Empirical Bounds on Linear Regions of Deep Rectifier Networks

We can compare the expressiveness of neural networks that use rectified linear units (ReLUs) by the number of linear regions, which reflect the number of pieces of the piecewise linear functions modeled by such networks. However, enumerating these regions is prohibitive and the known analytical bounds are identical for networks with same dimensions. In this work, we approximate the number of linear regions through empirical bounds based on features of the trained network and probabilistic inference. Our first contribution is a method to sample the activation patterns defined by ReLUs using universal hash functions. This method is based on a Mixed-Integer Linear Programming (MILP) formulation of the network and an algorithm for probabilistic lower bounds of MILP solution sets that we call MIPBound, which is considerably faster than exact counting and reaches values in similar orders of magnitude. Our second contribution is a tighter activation-based bound for the maximum number of linear regions, which is particularly stronger in networks with narrow layers. Combined, these bounds yield a fast proxy for the number of linear regions of a deep neural network.Comment: AAAI 202

High-Performance and Tunable Stereo Reconstruction

Traditional stereo algorithms have focused their efforts on reconstruction quality and have largely avoided prioritizing for run time performance. Robots, on the other hand, require quick maneuverability and effective computation to observe its immediate environment and perform tasks within it. In this work, we propose a high-performance and tunable stereo disparity estimation method, with a peak frame-rate of 120Hz (VGA resolution, on a single CPU-thread), that can potentially enable robots to quickly reconstruct their immediate surroundings and maneuver at high-speeds. Our key contribution is a disparity estimation algorithm that iteratively approximates the scene depth via a piece-wise planar mesh from stereo imagery, with a fast depth validation step for semi-dense reconstruction. The mesh is initially seeded with sparsely matched keypoints, and is recursively tessellated and refined as needed (via a resampling stage), to provide the desired stereo disparity accuracy. The inherent simplicity and speed of our approach, with the ability to tune it to a desired reconstruction quality and runtime performance makes it a compelling solution for applications in high-speed vehicles.Comment: Accepted to International Conference on Robotics and Automation (ICRA) 2016; 8 pages, 5 figure

Submodular Function Maximization for Group Elevator Scheduling

We propose a novel approach for group elevator scheduling by formulating it as the maximization of submodular function under a matroid constraint. In particular, we propose to model the total waiting time of passengers using a quadratic Boolean function. The unary and pairwise terms in the function denote the waiting time for single and pairwise allocation of passengers to elevators, respectively. We show that this objective function is submodular. The matroid constraints ensure that every passenger is allocated to exactly one elevator. We use a greedy algorithm to maximize the submodular objective function, and derive provable guarantees on the optimality of the solution. We tested our algorithm using Elevate 8, a commercial-grade elevator simulator that allows simulation with a wide range of elevator settings. We achieve significant improvement over the existing algorithms.Comment: 10 pages; 2017 International Conference on Automated Planning and Scheduling (ICAPS