arXiv subjects

Anish Gupta

Publications and source records attributed to Anish Gupta.

9 recordsLinked to original sources

arXiv2026

Pairwise edge correlations in random minimum spanning trees: a universal bound and complete-graph negative correlation

Let $G$ be a finite connected multigraph whose edges receive independent weights from one atomless law, and let $\operatorname{MST}(G)$ be the resulting random minimum spanning tree. Its law is not pairwise negatively correlated: Lyons, Peres and Schramm exhibited two positively correlated edges, and we give such an example on a simple graph. We prove that positive correlation is nevertheless uniformly controlled: $\mathbf{P}(e,f\in T)\leq 8\mathbf{P}(e\in T)\mathbf{P}(f\in T)$, answering a question of R. Lyons recorded by Tang and Zhang. After conditioning on all other weights, Harris's inequality gives conditional negative correlation; two bottleneck distances and a sharp second-moment estimate control the remaining environmental covariance. For $K_n$ we prove pairwise negative correlation for every $n\geq 3$. The key finite identity is $\mathbf{E}[\mathrm{deg}(x)^2]=10(n-1)/n-4\mathbf{E}[L_n]$, where $L_n$ is the total weight of the minimum spanning tree under rate-one exponential weights. Known expansions for $\mathbf{E}[L_n]$ then give the rate of convergence to $10-4\zeta(3)$ and the limits of both pair-correlation ratios. Finally, an explicit $K_4$ family shows that no universal constant survives when the independent edge laws need not be identical.

math.PR↗

arXiv2026

Tuza's conjecture for graphs of maximum degree at most seven

Tuza conjectured that every finite simple graph $G$ satisfies $\tau(G) \leq 2\nu(G)$, where $\nu(G)$ is the maximum number of pairwise edge-disjoint triangles and $\tau(G)$ is the minimum number of edges whose deletion makes $G$ triangle-free. Puleo proved the conjecture for every graph of maximum average degree less than $7$; this covers maximum degree at most $6$ but no $7$-regular graph. We prove the conjecture for maximum degree at most $7$. The proof uses Puleo's reducible-set framework. At average degree seven his discharging step no longer forces a reducible configuration. In a minimal $7$-regular counterexample every vertex link is a connected seven-vertex graph outside the weak Konig-Egervary class. An exhaustive census of such links supplies, at every vertex, an incident edge lying in four, five or six triangles. We prove that its endpoints form a reducible pair: codegrees five and six use a packing and covering template and Fano-plane witnesses, while codegree four uses an explicit catalogue of 1,144 machine-checked local certificates. We do not provide a human-readable proof of that catalogue; the certificates and their verifiers accompany the paper. The constant $2$ is sharp already at maximum degree three.

math.CO↗

arXiv2026

Balance Constants, Majority Cycles, and the Gold Partition Conjecture through Fourteen Elements

We determine the exact extremal balance data of all 1,338,193,159,771 unlabeled posets on fourteen elements. The least balance constant exceeding $1/3$ is $37/106$. The least over posets that are not nontrivial ordinal sums is $254/725$, attained by a ladder with broken rungs; this confirms a conjecture of Peczarski at order 14, while orders 12 and 13 reproduce De Loof, De Baets, and De Meyer. No balance constant lies in the gap above $1/3$ that Peczarski conjectures to be empty. Exactly 128 classes attain $1/3$, and every one is an ordinal sum of singletons and copies of the three-element poset with one relation, a family whose non-chain members are counted by $a(n)-1$, where $a(n)=a(n-1)+a(n-3)$. In the linear-extension-majority digraph the longest simple cycle has length 8, against 7 at order 13, and exactly 30 classes attain it; of the thirteen such classes whose witnesses the census retains, nine have a cycle spectrum containing no odd cycle at all. A second exhaustive pass over the same classes verifies Peczarski's Gold Partition Conjecture through fourteen elements, extending his order-11 frontier and implying in particular that the $1/3$-$2/3$ Conjecture holds through order 14. All arithmetic is exact and every extremal witness is recomputed by an independent program.

math.CO↗

arXiv2025

Forecasting Time Series with LLMs via Patch-Based Prompting and Decomposition

Recent advances in Large Language Models (LLMs) have demonstrated new possibilities for accurate and efficient time series analysis, but prior work often required heavy fine-tuning and/or ignored inter-series correlations. In this work, we explore simple and flexible prompt-based strategies that enable LLMs to perform time series forecasting without extensive retraining or the use of a complex external architecture. Through the exploration of specialized prompting methods that leverage time series decomposition, patch-based tokenization, and similarity-based neighbor augmentation, we find that it is possible to enhance LLM forecasting quality while maintaining simplicity and requiring minimal preprocessing of data. To this end, we propose our own method, PatchInstruct, which enables LLMs to make precise and effective predictions.

cs.LG↗

arXiv2024

Towards Efficient Risk-Sensitive Policy Gradient: An Iteration Complexity Analysis

Reinforcement Learning (RL) has shown exceptional performance across various applications, enabling autonomous agents to learn optimal policies through interaction with their environments. However, traditional RL frameworks often face challenges in terms of iteration efficiency and safety. Risk-sensitive policy gradient methods, which incorporate both expected return and risk measures, have been explored for their ability to yield safe policies, yet their iteration complexity remains largely underexplored. In this work, we conduct a rigorous iteration complexity analysis for the risk-sensitive policy gradient method, focusing on the REINFORCE algorithm with an exponential utility function. We establish an iteration complexity of $\mathcal{O}(\epsilon^{-2})$ to reach an $\epsilon$-approximate first-order stationary point (FOSP). Furthermore, we investigate whether risk-sensitive algorithms can achieve better iteration complexity compared to their risk-neutral counterparts. Our analysis indicates that risk-sensitive REINFORCE can potentially converge faster. To validate our analysis, we empirically evaluate the learning performance and convergence efficiency of the risk-neutral and risk-sensitive REINFORCE algorithms in multiple environments: CartPole, MiniGrid, and Robot Navigation. Empirical results confirm that risk-sensitive cases can converge and stabilize faster compared to their risk-neutral counterparts. More details can be found on our website https://anonymous.4open.science/w/riskrl.

cs.LG↗

arXiv2023

Pinned Billiard Balls Simulation (WXML Autumn 2023 report)

Systems of pinned billiard balls serve as simplified models of collisions, where all particles remain fixed in their positions while their (pseudo-)velocities evolve in accordance with the laws of conservation of energy and momentum. For some families of ball configurations, Athreya, Burdzy, and Duarte have established the maximum upper bound for the number of pseudo-collisions, thereby demonstrating that the number of collisions is finite. The result has been extended to all ball configurations. In this project, we do extensive simulations to study two specific configurations. First, we consider balls arranged in a half-space and assign a single ball an inward (pseudo-) velocity. Simulations suggest that in the long run, most of the energy is concentrated near the boundary. Second, when the balls are arranged on a flat torus, we find that in the stationary regime, the distributions of the velocity components are i.i.d. normal. Additionally, we find that the components of the velocities in the direction of impact between two touching balls are uncorrelated.

math.DS↗

arXiv2022

Leveraging Distributional Bias for Reactive Collision Avoidance under Uncertainty: A Kernel Embedding Approach

Many commodity sensors that measure the robot and dynamic obstacle's state have non-Gaussian noise characteristics. Yet, many current approaches treat the underlying-uncertainty in motion and perception as Gaussian, primarily to ensure computational tractability. On the other hand, existing planners working with non-Gaussian uncertainty do not shed light on leveraging distributional characteristics of motion and perception noise, such as bias for efficient collision avoidance. This paper fills this gap by interpreting reactive collision avoidance as a distribution matching problem between the collision constraint violations and Dirac Delta distribution. To ensure fast reactivity in the planner, we embed each distribution in Reproducing Kernel Hilbert Space and reformulate the distribution matching as minimizing the Maximum Mean Discrepancy (MMD) between the two distributions. We show that evaluating the MMD for a given control input boils down to just matrix-matrix products. We leverage this insight to develop a simple control sampling approach for reactive collision avoidance with dynamic and uncertain obstacles. We advance the state-of-the-art in two respects. First, we conduct an extensive empirical study to show that our planner can infer distributional bias from sample-level information. Consequently, it uses this insight to guide the robot to good homotopy. We also highlight how a Gaussian approximation of the underlying uncertainty can lose the bias estimate and guide the robot to unfavorable states with a high collision probability. Second, we show tangible comparative advantages of the proposed distribution matching approach for collision avoidance with previous non-parametric and Gaussian approximated methods of reactive collision avoidance.

cs.RO↗

arXiv2022

Multi-Modal Model Predictive Control through Batch Non-Holonomic Trajectory Optimization: Application to Highway Driving

Standard Model Predictive Control (MPC) or trajectory optimization approaches perform only a local search to solve a complex non-convex optimization problem. As a result, they cannot capture the multi-modal characteristic of human driving. A global optimizer can be a potential solution but is computationally intractable in a real-time setting. In this paper, we present a real-time MPC capable of searching over different driving modalities. Our basic idea is simple: we run several goal-directed parallel trajectory optimizations and score the resulting trajectories based on user-defined meta cost functions. This allows us to perform a global search over several locally optimal motion plans. Although conceptually straightforward, realizing this idea in real-time with existing optimizers is highly challenging from technical and computational standpoints. With this motivation, we present a novel batch non-holonomic trajectory optimization whose underlying matrix algebra is easily parallelizable across problem instances and reduces to computing large batch matrix-vector products. This structure, in turn, is achieved by deriving a linearization-free multi-convex reformulation of the non-holonomic kinematics and collision avoidance constraints. We extensively validate our approach using both synthetic and real data sets (NGSIM) of traffic scenarios. We highlight how our algorithm automatically takes lane-change and overtaking decisions based on the defined meta cost function. Our batch optimizer achieves trajectories with lower meta cost, up to 6x faster than competing baselines.

cs.RO↗

arXiv2022

Non Holonomic Collision Avoidance of Dynamic Obstacles under Non-Parametric Uncertainty: A Hilbert Space Approach

We consider the problem of an agent/robot with non-holonomic kinematics avoiding many dynamic obstacles. State and velocity noise of both the robot and obstacles as well as the robot's control noise are modelled as non-parametric distributions as often the Gaussian assumptions of noise models are violated in real-world scenarios. Under these assumptions, we formulate a robust MPC that samples robotic controls effectively in a manner that aligns the robot to the goal state while avoiding obstacles under the duress of such non-parametric noise. In particular, the MPC incorporates a distribution matching cost that effectively aligns the distribution of the current collision cone to a certain desired distribution whose samples are collision-free. This cost is posed as a distance function in the Hilbert Space, whose minimization typically results in the collision cone samples becoming collision-free. We compare and show tangible performance gain with methods that model the collision cone distribution by linearizing the Gaussian approximations of the original non-parametric state and obstacle distributions. We also show superior performance with methods that pose a chance constraint formulation of the Gaussian approximations of non-parametric noise without subjecting such approximations to further linearizations. The performance gain is shown both in terms of trajectory length and control costs that vindicates the efficacy of the proposed method. To the best of our knowledge, this is the first presentation of non-holonomic collision avoidance of moving obstacles in the presence of non-parametric state, velocity and actuator noise models.

cs.RO↗