Integrated gradients is prevalent within machine learning to address the black-box problem of neural networks. The explanations given by integrated gradients depend on a choice of base-point. The choice of base-point is not a priori obvious and can lead to drastically different explanations. There is a longstanding hypothesis that data lies on a low dimensional Riemannian manifold. The quality of explanations on a manifold can be measured by the extent to which an explanation for a point lies in its tangent space. In this work, we propose that the base-point should be chosen such that it maximises the tangential alignment of the explanation. We formalise the notion of tangential alignment and provide theoretical conditions under which a base-point choice will provide explanations lying in the tangent space. We demonstrate how to approximate the optimal base-point on several well-known image classification datasets. Furthermore, we compare the optimal base-point choice with common base-points and three gradient explainability models.

We present a decentralized control algorithm for a robotic swarm given the task of encapsulating static and moving targets in a bounded unknown environment. We consider minimalist robots without memory, explicit communication, or localization information. The state-of-the-art approaches generally assume that the robots in the swarm are able to detect the relative position of neighboring robots and targets in order to provide convergence guarantees. In this work, we propose a novel control law for the guaranteed encapsulation of static and moving targets while avoiding all collisions, when the robots do not know the exact relative location of any robot or target in the environment. We make use of the Lyapunov stability theory to prove the convergence of our control algorithm and provide bounds on the ratio between the target and robot speeds. Furthermore, our proposed approach is able to provide stochastic guarantees under the bounds that we determine on task parameters for scenarios where a target moves faster than a robot. Finally, we present an analysis of how the emergent behavior changes with different parameters of the task and noisy sensor readings.

I recently came across this remarkable randomized algorithm due to Seidel (1991) for linear programming in 2 dimensions that takes only time O(n) for n constraints (in expectation). It's definitely worth a close inspection, so I thought I would write a short article about it. Note that my presentation of this algorithm assumes that there are no redundant or colinear constraints, that the solution is bounded, and that the solution for each subproblem encountered is unique. These assumptions are easy to remove without slowing the algorithm down, but they complicate the presentation a little. OUTPUT: A point v in the polytope defined by H minimizing cTv.