arXiv subjects

Abhishek Rathod

Publications and source records attributed to Abhishek Rathod.

17 recordsLinked to original sources

arXiv2026

Learning Agile Intruder Interception using Differentiable Quadrotor Dynamics

This paper presents a methodology for learning a control policy to intercept an intruder using the 3D direction unit vector to the intruder and the interceptor state. Prior deep reinforcement learning approaches assume either relative position or distance to the intruder is available, but this information is not readily accessible in real-world applications that employ passive, monocular camera sensors. Instead, we propose a solution that leverages an analytical policy gradient method using differentiable quadrotor dynamics to learn agile interception at speeds up to 10 m/s. The proposed approach outperforms baseline methods that utilize simplified point mass dynamics by an average of 30%.

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arXiv2026

AcroRL: Learning Aggressive Quadrotor Inversion using Bidirectional Thrust

Bidirectional thrust grants quadrotors a second equilibrium condition and increased control authority, expanding the envelope of possible aggressive maneuvers and enabling inverted flight, perching, and sensing. Prior geometric control approaches extend differential flatness through Hopf fibration-based attitude representations to support bidirectional thrust, but struggle with actuator saturation and motor reversal delay during inversions, requiring heuristic thrust posture scheduling and waypoint tuning. We propose a learning-based framework that modulates a constant reference trajectory to perform compact, position-constrained quadrotor inversions while remaining compatible with traditional trajectory generation and tracking across flight regimes. Separate policies are trained via reinforcement learning for nominal-to-inverted and inverted-to-nominal transitions. In JAX-based simulation, the proposed method achieves the lowest position deviation and settling time across all evaluated baselines, reducing position root mean square error (RMSE) by 32% and settling time by 57% relative to the strongest optimization-based baseline. Hardware experiments demonstrate successful inversion across multiple yaw configurations with position RMSE below 0.35m, and compatibility with downstream trajectory generation and control through circular flight in both regimes. Additionally, we provide an open-source implementation of the proposed framework.

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arXiv2026

Bifunction and Interlevel Delaunay Trifiltrations

A key property of the Delaunay filtration is that it is topologically (i.e., weakly) equivalent to the offset (union-of-balls) filtration. Recently, this filtration has been extended to point clouds equipped with an $\mathbb{R}$-valued function, yielding a computable 2-parameter filtration that satisfies an analogous weak equivalence. Motivated in part by the study of time-varying data, we introduce a 3-parameter extension of the Delaunay filtration for point clouds equipped with an $\mathbb{R}^2$-valued function, also satisfying an analogous weak equivalence. For a point cloud $X \subset \mathbb{R}^d$, our trifiltration has size $O\bigl(|X|^{\lceil(d+1)/2\rceil+1}\bigr)$. We present an algorithm that computes this trifiltration in time $O\bigl(|X|^{\lceil d/2\rceil+2}\bigr)$, together with an implementation. Our experiments demonstrate that implementation can handle thousands of points in $\mathbb{R}^3$, with memory growth that is nearly linear.

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arXiv2026

Quadrotor Navigation using Reinforcement Learning with Privileged Information

This paper presents a reinforcement learning-based quadrotor navigation method that leverages efficient differentiable simulation, novel loss functions, and privileged information to navigate around large obstacles. Prior learning-based methods perform well in scenes that exhibit narrow obstacles, but struggle when the goal location is blocked by large walls or terrain. In contrast, the proposed method utilizes time-of-arrival (ToA) maps as privileged information and a yaw alignment loss to guide the robot around large obstacles. The policy is evaluated in photo-realistic simulation environments containing large obstacles, sharp corners, and dead-ends. Our approach achieves an 86% success rate and outperforms baseline strategies by 34%. We deploy the policy onboard a custom quadrotor in outdoor cluttered environments both during the day and night. The policy is validated across 20 flights, covering 589 meters without collisions at speeds up to 4 m/s.

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arXiv2025

Decentralized Uncertainty-Aware Active Search with a Team of Aerial Robots

Rapid search and rescue is critical to maximizing survival rates following natural disasters. However, these efforts are challenged by the need to search large disaster zones, lack of reliability in the communications infrastructure, and a priori unknown numbers of objects of interest (OOIs), such as injured survivors. Aerial robots are increasingly being deployed for search and rescue due to their high mobility, but there remains a gap in deploying multi-robot autonomous aerial systems for methodical search of large environments. Prior works have relied on preprogrammed paths from human operators or are evaluated only in simulation. We bridge these gaps in the state of the art by developing and demonstrating a decentralized active search system, which biases its trajectories to take additional views of uncertain OOIs. The methodology leverages stochasticity for rapid coverage in communication denied scenarios. When communications are available, robots share poses, goals, and OOI information to accelerate the rate of search. Detections from multiple images and vehicles are fused to provide a mean and covariance for each OOI location. Extensive simulations and hardware experiments in Bloomingdale, OH, are conducted to validate the approach. The results demonstrate the active search approach outperforms greedy coverage-based planning in communication-denied scenarios while maintaining comparable performance in communication-enabled scenarios. The results also demonstrate the ability to detect and localize all a priori unknown OOIs with a mean error of approximately 3m at flight altitudes between 50m-60m.

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arXiv2025

(Almost-)Optimal FPT Algorithm and Kernel for $T$-Cycle on Planar Graphs

Research of cycles through specific vertices is a central topic in graph theory. In this context, we focus on a well-studied computational problem, \textsc{$T$-Cycle}: given an undirected $n$-vertex graph $G$ and a set of $k$ vertices $T\subseteq V(G)$ termed \textit{terminals}, the objective is to determine whether $G$ contains a simple cycle $C$ through all the terminals. Our contribution is twofold: (i) We provide a $2^{O(\sqrt{k}\log k)}\cdot n$-time fixed-parameter deterministic algorithm for \textsc{$T$-Cycle} on planar graphs; (ii) We provide a $k^{O(1)}\cdot n$-time deterministic kernelization algorithm for \textsc{$T$-Cycle} on planar graphs where the produced instance is of size $k\log^{O(1)}k$. Both of our algorithms are optimal in terms of both $k$ and $n$ up to (poly)logarithmic factors in $k$ under the ETH. In fact, our algorithms are the first subexponential-time fixed-parameter algorithm for \textsc{$T$-Cycle} on planar graphs, as well as the first polynomial kernel for \textsc{$T$-Cycle} on planar graphs. This substantially improves upon/expands the known literature on the parameterized complexity of the problem.

cs.DS↗

arXiv2024

Rapid Quadrotor Navigation in Diverse Environments using an Onboard Depth Camera

Search and rescue environments exhibit challenging 3D geometry (e.g., confined spaces, rubble, and breakdown), which necessitates agile and maneuverable aerial robotic systems. Because these systems are size, weight, and power (SWaP) constrained, rapid navigation is essential for maximizing environment coverage. Onboard autonomy must be robust to prevent collisions, which may endanger rescuers and victims. Prior works have developed high-speed navigation solutions for autonomous aerial systems, but few have considered safety for search and rescue applications. These works have also not demonstrated their approaches in diverse environments. We bridge this gap in the state of the art by developing a reactive planner using forward-arc motion primitives, which leverages a history of RGB-D observations to safely maneuver in close proximity to obstacles. At every planning round, a safe stopping action is scheduled, which is executed if no feasible motion plan is found at the next planning round. The approach is evaluated in thousands of simulations and deployed in diverse environments, including caves and forests. The results demonstrate a 24% increase in success rate compared to state-of-the-art approaches.

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arXiv2024

Active Learning of Discrete-Time Dynamics for Uncertainty-Aware Model Predictive Control

Model-based control requires an accurate model of the system dynamics for precisely and safely controlling the robot in complex and dynamic environments. Moreover, in the presence of variations in the operating conditions, the model should be continuously refined to compensate for dynamics changes. In this paper, we present a self-supervised learning approach that actively models the dynamics of nonlinear robotic systems. We combine offline learning from past experience and online learning from current robot interaction with the unknown environment. These two ingredients enable a highly sample-efficient and adaptive learning process, capable of accurately inferring model dynamics in real-time even in operating regimes that greatly differ from the training distribution. Moreover, we design an uncertainty-aware model predictive controller that is heuristically conditioned to the aleatoric (data) uncertainty of the learned dynamics. This controller actively chooses the optimal control actions that (i) optimize the control performance and (ii) improve the efficiency of online learning sample collection. We demonstrate the effectiveness of our method through a series of challenging real-world experiments using a quadrotor system. Our approach showcases high resilience and generalization capabilities by consistently adapting to unseen flight conditions, while it significantly outperforms classical and adaptive control baselines.

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arXiv2024

Keeping it sparse: Computing Persistent Homology revisited

In this work, we study several variants of matrix reduction via Gaussian elimination that try to keep the reduced matrix sparse. The motivation comes from the growing field of topological data analysis where matrix reduction is the major subroutine to compute barcodes, the main invariant therein. We propose two novel variants of the standard algorithm, called swap and retrospective reductions. We test them on a large collection of data against other known variants to compare their efficiency, and we find that sometimes they provide a considerable speed-up. We also present novel output-sensitive bounds for the retrospective variant which better explain the discrepancy between the cubic worst-case complexity bound and the almost linear practical behavior of matrix reduction. Finally, we provide several constructions on which one of the variants performs strictly better than the others.

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arXiv2024

Fast Algorithms for Minimum Homology Basis

We study the problem of finding a minimum homology basis, that is, a lightest set of cycles that generates the $1$-dimensional homology classes with $\mathbb{Z}_2$ coefficients in a given simplicial complex $K$. This problem has been extensively studied in the last few years. For general complexes, the current best deterministic algorithm, by Dey et al., runs in $O(N m^{ω-1} + n m g)$ time, where $N$ denotes the total number of simplices in $K$, $m$ denotes the number of edges in $K$, $n$ denotes the number of vertices in $K$, $g$ denotes the rank of the $1$-homology group of $K$, and $ω$ denotes the exponent of matrix multiplication. In this paper, we present three conceptually simple randomized algorithms that compute a minimum homology basis of a general simplicial complex $K$. The first algorithm runs in $\tilde{O}(m^ω)$ time, the second algorithm runs in $O(N m^{ω-1})$ time and the third algorithm runs in $\tilde{O}(N^2 g + N m g{^2} + m g{^3})$ time which is nearly quadratic time when $g=O(1)$. We also study the problem of finding a minimum cycle basis in an undirected graph $G$ with $n$ vertices and $m$ edges. The best known algorithm for this problem runs in $O(m^ω)$ time. Our algorithm, which has a simpler high-level description, but is slightly more expensive, runs in $\tilde{O}(m^ω)$ time. We also provide a practical implementation of computing the minimum homology basis for general weighted complexes. The implementation is broadly based on the algorithmic ideas described in this paper, differing in its use of practical subroutines. Of these subroutines, the more costly step makes use of a parallel implementation, thus potentially addressing the issue of scale. We compare results against the currently known state of the art implementation (ShortLoop).

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arXiv2024

Geometric Localization of Homology Cycles

Computing an optimal cycle in a given homology class, also referred to as the homology localization problem, is known to be an NP-hard problem in general. Furthermore, there is currently no known optimality criterion that localizes classes geometrically and admits a stability property under the setting of persistent homology. We present a geometric optimization of the cycles that is computable in polynomial time and is stable in an approximate sense. Tailoring our search criterion to different settings, we obtain various optimization problems like optimal homologous cycle, minimum homology basis, and minimum persistent homology basis. In practice, the (trivial) exact algorithm is computationally expensive despite having a worst case polynomial runtime. Therefore, we design approximation algorithms for the above problems and study their performance experimentally. These algorithms have reasonable runtimes for moderate sized datasets and the cycles computed by these algorithms are consistently of high quality as demonstrated via experiments on multiple datasets.

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arXiv2024

Cup Product Persistence and Its Efficient Computation

It is well-known that the cohomology ring has a richer structure than homology groups. However, until recently, the use of cohomology in persistence setting has been limited to speeding up of barcode computations. Some of the recently introduced invariants, namely, persistent cup-length, persistent cup modules and persistent Steenrod modules, to some extent, fill this gap. When added to the standard persistence barcode, they lead to invariants that are more discriminative than the standard persistence barcode. In this work, we devise an $O(d n^4)$ algorithm for computing the persistent $k$-cup modules for all $k \in \{2, \dots, d\}$, where $d$ denotes the dimension of the filtered complex, and $n$ denotes its size. Moreover, we note that since the persistent cup length can be obtained as a byproduct of our computations, this leads to a faster algorithm for computing it for $d>3$. Finally, we introduce a new stable invariant called partition modules of cup product that is more discriminative than persistent $k$-cup modules and devise an $O(c(d)n^4)$ algorithm for computing it, where $c(d)$ is subexponential in $d$.

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arXiv2022

Parameterized inapproximability of Morse matching

We study the problem of minimizing the number of critical simplices from the point of view of inapproximability and parameterized complexity. We first show inapproximability of Min-Morse Matching within a factor of $2^{\log^{(1-ε)}n}$. Our second result shows that Min-Morse Matching is ${\bf W{[P]}}$-hard with respect to the standard parameter. Next, we show that Min-Morse Matching with standard parameterization has no FPT approximation algorithm for any approximation factor $ρ$. The above hardness results are applicable to complexes of dimension $\ge 2$. On the positive side, we provide a factor $O(\frac{n}{\log n})$ approximation algorithm for Min-Morse Matching on $2$-complexes, noting that no such algorithm is known for higher dimensional complexes. Finally, we devise discrete gradients with very few critical simplices for typical instances drawn from a fairly wide range of parameter values of the Costa-Farber model of random complexes.

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arXiv2021

The complexity of high-dimensional cuts

Cut problems form one of the most fundamental classes of problems in algorithmic graph theory. For instance, the minimum cut, the minimum $s$-$t$ cut, the minimum multiway cut, and the minimum $k$-way cut are some of the commonly encountered cut problems. Many of these problems have been extensively studied over several decades. In this paper, we initiate the algorithmic study of some cut problems in high dimensions. The first problem we study, namely, Topological Hitting Set (THS), is defined as follows: Given a nontrivial $r$-cycle $ζ$ in a simplicial complex $\mathsf{K}$, find a set $\mathcal{S}$ of $r$-dimensional simplices of minimum cardinality so that $\mathcal{S}$ meets every cycle homologous to $ζ$. Our main result is that this problem admits a polynomial-time solution on triangulations of closed surfaces. Interestingly, the optimal solution is given in terms of the cocycles of the surface. For general complexes, we show that THS is W[1]-hard with respect to the solution size $k$. On the positive side, we show that THS admits an FPT algorithm with respect to $k+d$, where $d$ is the maximum degree of the Hasse graph of the complex $\mathsf{K}$. We also define a problem called Boundary Nontrivialization (BNT): Given a bounding $r$-cycle $ζ$ in a simplicial complex $\mathsf{K}$, find a set $\mathcal{S}$ of $(r+1)$-dimensional simplices of minimum cardinality so that the removal of $\mathcal{S}$ from $\mathsf{K}$ makes $ζ$ non-bounding. We show that BNT is W[1]-hard with respect to the solution size as the parameter, and has an $O(\log n)$-approximation FPT algorithm for $(r+1)$-dimensional complexes with the $(r+1)$-th Betti number $β_{r+1}$ as the parameter. Finally, we provide randomized (approximation) FPT algorithms for the global variants of THS and BNT.

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arXiv2019

Parametrized Complexity of Expansion Height

Deciding whether two simplicial complexes are homotopy equivalent is a fundamental problem in topology, which is famously undecidable. There exists a combinatorial refinement of this concept, called simple-homotopy equivalence: two simplicial complexes are of the same simple-homotopy type if they can be transformed into each other by a sequence of two basic homotopy equivalences, an elementary collapse and its inverse, an elementary expansion. In this article we consider the following related problem: given a 2-dimensional simplicial complex, is there a simple-homotopy equivalence to a 1-dimensional simplicial complex using at most p expansions? We show that the problem, which we call the erasability expansion height, is W[P]-complete in the natural parameter p.

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arXiv2018

Hardness of Approximation for Morse Matching

Discrete Morse theory has emerged as a powerful tool for a wide range of problems, including the computation of (persistent) homology. In this context, discrete Morse theory is used to reduce the problem of computing a topological invariant of an input simplicial complex to computing the same topological invariant of a (significantly smaller) collapsed cell or chain complex. Consequently, devising methods for obtaining gradient vector fields on complexes to reduce the size of the problem instance has become an emerging theme over the last decade. While computing the optimal gradient vector field on a simplicial complex is NP-hard, several heuristics have been observed to compute near-optimal gradient vector fields on a wide variety of datasets. Understanding the theoretical limits of these strategies is therefore a fundamental problem in computational topology. In this paper, we consider the approximability of maximization and minimization variants of the Morse matching problem, posed as open problems by Joswig and Pfetsch. We establish hardness results for Max-Morse matching and Min-Morse matching. In particular, we show that, for a simplicial complex with n simplices and dimension $d \leq 3$, it is NP-hard to approximate Min-Morse matching within a factor of $O(n^{1-ε})$, for any $ε> 0$. Moreover, using an L-reduction from Degree 3 Max-Acyclic Subgraph to Max-Morse matching, we show that it is both NP-hard and UGC-hard to approximate Max-Morse matching for simplicial complexes of dimension $d \leq 2$ within certain explicit constant factors.

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arXiv2016

Approximation Algorithms for Max-Morse Matching

In this paper, we prove that the Max-Morse Matching Problem is approximable, thus resolving an open problem posed by Joswig and Pfetsch. We describe two different approximation algorithms for the Max-Morse Matching Problem. For $D$-dimensional simplicial complexes, we obtain a $\frac{(D+1)}{(D^2+D+1)}$-factor approximation ratio using a simple edge reorientation algorithm that removes cycles. Our second result is an algorithm that provides a $\frac{2}{D}$-factor approximation for simplicial manifolds by processing the simplices in increasing order of dimension. One application of these algorithms is towards efficient homology computation of simplicial complexes. Experiments using a prototype implementation on several datasets indicate that the algorithm computes near optimal results.

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