Compared to conventional cars, electric vehicles (EVs) still suffer from considerably shorter cruising ranges. Combined with the sparsity of battery loading stations, the complete transition to E-mobility still seems a long way to go. In this paper, we consider the problem of placing as few loading stations as possible so that on any shortest path there are sufficiently many not to run out of energy. We show how to model this problem and introduce heuristics which provide close-to-optimal solutions even in large road networks.

We propose a globally convergent alternating minimization (AM) algorithm for image reconstruction in transmission tomography, which extends automatic relevance determination (ARD) to Poisson noise models with Beer's law. The algorithm promotes solutions that are sparse in the pixel/voxel-differences domain by introducing additional latent variables, one for each pixel/voxel, and then learning these variables from the data using a hierarchical Bayesian model. Importantly, the proposed AM algorithm is free of any tuning parameters with image quality comparable to standard penalized likelihood methods. Our algorithm exploits optimization transfer principles which reduce the problem into parallel 1D optimization tasks (one for each pixel/voxel), making the algorithm feasible for large-scale problems. This approach considerably reduces the computational bottleneck of ARD associated with the posterior variances. Positivity constraints inherent in transmission tomography problems are also enforced. We demonstrate the performance of the proposed algorithm for x-ray computed tomography using synthetic and real-world datasets. The algorithm is shown to have much better performance than prior ARD algorithms based on approximate Gaussian noise models, even for high photon flux.

We present a sparse knowledge gradient (SpKG) algorithm for adaptively selecting the targeted regions within a large RNA molecule to identify which regions are most amenable to interactions with other molecules. Experimentally, such regions can be inferred from fluorescence measurements obtained by binding a complementary probe with fluorescence markers to the targeted regions. We use a biophysical model which shows that the fluorescence ratio under the log scale has a sparse linear relationship with the coefficients describing the accessibility of each nucleotide, since not all sites are accessible (due to the folding of the molecule). The SpKG algorithm uniquely combines the Bayesian ranking and selection problem with the frequentist $\ell_1$ regularized regression approach Lasso. We use this algorithm to identify the sparsity pattern of the linear model as well as sequentially decide the best regions to test before experimental budget is exhausted. Besides, we also develop two other new algorithms: batch SpKG algorithm, which generates more suggestions sequentially to run parallel experiments; and batch SpKG with a procedure which we call length mutagenesis. It dynamically adds in new alternatives, in the form of types of probes, are created by inserting, deleting or mutating nucleotides within existing probes. In simulation, we demonstrate these algorithms on the Group I intron (a mid-size RNA molecule), showing that they efficiently learn the correct sparsity pattern, identify the most accessible region, and outperform several other policies.

Approximate Newton methods are a standard optimization tool which aim to maintain the benefits of Newton's method, such as a fast rate of convergence, whilst alleviating its drawbacks, such as computationally expensive calculation or estimation of the inverse Hessian. In this work we investigate approximate Newton methods for policy optimization in Markov Decision Processes (MDPs). We first analyse the structure of the Hessian of the objective function for MDPs. We show that, like the gradient, the Hessian exhibits useful structure in the context of MDPs and we use this analysis to motivate two Gauss-Newton Methods for MDPs. Like the Gauss-Newton method for non-linear least squares, these methods involve approximating the Hessian by ignoring certain terms in the Hessian which are difficult to estimate. The approximate Hessians possess desirable properties, such as negative definiteness, and we demonstrate several important performance guarantees including guaranteed ascent directions, invariance to affine transformation of the parameter space, and convergence guarantees. We finally provide a unifying perspective of key policy search algorithms, demonstrating that our second Gauss-Newton algorithm is closely related to both the EM-algorithm and natural gradient ascent applied to MDPs, but performs significantly better in practice on a range of challenging domains.

Gaussian process (GP) regression is a powerful tool for building predictive models for spatial systems. However, it does not scale efficiently for large datasets. Particularly, for high-dimensional spatial datasets, i.e., spatial datasets that contain exogenous variables, the performance of GP regression further deteriorates. This paper presents the Sparse Pseudo-input Local Kriging (SPLK) which approximates the full GP for spatial datasets with exogenous variables. SPLK employs orthogonal cuts which decompose the domain into smaller subdomains and then applies a sparse approximation of the full GP in each subdomain. We obtain the continuity of the global predictor by imposing continuity constraints on the boundaries of the neighboring subdomains. The domain decomposition scheme applies independent covariance structures in each region, and as a result, SPLK captures heterogeneous covariance structures. SPLK achieves computational efficiency by utilizing sparse approximation in each subdomain which enables SPLK to accommodate large subdomains that contain many data points and possess a homogenous covariance structure. We Apply the proposed method to real and simulated datasets. We conclude that the combination of orthogonal cuts and sparse approximation makes the proposed method an efficient algorithm for high-dimensional large spatial datasets.

Calibration refers to the process of adjusting features of a computational model that are not observed in the physical process so that the model matches the real process. We propose a framework for calibration when the unobserved features, i.e. calibration parameters, do not assume a single value, but are functionally dependent on other inputs. We demonstrate that this problem is curve to surface matching where the matched curve does not possess the same length as the original curve. Therefore, we perform non-isometric matching of a curve to a surface. Since in practical applications we do not observe a continuous curve but a sample of data points, we use a graph-theoretic approach to solve this matching of incomplete data. We define a graph structure in which the nodes are selected from the incomplete surface and the weights of the edges are decided based on the response values of the curve and surface. We show that the problem of non-isometric incomplete curve to surface matching is a shortest path problem in a directed acyclic graph. We apply the proposed method, graph-theoretic non-isometric matching, to real and synthetic data and demonstrate that the proposed method improves the prediction accuracy in functional calibration.

We consider the following multi-component sparse PCA problem: given a set of data points, we seek to extract a small number of sparse components with disjoint supports that jointly capture the maximum possible variance. These components can be computed one by one, repeatedly solving the single-component problem and deflating the input data matrix, but as we show this greedy procedure is suboptimal. We present a novel algorithm for sparse PCA that jointly optimizes multiple disjoint components. The extracted features capture variance that lies within a multiplicative factor arbitrarily close to 1 from the optimal. Our algorithm is combinatorial and computes the desired components by solving multiple instances of the bipartite maximum weight matching problem. Its complexity grows as a low order polynomial in the ambient dimension of the input data matrix, but exponentially in its rank. However, it can be effectively applied on a low-dimensional sketch of the data; this allows us to obtain polynomial-time approximation guarantees via spectral bounds. We evaluate our algorithm on real data-sets and empirically demonstrate that in many cases it outperforms existing, deflation-based approaches.

We study the convergence properties of the VR-PCA algorithm introduced by \cite{shamir2015stochastic} for fast computation of leading singular vectors. We prove several new results, including a formal analysis of a block version of the algorithm, and convergence from random initialization. We also make a few observations of independent interest, such as how pre-initializing with just a single exact power iteration can significantly improve the runtime of stochastic methods, and what are the convexity and non-convexity properties of the underlying optimization problem.

This paper is concerned with a lesser-studied problem in the context of model-based, uncertainty quantification (UQ), that of optimization/design/control under uncertainty. The solution of such problems is hindered not only by the usual difficulties encountered in UQ tasks (e.g. the high computational cost of each forward simulation, the large number of random variables) but also by the need to solve a nonlinear optimization problem involving large numbers of design variables and potentially constraints. We propose a framework that is suitable for a large class of such problems and is based on the idea of recasting them as probabilistic inference tasks. To that end, we propose a Variational Bayesian (VB) formulation and an iterative VB-Expectation-Maximization scheme that is also capable of identifying a low-dimensional set of directions in the design space, along which, the objective exhibits the largest sensitivity. We demonstrate the validity of the proposed approach in the context of two numerical examples involving $\mathcal{O}(10^3)$ random and design variables. In all cases considered the cost of the computations in terms of calls to the forward model was of the order $\mathcal{O}(10^2)$. The accuracy of the approximations provided is assessed by appropriate information-theoretic metrics.

In this era of large-scale data, distributed systems built on top of clusters of commodity hardware provide cheap and reliable storage and scalable processing of massive data. Here, we review recent work on developing and implementing randomized matrix algorithms in large-scale parallel and distributed environments. Randomized algorithms for matrix problems have received a great deal of attention in recent years, thus far typically either in theory or in machine learning applications or with implementations on a single machine. Our main focus is on the underlying theory and practical implementation of random projection and random sampling algorithms for very large very overdetermined (i.e., overconstrained) $\ell_1$ and $\ell_2$ regression problems. Randomization can be used in one of two related ways: either to construct sub-sampled problems that can be solved, exactly or approximately, with traditional numerical methods; or to construct preconditioned versions of the original full problem that are easier to solve with traditional iterative algorithms. Theoretical results demonstrate that in near input-sparsity time and with only a few passes through the data one can obtain very strong relative-error approximate solutions, with high probability. Empirical results highlight the importance of various trade-offs (e.g., between the time to construct an embedding and the conditioning quality of the embedding, between the relative importance of computation versus communication, etc.) and demonstrate that $\ell_1$ and $\ell_2$ regression problems can be solved to low, medium, or high precision in existing distributed systems on up to terabyte-sized data.