arXiv subjects

Aditya Acharya

Publications and source records attributed to Aditya Acharya.

7 recordsLinked to original sources

arXiv2026

On Voronoi diagrams in the Funk Conical Geometry

The forward and reverse Funk weak metrics are fundamental distance functions on convex bodies that serve as the building blocks for the Hilbert and Thompson metrics. In this paper we study Voronoi diagrams under the forward and reverse Funk metrics in polygonal and elliptical cones. We establish several key geometric properties: (1) bisectors consist of a set of rays emanating from the apex of the cone, and (2) Voronoi diagrams in the $d$-dimensional forward (or reverse) Funk metrics are equivalent to additively-weighted Voronoi diagrams in the $(d-1)$-dimensional forward (or reverse) Funk metrics on bounded cross sections of the cone. Leveraging this, we provide an $O\big(n^{\ceil{\frac{d-1}{2}}+1}\big)$ time algorithm for creating these diagrams in $d$-dimensional elliptical cones using a transformation to and from Apollonius diagrams, and an $O(mn\log(n))$ time algorithm for 3-dimensional polygonal cones with $m$ facets via a reduction to abstract Voronoi diagrams. We also provide a complete characterization of when three sites have a circumcenter in 3-dimensional cones. This is one of the first algorithmic studies of the Funk metrics.

cs.CG↗

arXiv2026

Classifiers in High Dimensional Hilbert Metrics

Classifying points in high dimensional spaces is a fundamental geometric problem in machine learning. In this paper, we address classifying points in the $d$-dimensional Hilbert polygonal metric. The Hilbert metric is a generalization of the Cayley-Klein hyperbolic distance to arbitrary convex bodies and has a diverse range of applications in machine learning and convex geometry. We first present an efficient LP-based algorithm in the metric for the large-margin SVM problem. Our algorithm runs in time polynomial to the number of points, bounding facets, and dimension. This is a significant improvement on previous works, which either provide no theoretical guarantees on running time, or suffer from exponential runtime. We also consider the closely related Funk metric. We also present efficient algorithms for the soft-margin SVM problem and for nearest neighbor-based classification in the Hilbert metric.

cs.CG↗

arXiv2024

Evolving Distributions Under Local Motion

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 $\Omega(n)$, implying that our algorithm is asymptotically optimal.

cs.CG↗

arXiv2023

Tracking Evolving labels using Cone based Oracles

The evolving data framework was first proposed by Anagnostopoulos et al., where an evolver makes small changes to a structure behind the scenes. Instead of taking a single input and producing a single output, an algorithm judiciously probes the current state of the structure and attempts to continuously maintain a sketch of the structure that is as close as possible to its actual state. There have been a number of problems that have been studied in the evolving framework including our own work on labeled trees. We were motivated by the problem of maintaining a labeling in the plane, where updating the labels require physically moving them. Applications involve tracking evolving disease hot-spots via mobile testing units , and tracking unmanned aerial vehicles. To be specific, we consider the problem of tracking labeled nodes in the plane, where an evolver continuously swaps labels of any two nearby nodes in the background unknown to us. We are tasked with maintaining a hypothesis, an approximate sketch of the locations of these labels, which we can only update by physically moving them over a sparse graph. We assume the existence of an Oracle, which when suitably probed, guides us in fixing our hypothesis.

cs.DS↗

arXiv2022

Optimally Tracking Labels on an Evolving Tree

Motivated by the problem of maintaining data structures for a large sets of points that are evolving over the course of time, we consider the problem of maintaining a set of labels assigned to the vertices of a tree, where the locations of these labels change over time. We study the problem in the evolving data framework, where labels change over time due to the action of an agent called the evolver. The algorithm can only track these changes by explicitly probing individual nodes of the tree. This framework captures the tradeoff between the complexity of maintaining an up-to-date view of the structure and the quality of results computed with the available view. Our results allow for both randomized and adversarial evolution of the data, subject to allowing different speedup factors between the algorithm and the evolver. We show that in the limit, our algorithm maintains labels to within an average distance of $O(1)$ of their actual locations. We also present nearly matching lower bounds.

cs.DS↗

arXiv2022

Rediscovering Affordance: A Reinforcement Learning Perspective

Affordance refers to the perception of possible actions allowed by an object. Despite its relevance to human-computer interaction, no existing theory explains the mechanisms that underpin affordance-formation; that is, how affordances are discovered and adapted via interaction. We propose an integrative theory of affordance-formation based on the theory of reinforcement learning in cognitive sciences. The key assumption is that users learn to associate promising motor actions to percepts via experience when reinforcement signals (success/failure) are present. They also learn to categorize actions (e.g., "rotating" a dial), giving them the ability to name and reason about affordance. Upon encountering novel widgets, their ability to generalize these actions determines their ability to perceive affordances. We implement this theory in a virtual robot model, which demonstrates human-like adaptation of affordance in interactive widgets tasks. While its predictions align with trends in human data, humans are able to adapt affordances faster, suggesting the existence of additional mechanisms.

cs.HC↗

arXiv2015

On the Complexity of Slide-and-Merge Games

We study the complexity of a particular class of board games, which we call `slide and merge' games. Namely, we consider 2048 and Threes, which are among the most popular games of their type. In both games, the player is required to slide all rows or columns of the board in one direction to create a high value tile by merging pairs of equal tiles into one with the sum of their values. This combines features from both block pushing and tile matching puzzles, like Push and Bejeweled, respectively. We define a number of natural decision problems on a suitable generalization of these games and prove NP-hardness for 2048 by reducing from 3SAT. Finally, we discuss the adaptation of our reduction to Threes and conjecture a similar result.

cs.CC↗