In this task, you are given a part of an article. Your task is to generate headline (title) for this text. Preferred headlines are under fifteen words.

Input: Consider Input: Consider a weighted or unweighted k-nearest neighbor graph that has been built on n data points drawn randomly according to some density p on R. We study the convergence of the shortest path distance in such graphs as the sample size tends to infinity. We prove that for unweighted kNN graphs, this distance converges to an unpleasant distance function on the underlying space whose properties are detrimental to machine learning. We also study the behavior of the shortest path distance in weighted kNN graphs.

Output: Shortest path distance in random k-nearest neighbor graphs


Input: Consider Input: Path planning for multiple robots is well studied in the AI and robotics communities. For a given discretized environment, robots need to find collision-free paths to a set of specified goal locations. Robots can be fully anonymous, non-anonymous, or organized in groups. Although powerful solvers for this abstract problem exist, they make simplifying assumptions by ignoring kinematic constraints, making it difficult to use the resulting plans on actual robots. In this paper, we present a solution which takes kinematic constraints, such as maximum velocities, into account, while guaranteeing a user-specified minimum safety distance between robots. We demonstrate our approach in simulation and on real robots in 2D and 3D environments.

Output: Path Planning With Kinematic Constraints For Robot Groups


Input: Consider Input: This is an overview paper written in style of research proposal. In recent years we introduced a general framework for large-scale unconstrained optimization – Sequential Subspace Optimization (SESOP) and demonstrated its usefulness for sparsity-based signal/image denoising, deconvolution, compressive sensing, computed tomography, diffraction imaging, support vector machines. We explored its combination with Parallel Coordinate Descent and Separable Surrogate Function methods, obtaining state of the art results in above-mentioned areas. There are several methods, that are faster than plain SESOP under specific conditions: Trust region Newton method for problems with easily invertible Hessian matrix; Truncated Newton method when fast multiplication by Hessian is available; Stochastic optimization methods for problems with large stochastic-type data; Multigrid methods for problems with nested multilevel structure. Each of these methods can be further improved by merge with SESOP. One can also accelerate Augmented Lagrangian method for constrained optimization problems and Alternating Direction Method of Multipliers for problems with separable objective function and non-separable constraints.


Output: Speeding-Up Convergence via Sequential Subspace Optimization: Current State and Future Directions