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In this paper, we consider the problem of determining the presence of a given signal in a high-dimensional observation with unknown covariance matrix by using an adaptive matched filter. Traditionally such filters are formed from the sample covariance matrix of some given training data, but, as is well-known, the performance of such filters is poor when the number of training data $n$ is not much larger than the data dimension $p$. We thus seek a covariance estimator to replace sample covariance. To account for the fact that $n$ and $p$ may be of comparable size, we adopt the "large-dimensional asymptotic model" in which $n$ and $p$ go to infinity in a fixed ratio. Under this assumption, we identify a covariance estimator that is asymptotically detection-theoretic optimal within a general shrinkage class inspired by C. Stein, and we give consistent estimates for conditional false-alarm and detection rate of the corresponding adaptive matched filter.

In this paper, we investigate the homothetic point enclosure problem: given a set $S$ of $n$ triangles with sides parallel to three fixed directions, find a data structure for $S$ that can report all the triangles of $S$ that contain a query point efficiently. The problem is "inverse" of the homothetic range search problem. We present an $O(n\log n)$ space solution that supports the queries in $O(\log n + k)$ time, where $k$ is the output size. The preprocessing time is $O(n\log n)$. The same results also hold for homothetic polygons.

With technological advancement, drone has emerged as unmanned aerial vehicle that can be controlled by humans to fly or reach a destination. This may be autonomous as well, where the drone itself is intelligent enough to find a shortest obstacle-free path to reach the destination from a designated source. Be it a planned smart city or even a wreckage site affected by natural calamity, we may imagine the buildings, any surface-erected structure or other blockage as obstacles for the drone to fly in a straight line-of-sight path. To address such path-planning of drones, the bird's eye-view of the whole landscape is first transformed to a graph of grid-cells, where some are occupied to indicate the obstacles and some are free to indicate the free path. We propose a method to find out the shortest obstacle-free path in the coordinate system guided by GPS. The autonomous drone (AutoDrone) will thus be able to move from one place to another along the shortest path, without colliding into hindrances, while traveling in a two-dimensional space. Heuristics to extend this to long journeys and 3D space are also elaborated. Our approach can be especially beneficial in rescue operations and fast delivery or pick-up in an energy-efficient way, where our algorithm will help in finding out the shortest path and angle along which it should fly. Experiments are done on different scenarios of map layouts and obstacle positions to understand the shortest feasible route, computed by the autonomous drone.

In this paper, we formulate and study substructuring type algorithm for the Cahn-Hilliard (CH) equation, which was originally proposed to describe the phase separation phenomenon for binary melted alloy below the critical temperature and since then it has appeared in many fields ranging from tumour growth simulation, image processing, thin liquid films, population dynamics etc. Being a non-linear equation, it is important to develop robust numerical techniques to solve the CH equation. Here we present the formulation of Dirichlet-Neumann (DN) and Neumann-Neumann (NN) methods applied to CH equation and study their convergence behaviour. We consider the domain-decomposition based DN and NN methods in one and two space dimension for two subdomains and extend the study for multi-subdomain setting for NN method. We verify our findings with numerical results.

The conditions for a Runge--Kutta method to be of order $p$ with $p\ge 5$ for a scalar non-autonomous problem are a proper subset of the order conditions for a vector problem. Nevertheless, Runge--Kutta methods that were derived historically only for scalar problems happened to be of the same order for vector problems. We relate the order conditions for scalar problems to factorisations of the Runge--Kutta trees into "atomic stumps" and enumerate those conditions up to $p=20$. Using a special search procedure over unsatisfied order conditions, new Runge--Kutta methods of "ambiguous orders" five and six are constructed. These are used to verify the validity of the results.

Imitation learning seeks to circumvent the difficulty in designing proper reward functions for training agents by utilizing expert behavior. With environments modeled as Markov Decision Processes (MDP), most of the existing imitation algorithms are contingent on the availability of expert demonstrations in the same MDP as the one in which a new imitation policy is to be learned. In this paper, we study the problem of how to imitate tasks when there exist discrepancies between the expert and agent MDP. These discrepancies across domains could include differing dynamics, viewpoint, or morphology; we present a novel framework to learn correspondences across such domains. Importantly, in contrast to prior works, we use unpaired and unaligned trajectories containing only states in the expert domain, to learn this correspondence. We utilize a cycle-consistency constraint on both the state space and a domain agnostic latent space to do this. In addition, we enforce consistency on the temporal position of states via a normalized position estimator function, to align the trajectories across the two domains. Once this correspondence is found, we can directly transfer the demonstrations on one domain to the other and use it for imitation. Experiments across a wide variety of challenging domains demonstrate the efficacy of our approach.

Graph convolution is the core of most Graph Neural Networks (GNNs) and usually approximated by message passing between direct (one-hop) neighbors. In this work, we remove the restriction of using only the direct neighbors by introducing a powerful, yet spatially localized graph convolution: Graph diffusion convolution (GDC). GDC leverages generalized graph diffusion, examples of which are the heat kernel and personalized PageRank. It alleviates the problem of noisy and often arbitrarily defined edges in real graphs. We show that GDC is closely related to spectral-based models and thus combines the strengths of both spatial (message passing) and spectral methods. We demonstrate that replacing message passing with graph diffusion convolution consistently leads to significant performance improvements across a wide range of models on both supervised and unsupervised tasks and a variety of datasets. Furthermore, GDC is not limited to GNNs but can trivially be combined with any graph-based model or algorithm (e.g. spectral clustering) without requiring any changes to the latter or affecting its computational complexity. Our implementation is available online.

We consider the task of learning the parameters of a {\em single} component of a mixture model, for the case when we are given {\em side information} about that component, we call this the "search problem" in mixture models. We would like to solve this with computational and sample complexity lower than solving the overall original problem, where one learns parameters of all components. Our main contributions are the development of a simple but general model for the notion of side information, and a corresponding simple matrix-based algorithm for solving the search problem in this general setting. We then specialize this model and algorithm to four common scenarios: Gaussian mixture models, LDA topic models, subspace clustering, and mixed linear regression. For each one of these we show that if (and only if) the side information is informative, we obtain parameter estimates with greater accuracy, and also improved computation complexity than existing moment based mixture model algorithms (e.g. tensor methods). We also illustrate several natural ways one can obtain such side information, for specific problem instances. Our experiments on real data sets (NY Times, Yelp, BSDS500) further demonstrate the practicality of our algorithms showing significant improvement in runtime and accuracy.

A recommender system aims to recommend items that a user is interested in among many items. The need for the recommender system has been expanded by the information explosion. Various approaches have been suggested for providing meaningful recommendations to users. One of the proposed approaches is to consider a recommender system as a Markov decision process (MDP) problem and try to solve it using reinforcement learning (RL). However, existing RL-based methods have an obvious drawback. To solve an MDP in a recommender system, they encountered a problem with the large number of discrete actions that bring RL to a larger class of problems. In this paper, we propose a novel RL-based recommender system. We formulate a recommender system as a gridworld game by using a biclustering technique that can reduce the state and action space significantly. Using biclustering not only reduces space but also improves the recommendation quality effectively handling the cold-start problem. In addition, our approach can provide users with some explanation why the system recommends certain items. Lastly, we examine the proposed algorithm on a real-world dataset and achieve a better performance than the widely used recommendation algorithm.

In this paper, we study the optimal convergence rate for distributed convex optimization problems in networks. We model the communication restrictions imposed by the network as a set of affine constraints and provide optimal complexity bounds for four different setups, namely: the function $F(\xb) \triangleq \sum_{i=1}^{m}f_i(\xb)$ is strongly convex and smooth, either strongly convex or smooth or just convex. Our results show that Nesterov's accelerated gradient descent on the dual problem can be executed in a distributed manner and obtains the same optimal rates as in the centralized version of the problem (up to constant or logarithmic factors) with an additional cost related to the spectral gap of the interaction matrix. Finally, we discuss some extensions to the proposed setup such as proximal friendly functions, time-varying graphs, improvement of the condition numbers.