arXiv subjects

Ghurumuruhan Ganesan

Publications and source records attributed to Ghurumuruhan Ganesan.

At least 19 recordsLinked to original sources

arXiv2026

Criticality in Dissimilar Decomposition and Undersampling of Random Datasets with Anomalies

Training datasets for upcoming LLMs would include a significant amount of AI text/image data generated from current LLMs. In such a scenario, it is important to understand how this affects batch decompositions and thereby, the performance of the resultant new LLM. In this paper, we consider AI generated data as anomalies ``linked" to main data points and study decomposition and undersampling properties of the overall random dataset. We use redundancy graphs and iteration techniques to obtain bounds for the minimum size of a strongly dissimilar (SD) decomposition and demonstrate a phase transition phenomena, wherein the minimum size is essentially determined by the \emph{main} data points when the number of anomalies is small and is ``taken" over by the anomalies above a certain threshold. We also establish a size criticality result for the strong similarity of a randomly undersampled dataset and illustrate our results with examples involving categorical datasets, whose overall space size is much larger than the size of the dataset.

cs.LG↗

arXiv2026

Maximum Spanning Trees of Random Geometric Graphs With Independent Edge Weights

In this paper, we study maximum weight spanning trees of the random geometric graph (RGG)~$G$ formed by~$n$ vertices where each edge is independently either open or closed with a certain probability and is also equipped with an independent random positive weight. We use segmentation and iterative path construction to obtain deviation bounds for the order of growth of the maximum weight of a spanning tree in terms of an inverse of the edge weight complementary cumulative distribution function (ccdf) and also illustrate our results for the special cases of power law and exponential decay. We then use martingale difference methods to individually estimate the variance contribution due to randomness in vertex locations and edge states/weights and determine sufficient conditions for~$L^2-$convergence of the maximum weight, appropriately scaled and centred.

math.PR↗

arXiv2026

Constrained Maximum Weight Paths in Random Geometric Graphs

In this paper, we consider a random geometric graph (RGG)~$G$ formed by~$n$ vertices distributed uniformly in the unit square~$S$ on the plane and equip each edge of~$G$ with an independent weight. We assume that the adjacency distance between vertices is larger than the connectivity threshold and estimate the maximum weight of a path connecting two fixed points~$O_1$ and~$O_2$ in~$S,$ using edges of~$G$ that satisfy length and weight constraints. Our strategy is to first divide the space between~$O_1$ and~$O_2$ into small squares and identify nice squares containing heavy edges. We then use an iterative stitching procedure to connect these heavy edges and estimate the weight of the resulting path~$P.$ Finally, we invoke a multi-level scaling procedure along with weight segmentation to establish an upper bound for the maximum weight of \emph{any} path between~$O_1$ and~$O_2$ and thereby demonstrate the near-optimality of~$P.$ We also illustrate our results using examples involving edge weights satisfying power law and exponential decay.

math.PR↗

arXiv2026

Extremal Spanning Trees of Random Marked Graphs with Independent Edge Weights

In this paper, we consider a Bernoulli random graph~$G$ on~$n$ vertices with non-uniform edge probabilities, where each vertex has an independent mark and each edge is equipped with an independent positive weight. The cost of an edge depends on the weight as well the marks of the endvertices and we estimate the growth of the maximum and minimum cost of a spanning tree containing all the vertices. For edge weights with heavy tails and vertex marks distributed uniformly in the unit square, we obtain a phase transition in terms of the tail decay exponent~$s:$ If~$s$ is large, then the maximum cost is essentially determined by the vertex locations and if~$s$ is small, then the edge weights crucially influence the maximum cost. We derive a similar result for minimum cost spanning trees and use martingale difference based methods to establish the~$L^2-$convergence of the extremal cost, appropriately scaled and centred.

math.PR↗

arXiv2025

Dissimilar Batch Decompositions of Random Datasets

For better learning, large datasets are often split into small batches and fed sequentially to the predictive model. In this paper, we study such batch decompositions from a probabilistic perspective. We assume that data points (possibly corrupted) are drawn independently from a given space and define a concept of similarity between two data points. We then consider decompositions that restrict the amount of similarity within each batch and obtain high probability bounds for the minimum size. We demonstrate an inherent tradeoff between relaxing the similarity constraint and the overall size and also use martingale methods to obtain bounds for the maximum size of data subsets with a given similarity.

cs.LG↗

arXiv2025

Extremal Planar Matchings of Inhomogenous Random Bipartite Graphs

In this paper we study maximum size and minimum weight planar matchings of inhomogenous random bipartite graphs. Our motivation for this study comes from efficient usage of cross edges in relay networks for overall improvement in network performance. We first consider Bernoulli planar matchings with a constraint on the edge length and obtain deviation estimates for the maximum size of a planar matching. We then equip each edge of the complete bipartite graph with a positive random weight and obtain bounds on the minimum weight of a planar matching containing a given number of edges. We also use segmentation and martingale methods to obtain~$L^2-$convergence of the minimum weight, appropriately scaled and centred.

math.PR↗

arXiv2025

Deviation Estimates for Extremal Relay Random Geometric Graphs

In this paper, we consider a deterministic graph~$Γ$ drawn on the unit square with straight line segments as edges and connect vertices of~$Γ$ using edges of a random geometric graph (RGG)~$G$ with adjacency distance~$r_n$ as relays. We call the resulting graph as a \emph{relay} RGG and determine sufficient conditions under such relay RGGs exist and are also near optimal, in terms of the graph parameters of~$Γ.$ We then equip edges of~$G$ with independent, exponentially distributed weights and obtain bounds for the maximum possible weight~$W_n$ of a relay RGG with a given length~$L_n.$

math.PR↗

arXiv2024

Classification and Regression Error Bounds for Inhomogenous Data With Applications to Wireless Networks

In this paper, we study classification and regression error bounds for inhomogenous data that are independent but not necessarily identically distributed. First, we consider classification of data in the presence of non-stationary noise and establish ergodic type sufficient conditions that guarantee the achievability of the Bayes error bound, using universal rules. We then perform a similar analysis for $k$-nearest neighbour regression and obtain optimal error bounds for the same. Finally, we illustrate applications of our results in the context of wireless networks.

cs.IT↗

arXiv2024

Tracking the Mean of a Piecewise Stationary Sequence

In this paper we study the problem of tracking the mean of a piecewise stationary sequence of independent random variables. First we consider the case where the transition times are known and show that a direct running average performs the tracking in short time and with high accuracy. We then use a single valued weighted running average with a tunable parameter for the case when transition times are unknown and establish deviation bounds for the tracking accuracy. Our result has applications in choosing the optimal rewards for the multiarmed bandit scenario.

math.PR↗

arXiv2024

Dirac-type Theorems for Inhomogenous Random Graphs

In this paper, we study Dirac-type theorems for an inhomogenous random graph (G) whose edge probabilities are not necessarily all the same. We obtain sufficient conditions for the existence of Hamiltonian paths and perfect matchings, in terms of the \emph{sum} of edge probabilities. For edge probability assignments with two-sided bounds, we use Pósa rotation and single vertex exclusion techniques to show that (G) is Hamiltonian with high probability. For weaker one-sided bounds, we use bootstrapping techniques to obtain a perfect matching in (G,) with high probability. We also highlight an application of our results in the context of channel assignment problem in wireless networks.

math.PR↗

arXiv2023

Critical Exponent for the Acyclic Chromatic Number of Random Graphs

In this paper we study acyclic colouring in the random subgraph $\mathit{G}$ of the complete graph $\mathit{K}_n$ on $\mathit{n}$ vertices where each edge is present with probability $\mathit{p}$; independent of the other edges. We show that the acyclic chromatic number exhibits a phase transition from sublinear to linear growth as the edge probability increases, even in the sparse regime and obtain estimates for the critical exponent. Next, we introduce a relaxation by allowing for a small fraction of "bad" cycles to violate the acyclic colouring condition and show that the critical exponent in this case is in fact zero, no matter how small the fraction.

math.PR↗

arXiv2023

Linearity Property of Unique Colourings in Random Graphs

In this paper, we study unique colourings in random graphs as a generalization of both conflict-free and injective colourings. Specifically, we impose the condition that a fraction of vertices in the neighbourhood of any vertex are assigned unique colours and use vertex partitioning and the probabilistic method to show that the minimum number of colours needed grows linearly with the uniqueness parameter, unlike both conflict-free and injective colourings. We argue how the unboundedness of the vertex neighbourhoods influences the linearity property and illustrate our case with an example involving treeunique colourings in random graphs.

math.CO↗

arXiv2023

Linearity and Local Correctness in Weighted Colourings of Random Graphs

In this paper, we consider a weighted generalization of the chromatic number of a Binomial random graph~$G.$ We equip each edge with a random weight and then colour the vertices in such a way that the absolute colour difference between any two adjacent vertices is at least as large as their edge weight. We show that with high probability, the weighted colouring number grows linearly with the maximum vertex degree if the edge weights have sufficiently large moments. Conversely, if the edge weight moments are unbounded then the weighted chromatic number is much larger than the maximum vertex degree, with high probability. We also obtain a sharp threshold result for locally correct weighted colourings for balanced subgraphs of~$G.$

math.CO↗

arXiv2023

Probabilistic Bounds for Data Storage with Feature Selection and Undersampling

In this paper we consider data storage from a probabilistic point of view and obtain bounds for efficient storage in the presence of feature selection and undersampling, both of which are important from the data science perspective. First, we consider encoding of correlated sources for nonstationary data and obtain a Slepian-Wolf type result for the probability of error. We then reinterpret our result by allowing one source to be the set of features to be discarded and other source to be remaining data to be encoded. Next, we consider neighbourhood domination in random graphs where we impose the condition that a fraction of neighbourhood must be present for each vertex and obtain optimal bounds on the minimum size of such a set. We show how such sets are useful for data undersampling in the presence of imbalanced datasets and briefly illustrate our result using~$k-$nearest neighbours type classification rules as an example.

cs.IT↗

arXiv2023

A Probabilistic Model for Data Redundancy in the Feature Domain

In this paper, we use a probabilistic model to estimate the number of uncorrelated features in a large dataset. Our model allows for both pairwise feature correlation (collinearity) and interdependency of multiple features (multicollinearity) and we use the probabilistic method to obtain upper and lower bounds of the same order, for the size of a feature set that exhibits low collinearity and low multicollinearity. We also prove an auxiliary result regarding mutually good constrained sets that is of independent interest.

cs.LG↗

arXiv2023

Euclidean minimum spanning trees with location dependent and power weighted edges

Consider~$n$ nodes~$\{X_i\}_{1 \leq i \leq n}$ independently distributed in the unit square~$S,$ each according to a distribution~$f$ and let~$K_n$ be the complete graph formed by joining each pair of nodes by a straight line segment. For every edge~$e$ in~$K_n$ we associate a weight~$w(e)$ that may depend on the \emph{individual locations} of the endvertices of~$e.$ Denoting~$MST_n$ to be the minimum weight of a spanning tree of~$K_n$ and assuming an equivalence condition on the weight function~$w(.),$ we prove that~$MST_n$ appropriately scaled and centred converges to zero a.s.\ and in mean as~$n \rightarrow \infty.$ We also obtain upper and lower bound deviation estimates for~$MST_n.$

math.PR↗

arXiv2023

Linear Eulerian Extensions of Inhomogenous Random Graphs

The Eulerian extension number of any graph~$H$ (i.e. the minimum number of edges needed to be added to make~$H$ Eulerian) is at least~$t(H),$ half the number of odd degree vertices of~$H.$ In this paper we consider an inhomogenous random graph~$G$ whose edge probabilities need not all be the same and use an iterative probabilistic method to obtain sufficient conditions for the Eulerian extension number of~$G$ to grow \emph{linearly} with~$t(G).$ We derive our conditions in terms of the average edge probabilities and edge density and also briefly illustrate our result with an example.

math.PR↗

arXiv2023

Linear Codes with Prescribed Hull Dimension and Minimum Distance

The hull of a linear code (i.e., a finite field vector space)~${\mathcal C}$ is defined to be the vector space formed by the intersection of~${\mathcal C}$ with its dual~${\mathcal C}^{\perp}.$ Constructing vector spaces with a specified hull dimension has important applications and it is therefore of interest to study minimum distance properties of such spaces. In this paper, we use the probabilistic method to obtain spaces with a given hull dimension and minimum distance and also derive Gilbert-Varshamov type sufficient conditions for their existence.

cs.IT↗