Evolving Distributions Under Local Motion
Aditya Acharya, David M. Mount
TL;DR
This work extends the evolving data framework to track moving points in $\mathbb{R}^d$ by modeling each object as a ground-truth distribution $P_i$ supported on a local ball and maintaining a probabilistic hypothesis $H_i$, with tracking distance measured by the sum of KL divergences $\sum_i D_{KL}(P_i\parallel H_i)$. It introduces a locality-based motion model where each step moves a point by at most $\alpha N_i$ (nearest-neighbor distance) and uses a weak containment oracle to update the hypothesis. The proposed TrackByZoom algorithm represents each hypothesis as an independent $d$-dimensional Cauchy distribution, and uses a potential function to bound the total KL-distance, achieving $\mathcal D(\mathcal P,\mathcal H)=O(n)$ in steady state after a burn-in time $t_0=O(n\log\Lambda_0\log\log\Lambda_0)$ with a constant speedup $\sigma$. The paper proves a matching lower bound $\Omega(n)$, demonstrating asymptotic optimality, and discusses extensions to broader distribution families and potential refinements (e.g., using $k$-nearest-neighbor scales or unbounded support). This framework provides a theoretically grounded method for maintaining imprecise, dynamically evolving geometric summaries with limited oracle access, relevant for large-scale motion tracking and spatial data management.
Abstract
Geometric data sets arising in modern applications are often very large and change dynamically over time. A popular framework for dealing with such data sets is the evolving data framework, where a discrete structure continuously varies over time due to the unseen actions of an evolver, which makes small changes to the data. An algorithm probes the current state through an oracle, and the objective is to maintain a hypothesis of the data set's current state that is close to its actual state at all times. In this paper, we apply this framework to maintaining a set of $n$ point objects in motion in $d$-dimensional Euclidean space. To model the uncertainty in the object locations, both the ground truth and hypothesis are based on spatial probability distributions, and the distance between them is measured by the Kullback-Leibler divergence (relative entropy). We introduce a simple and intuitive motion model where with each time step, the distance that any object can move is a fraction of the distance to its nearest neighbor. We present an algorithm that, in steady state, guarantees a distance of $O(n)$ between the true and hypothesized placements. We also show that for any algorithm in this model, there is an evolver that can generate a distance of $Ω(n)$, implying that our algorithm is asymptotically optimal.