley (CV) hydrological system is challenging due to low-quality well data which is sparsely and noisily sampled across time and space. The lack of consistent well data makes it difficult to evaluate the impact of 2017 and 2019 wet years on CV groundwater following a severe drought during 2012-2015. A novel machine learning method is formulated for modeling groundwater levels by learning from a 3D lithological texture model of the CV aquifer. The proposed formulation performs multivariate regression by combining Gaussian processes (GP) and deep neural networks (DNN). The hierarchical modeling approach constitutes training the DNN to learn a lithologically informed latent space where non-parametric regression with GP is performed. We demonstrate the efficacy of GP-DNN regression for modeling non-stationary features in the well data with fast and reliable uncertainty quantification, as validated to be statistically consistent with the empirical data distribution from 90 blind wells across CV. We show how the model predictions may be used to supplement hydrological understanding of aquifer responses in basins with irregular well data. Our results indicate that on average the 2017 and 2019 wet years in California were largely ineffective in replenishing the groundwater loss caused during previous drought years.

Eray Unsal Atay, Venkat Chandrasekaran, Victoria Kostina

We study the rate-cost tradeoff in rate-limited control of general stochastic control systems, including nonlinear systems, over a finite horizon. At each time step, an encoder observes the state and transmits a description to a controller, which then selects the control action. For an average control-cost threshold $D$, we characterize the minimum achievable communication rate $R_n(D)$ via a nonasymptotic bound: $R_n(D)$ lies within an additive logarithmic gap of the optimal value of a directed-information minimization $F_n(D)$, namely, we show that $F_n(D) \le R_n(D) \le F_n(D)+\log \bigl(F_n(D)+3.4\bigr)+2+\frac{1}{n}$, in bits. This establishes directed information as the operationally relevant quantity governing rate-limited control, thereby broadening its utility beyond its previously established roles in causal source coding and linear quadratic Gaussian (LQG) control to general nonlinear control systems. We prove the upper bound constructively by building an encoding-and-control policy using the strong functional representation lemma at each time step. As special cases of our setting, our framework yields nonasymptotic bounds for sequential (causal) rate-distortion and LQG control.

Eliza O'Reilly, Venkat Chandrasekaran

Convex regression is the problem of fitting a convex function to a data set consisting of input-output pairs. We present a new approach to this problem called spectrahedral regression, in which we fit a spectrahedral function to the data, i.e. a function that is the maximum eigenvalue of an affine matrix expression of the input. This method represents a significant generalization of polyhedral (also called max-affine) regression, in which a polyhedral function (a maximum of a fixed number of affine functions) is fit to the data. We prove bounds on how well spectrahedral functions can approximate arbitrary convex functions via statistical risk analysis. We also analyze an alternating minimization algorithm for the non-convex optimization problem of fitting the best spectrahedral function to a given data set. We show that this algorithm converges geometrically with high probability to a small ball around the optimal parameter given a good initialization. Finally, we demonstrate the utility of our approach with experiments on synthetic data sets as well as real data arising in applications such as economics and engineering design.

Armeen Taeb, Parikshit Shah, Venkat Chandrasekaran

Fitting a graphical model to a collection of random variables given sample observations is a challenging task if the observed variables are influenced by latent variables, which can induce significant confounding statistical dependencies among the observed variables. We present a new convex relaxation framework based on regularized conditional likelihood for latent-variable graphical modeling in which the conditional distribution of the observed variables conditioned on the latent variables is given by an exponential family graphical model. In comparison to previously proposed tractable methods that proceed by characterizing the marginal distribution of the observed variables, our approach is applicable in a broader range of settings as it does not require knowledge about the specific form of distribution of the latent variables and it can be specialized to yield tractable approaches to problems in which the observed data are not well-modeled as Gaussian. We demonstrate the utility and flexibility of our framework via a series of numerical experiments on synthetic as well as real data.

Yong Sheng Soh, Venkat Chandrasekaran

Regularization techniques are widely employed in optimization-based approaches for solving ill-posed inverse problems in data analysis and scientific computing. These methods are based on augmenting the objective with a penalty function, which is specified based on prior domain-specific expertise to induce a desired structure in the solution. We consider the problem of learning suitable regularization functions from data in settings in which precise domain knowledge is not directly available. Previous work under the title of `dictionary learning' or `sparse coding' may be viewed as learning a regularization function that can be computed via linear programming. We describe generalizations of these methods to learn regularizers that can be computed and optimized via semidefinite programming. Our framework for learning such semidefinite regularizers is based on obtaining structured factorizations of data matrices, and our algorithmic approach for computing these factorizations combines recent techniques for rank minimization problems along with an operator analog of Sinkhorn scaling. Under suitable conditions on the input data, our algorithm provides a locally linearly convergent method for identifying the correct regularizer that promotes the type of structure contained in the data. Our analysis is based on the stability properties of Operator Sinkhorn scaling and their relation to geometric aspects of determinantal varieties (in particular tangent spaces with respect to these varieties). The regularizers obtained using our framework can be employed effectively in semidefinite programming relaxations for solving inverse problems.

Nikolai Matni, Venkat Chandrasekaran

When designing controllers for large-scale systems, the architectural aspects of the controller such as the placement of actuators, sensors, and the communication links between them can no longer be taken as given. The task of designing this architecture is now as important as the design of the control laws themselves. By interpreting controller synthesis (in a model matching setup) as the solution of a particular linear inverse problem, we view the challenge of obtaining a controller with a desired architecture as one of finding a structured solution to an inverse problem. Building on this conceptual connection, we formulate and analyze a framework called \textit{Regularization for Design (RFD)}, in which we augment the variational formulations of controller synthesis problems with convex penalty functions that induce a desired controller architecture. The resulting regularized formulations are convex optimization problems that can be solved efficiently, these convex programs provide a unified computationally tractable approach for the simultaneous co-design of a structured optimal controller and the actuation, sensing and communication architecture required to implement it. Further, these problems are natural control-theoretic analogs of prominent approaches such as the Lasso, the Group Lasso, the Elastic Net, and others that are employed in statistical modeling. In analogy to that literature, we show that our approach identifies optimally structured controllers under a suitable condition on a "signal-to-noise" type ratio.

Venkat Chandrasekaran, Nathan Srebro, Prahladh Harsha

It is well-known that inference in graphical models is hard in the worst case, but tractable for models with bounded treewidth. We ask whether treewidth is the only structural criterion of the underlying graph that enables tractable inference. In other words, is there some class of structures with unbounded treewidth in which inference is tractable? Subject to a combinatorial hypothesis due to Robertson et al. (1994), we show that low treewidth is indeed the only structural restriction that can ensure tractability. Thus, even for the "best case" graph structure, there is no inference algorithm with complexity polynomial in the treewidth.