Dissertation and Thesis

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PORTRAIT: Holistic Data Visualization using Neural Networks
(2026-09-16) Maitra, Chayan
With the exponential growth of complex data across domains, effective visualization has become increasingly crucial for understanding relationships hidden within high-dimensional spaces. However, existing visualization techniques often struggle to effectively capture and represent such high-dimensional data. Motivated by this challenge, we have developed NeuroDAVIS, a neural network model designed to visualize high-dimensional data by extracting meaningful latent representations through deep feature extraction. While NeuroDAVIS has successfully addressed the visualization aspect, we have soon recognized the necessity of identifying the most relevant features that contribute to the visualization and downstream analysis. To address this issue, we have extended our framework and introduced NeuroDAVIS-FS, a feature selection model built upon the NeuroDAVIS architecture. This extension not only visualizes but also selects the most informative features from complex datasets. As data collection techniques have evolved, it has become increasingly common to encounter datasets that capture multiple aspects—or modalities—of the same set of samples. Visualizing and understanding such multi-modal datasets require a more advanced approach that is capable of integrating and producing a joint embedding of these heterogeneous data sources. Inspired by this need, we have enhanced our model to handle multiple modalities, resulting in NeuroMDAVIS, an extended version of NeuroDAVIS, capable of joint visualization across modalities. In addition, we have also developed a feature selection framework that operates on multi-modal data, enabling the identification of key features across different data types. These key features have been found to be effective in predicting the survival of lung cancer patients. Despite these advancements, another significant limitation remained—while the learned embeddings provided meaningful visualization and feature selection, they did not have generalization capabilities. Some regions of the embedding space remained sparse or unused, limiting the model’s ability to generate new, realistic samples. In order to overcome this issue, we have further extended our approach to a generative framework, giving rise to G-NeuroDAVIS. This generative variant produces a smooth, well-distributed latent embedding that is capable of not only capturing the complete data distribution but also generating high-quality.
Some Contributions to Inference and Model/Variable Selection in High-Dimensional Problems
(2026-08-19) Paul, Sayantan
In today’s world, we often come across situations where we need to infer about several parameters simultaneously based on a single dataset. This is known as the problem of simultaneous statistical inference and the need for such inference arises for data from the fields of biology, astronomy, genomics, bioinformatics, medicine, economics, finance, and image processing, just to name a few. As the name suggests, instead of being concerned about the optimality of inference for the individual parameters, the main interest here is in proposing procedures which can provide satisfactory results in the overall inferential problem involving all parameters of interest. The need for simultaneous inference may arise in the contexts of hypothesis testing, point estimation of several parameters, and building confidence intervals. Such inference becomes challenging when the number of parameters increases with sample size at the same or at a higher rate, i.e., when the problem becomes the high-dimensional. Equally challenging are the problems of model selection or variable selection for data of such kind. In this thesis, our interest is in simultaneous statistical inference and model/variable selection in the high-dimensional setting. One of our main goals would be to propose new inference and model/variable selection procedures and study their properties through theory and simulations. We would also employ some of our proposed methods in suitable real-life examples to understand how they perform in actual applications. We would also like to theoretically address interesting questions about existing procedures widely used in such contexts. An important focus of this work will be the special situations when the number of significant (which may mean nonzero or of sufficiently large magnitude in appropriate contexts) parameters is small compared to the total number of parameters under consideration. These are the so-called “sparse” situations. The assumption of sparsity is natural and very common in the literature, e.g., in the high-dimensional regression setting and in the problem of inference on a high dimensional mean vector. Our approaches here will be built under the Bayesian setting where the unknown parameters are assumed to follow some prior probability distributions. A big focus of our theoretical work will be on studying the optimality (in the frequentist or Bayesian decision theoretic sense) of the new methods proposed or of existing methods. A natural Bayesian approach under sparse settings is to model the parameters by two-group spikeand-slab priors. These priors are expressed as mixtures of a distribution degenerate at 0 (or highly concentrated near 0) and a distribution with high spread. However, as noted by many authors, inference with such priors can be computationally very challenging in high-dimensional situations and complex parametric frameworks. As an alternative to two-group mixture priors, there have been proposals in the literature to consider unimodal, continuous priors having sufficient mass around zero while possessing sufficiently heavy tails. Such priors are named as one-group “global-local” priors, the name “global-local” originating from the use of two sets of parameters to simultaneously enforce and modulate shrinkage at the global level and at the levels of individual parameters respectively. These are called the global shrinkage parameter and the local shrinkage parameters respectively. Although serious efforts have gone into studying optimality of inference using one-group priors in sparse parametric settings, several interesting questions in this connection are still unanswered and several areas somewhat less trodden in the current literature. This thesis is a modest attempt to address some of these in the context of specific models. A large part of our work will focus on inference on the famous normal means model and the linear regression model. The thesis concludes with a study of inference on non-normal count data. Our study is based on a broad class of one-group global-local shrinkage priors covering a lot of popularly used priors including the horseshoe. When the mean parameter of the sparse normal means model is modeled by a spike-and-slab prior, it has been shown in the literature that the corresponding posterior distribution contracts around the truth at a near minimax rate when the level of sparsity is unknown. This raises the question of whether the same phenomenon works when one uses a one-group prior instead of its two-group counterpart for the same problem. We provide an answer to this question in Chapter 2 of the thesis. In order to handle the unknown level of sparsity, we either estimate the global shrinkage parameter based on the data, an empirical Bayes approach, or model it by a non-degenerate absolutely continuous prior distribution on a suitable support in a full Bayes approach. We establish that the posterior distributions of the mean parameter vector, when modelled by broad classes of one-group priors, contract around the truth at a near minimax rate, for both the empirical Bayes and full Bayes approaches. Another interesting question is whether one-group priors can be used to form a good decision rule for the simultaneous testing problem of whether the means are zero or not when the means are truly generated from a two-group prior. In the latter half of this chapter, we are interested in answering this question, assuming that the level of sparsity is unknown. Considering a full Bayes approach, we are to establish that the Bayes risk of a decision rule using a broad class of one-group priors attains the Bayes risk of the optimal rule for the two-group settings asymptotically for a wide range of sparsity levels. The loss function considered is the additive symmetric 0 − 1 loss measuring the number of misclassifications made by a multiple testing rule. One of the main goals in a multiple hypothesis testing problem is to propose a decision rule which can control some overall measure of type I error rate, e.g., the False Discovery Rate (FDR). Given that a testing rule controls the FDR at some desired level, the next obvious question is whether the same rule can provide any control over the False Negative Rate (FNR). In this context, researchers are often interested in the optimal multiple testing rules in terms of controlling sum of certain type I and type II error measures. This can be answered by studying the minimax risk of multiple testing rules with respect to the corresponding loss functions. Very recently, for the normal means model, the expressions for the minimax risks based on misclassification (or Hamming) loss and the loss defined as the sum of FDP and FNP have been derived. It has also been proved that the famous Benjamini-Hochberg (BH) procedure and an ℓ−value based procedure using spike-and-slab priors attain the minimax risk asymptotically adaptively over broad sparsity levels. This motivates us to study whether decision rules based on one-group priors, if any, can enjoy such asymptotic optimality. When the level of sparsity is known, by choosing the global shrinkage parameter appropriately based on the knowledge of sparsity, we prove that the corresponding decision rule based on the broad class of one-group priors mentioned earlier achieves the minimax risk for both loss functions stated earlier. When the level of sparsity is unknown, some empirical Bayes and full Bayes versions of our decision rules can also attain the minimax risk. These results are proved in Chapter 3 of the thesis. Another important problem of interest is variable/model selection in a high-dimensional normal linear regression model. In this thesis, we are interested in a situation where covariates under study inregression model. In Chapter 4 of this thesis, motivated by the existing literature, we are interested in proposing a decision rule and resulting estimators, based on a general class of global-local priors, which have the “Oracle property”, in the sense that, they achieve variable selection consistency and optimal estimation rate, respectively. We propose a decision rule which declares a group to be active if the ratio of the ℓ2 norm of the posterior mean of the group regression coefficient to that of the least square estimate exceeds half. When the design matrix is block-orthogonal, we are able to establish that the global shrinkage parameter can be chosen in such a way that our proposed inference procedures have both selection consistency and optimal estimation rate, provided the level of sparsity is known. Even if the sparsity pattern is unknown, our modified decision rules, by either estimating the global shrinkage parameter from the data or by modeling a prior on it, can still enjoy the oracle property. In the simulation studies, our rules perform favorably compared to many existing methods in a variety of sparsity settings. Our methods, when applied to real datasets, also return encouraging results. In Chapters 2 through 4 of the thesis, our main areas of interest were the situations when the observed data are generated from normal distributions of various parametric forms. However, depending on the problem of interest, the Gaussianity assumption is not always proper. In Chapter 5 of this thesis, we are interested in one such example where the data consists of counts of events, most counts being close to zero while some are moderate or large. The data is modelled by Poisson distribution with unknown means. Clearly the natural prior for such means would be a two-group mixture with large mass on the component concentrated near zero. Our interest is in finding at if one-group modelling is still a good alternative in this context, as seen in previously in Chapters 2 through 4 of this thesis. Specifically, one of the questions this leads us to is whether any decision rule for multiple testing based on one-group priors can approximate the optimal rule with respect to two-group priors in terms of risk when the sample size gets large. Towards that we first obtain the asymptotic expression for the optimal Bayes risk under two-group prior in appropriate asymptotic framework. The loss is taken to be additive symmetric 0 − 1 loss. Next, irrespective of the sparsity pattern to be known or unknown, we establish that the Bayes risks corresponding to our proposed decision rules based on one-group priors attain the optimal Bayes risk, up to some multiplicative constant. Finally, the theoretical results are verified using simulation studies followed by a real data analysis. Many of the theoretical results derived in this thesis are the first of their kind in the literature in their specific contexts. Last but not the least, our theoretical results, simulations and to some extent real data analyses reinforce the logic of using appropriately chosen one-group priors as alternatives to their two-group counterparts in high-dimensional sparse parametric settings.
Bivariate Current Status Data with Competing Risks using Frailty Models
(Indian Statistical Institute, 2026-07-21) Ghosh, Biswadeep
Bivariate current status data with competing risks arises in many application areas, for example, carcinogenicity studies, epidemiological studies, reliability studies, etc..As par of our knowledge, bivariate current status data with competing risks has not received much attention in literature. We initiate this research focusing on modeling and estimation of quantities of interest in absence of any covariate. We model bivariate current status data with competing risks through different frailty structures in order to describe different kinds of association. We consider multiplicative frailty effect on different cause-specific baseline hazard functions. One important part of our analysis is to investigate the identifiability of a proposed model. It has been shown that, when both the cause-specific baseline hazard function and frailty distribution are non-parametric, then the model is not identifiable. Motivated by the previous result, we consider the analysis of following three different combinations of cause-specific baseline hazard and frailty distributions, namely, (a) both the cause-specific baseline hazard and frailty distribution are parametric, (b) the cause-specific baseline hazard functions are parametric and the frailty distribution is non-parametric, (c) the cause-specific baseline hazard functions are non-parametric and the frailty distribution is parametric. For the parametric frailty distribution, we have considered Gamma distribution of different kinds. We have used the maximum likelihood estimation method to estimate the parameters when both the cause-specific baseline hazard functions and frailty distribution are assumed to be parametric. In case of parametric cause-specific baseline hazard and non-parametric frailty, we have implemented an algorithm to estimate mixing distribution in order to obtain the NPMLE of frailty distribution. We have used the Bernstein polynomials to model the non-parametric cause-specific baseline hazard functions, and using sieve maximum likelihood estimation, we have estimated the non-parametric cause-specific baseline hazard function and relevant frailty parameter(s). In every case, simulation studies have been carried out in order to investigate the finite sample properties of the proposed methodologies. We have illustrated the proposed methods through the analyses of a hearing loss data.
Large-scale Asymptotics in Fixed-sample and Sequential Multiple Testing
(Indian Statistical Institute, 2026-08-26) Roy, Rahul
The advent of large-scale data acquisition technologies has led to the routine emergence of massive, dynamically evolving datasets. This has created a pressing need for statistical methodologies capable of operating effectively in both static and sequential data environments. Multiple hypotheses testing, a central tool in such settings, must therefore be adapted to both fixed-sample and sequential frameworks to ensure effective decision-making while controlling error rates. This thesis comprises two main parts. The first part addresses fixed-sample multiple testing under dependence in a Bayesian framework. Building upon the two-group mixture normal model of Bogdan et al. (2011), we extend their methodology to exchangeable multivariate normal test statistics, thereby accommodating realistic dependency structures frequently encountered in high-dimensional applications. We derive sufficient conditions under which multiple testing procedures satisfy the Asymptotic Bayes Optimality under Sparsity (ABOS) property in the presence of dependence. Furthermore, we show that several classical procedures, including those of Bonferroni (1936), Šidák (1967), and Benjamini and Hochberg (1995), retain the ABOS property under suitable sparsity and dependence assumptions. The second part focuses on large-scale multiple testing in sequential settings, where data vectors or multiple data streams are observed over time rather than being available in full initially. Motivated by diverse applications, we investigate two distinct sampling termination schemes: (i) synchronous termination, in which sampling and testing for all hypotheses stop simultaneously; and (ii) asynchronous termination, in which different hypotheses may stop at different times. For the synchronous case, we develop the Oracle Intersection (OI) test, based on the local false discovery rate statistic under a two-group mixture model. The OI test guarantees exact control of both the False Discovery Rate (FDR) and the False Non-Discovery Rate (FNR) at prespecified levels. We further propose a fully datadriven version that achieves asymptotic simultaneous control of FDR and FNR as the number of hypotheses m → ∞. While existing sequential tests use fixed stopping boundaries, the proposed tests employ stopping boundaries that adapt quickly with the samplesize, ensuring shrinkage of the continue-sampling region as the trial progresses. As a result, stopping times for both the oracle and data-driven tests converge to a finite constant as m → ∞. Moreover, the ratio of the expected sample size of the OI test to that of the Gap rule (He and Bartroff, 2021) converges to zero as m →∞. Extensive analyses of real and simulated datasets demonstrate the superiority of the proposed methods over existing approaches. Finally, we address the case of asynchronous termination and introduce the Oracle Stagewise (OS) test, constructed from the local false discovery rate statistic under a two-group mixture model. Employing adaptive stopping boundaries, the OS test drops hypotheses by accepting or rejecting them at interim stages of sampling until all hypotheses are decided. This stagewise procedure achieves simultaneous control of the FDR and FNR, while substantially reducing the total sample size relative to existing sequential methods (Bartroff and Song, 2020). To address challenges arising from composite hypotheses and the instability of parameter estimation caused by shrinking active sets, we further propose the Oracle Composite (OC) and Data-driven Composite (DC) tests. These hybrid procedures combine stagewise elimination with intersection-based testing, ensuring reliable estimation and improved practical applicability. Simulation studies demonstrate that the proposed methods substantially reduce the total sample size relative to existing sequential procedures while maintaining rigorous error control. Overall, this thesis advances the theory of large-scale multiple testing in both fixed-sample and sequential settings by establishing new optimality results under dependence and developing adaptive procedures that simultaneously achieve rigorous error control, finite stopping times, and improved sampling efficiency.
On Universal C∗ Algebras associated to Operator Spaces and Generalized Crossed Products
(2026-07-08) Banik, Sayan Kansa
In my thesis, we study generalized crossed product constructions of the group $ C^* $-algebra $ C^*(G) $ with respect to certain completely positive maps, where $ G $ is assumed to be a discrete amenable group. We also investigate the universal $ C^* $-algebra $ \mathcal{E}_\alpha $ introduced by Hirshberg, which is constructed from $ C^*(G) $ and a pure injective homomorphism $ \alpha \colon G \to G $. In particular, we analyze its relationship with Exel's construction of generalized crossed products associated with the endomorphism of $ C^*(G) $ induced by $ \alpha $, together with an appropriate choice of transfer operator. In addition, we study crossed products of $ C^*(G) $ arising from states and conditional expectations of the form $ E_H \colon C^*(G) \to C^*(H) $, where $ H < G $ is a proper subgroup. We examine how generalized crossed product constructions change when passing from endomorphism-based framework to to the case of completely positive maps. We study crossed products of $ C^* $-algebras with respect to states, focusing in particular on $ C^*(G) $ equipped with its canonical trace and construct a spatial representation of the system isomorphic to the universal crossed product construction. Further, we show that when $ G $ is virtually abelian, there exists no spatial representation of the system $ (C^*(G), E_H) $ inside $ B(\ell^2(G)) $. Finally, we construct a specific spatial representation of this system and show that, when $[G:H]=\infty$ , the resulting spatial $ C^* $-algebra is isomorphic to the corresponding universal crossed product In the other half, we make a detailed study of operator spaces associated to Brown's noncommutative unitary $C^\ast$-algebra $\mathcal{U}^{nc}_n$ and related $C^\ast$-algebras. Specifically, we identify the universal $C^{\ast}$-algebra $C^{\ast}\langle M_n(\C)^{\ast}\rangle$ of the operator space $M_n(\C)^{\ast}$ with the non-commutative $C^{\ast}$-algebra $\Y^{nc}_n$, the universal unital $C^{\ast}$-algebra generated by elements $u_{ij}$, $1\leq i,j\leq n$ satisfying the relations which make $[u_{ij}]$ a contractive matrix. We also exhibited several operator algebraic properties of $C^{\ast}\langle M_n(\C)^{\ast}\rangle$- in particular, we study the Lifting property (LP), residual finite dimensionality and primitivity of $C^{\ast}\langle M_n(\C)^{\ast}\rangle$. Further, we study the maximal and minimal operator space structure of the standard generators of $\U^{nc}_n$ as well as $\U^{nc}_{n, red}$. Finally, we discuss a natural compact quantum semigroup structure on $C^{\ast}\langle M_n(\C)^{\ast}\rangle$, characterizing invertible elements in its state space under convolution.
Consumer Welfare and Privatization in Mixed Markets: A Developing Country Perspective
(2026-07-14) Dutta, Priyanka
This thesis is presented in four chapters, each examining a distinct aspect of mixed market structures, with a consistent focus on consumer surplus as the primary metric for evaluating the desirability of privatization or private provision in the context of developing countries. The first chapter introduces a symmetric Cournot oligopoly model in which both public and private firms produce a homogeneous good and compete in quantities. We find that consumer surplus is maximized at the two extremes: when the market consists solely of public firms or solely of private firms. In contrast, mixed regimes consistently yield intermediate outcomes, never achieving either the highest or the lowest level of consumer surplus. This result is robust to the level of competition, the specific objective functions assigned to public firms, and holds across all log-concave demand functions and convex cost functions. Importantly, when cost functions are strictly convex, we show that, contrary to conventional wisdom, an increase in the number of firms does not necessarily support privatization; in fact, it may weaken the case for introducing private firms. The first chapter further extends the analysis to Bertrand competition with differentiated products, where firms compete in prices rather than quantities. Unlike in Cournot settings, firms' strategies are strategic complements under Bertrand competition. Despite this fundamental difference in the nature of competition, the key finding persists: mixed oligopolies never maximize consumer surplus. Moreover, in some cases, mixed markets can yield the lowest consumer surplus, even compared to fully public or private regimes. This counterintuitive outcome stems from the regime-contingent behavior of public firms. That is, an inefficient public firm with welfare concerns may respond to rival pricing by setting a higher price in a mixed regime than it would in a fully public one, thereby dampening consumer surplus. While the first chapter shows that mixed markets never yield the highest or lowest consumer surplus, this finding appears at odds with their widespread existence and institutional support across the globe. Several explanations may account for this disconnect. First, privatization decisions are often driven by objectives such as profitability or broader welfare considerations, rather than consumer surplus alone. Second, governments concerned with consumer surplus may still prefer a mixed regime in settings with weak competition policy, where full privatization could increase the risk of collusion. However, in the next two chapters, we demonstrate that mixed regimes can, in fact, top the consumer surplus ranking without resorting to alternative welfare metrics or relying on collusion-based explanations. What is required is to move beyond the symmetric, single-stage oligopoly framework used in the first chapter. The second chapter relaxes the assumption of symmetric firms by introducing cost heterogeneity, a key real-world feature, as firms often differ in cost structures for reasons unrelated to ownership. While privatization can improve efficiency, it may not fully eliminate these underlying cost differences. In this setting, we show that mixed regimes can deliver the highest consumer surplus. For instance, in a duopoly, privatizing the inefficient firm while retaining public ownership of the efficient one can outperform both fully public and fully private regimes. Conversely, if privatization targets the more efficient firm, leaving the less efficient one public, consumer surplus can be the lowest among all ownership structures. These results suggest that ownership design should account for firm-specific efficiency differences, and that optimal privatization policy may be highly context-dependent. The third chapter addresses another limitation of earlier models by considering a vertically related market, with upstream and downstream monopolists interacting in a two stage production process. Public firms in both sectors introduce two layers of inefficiency, while private firms in both sectors generate two rounds of markups, a classic double marginalization problem. We show that a mixed regime can yield the highest consumer surplus by eliminating one layer of inefficiency and one round of markup. This outcome is most likely when markups are moderate and inefficiencies are unevenly distributed across the two sectors. If the inefficiency of public firms is symmetric across the upstream and downstream sectors, mixed regimes tend to perform intermediately or even poorly in terms of consumer surplus. However, when public firm inefficiencies differ sufficiently across sectors, and markup levels are not too high, a mixed structure can outperform both extremes. Extending the model to a richer setting with both upstream and downstream oligopoly shows that the desirability of mixed regimes persists and, in fact, becomes stronger as competition intensifies. Thus, the interaction between vertical structure, firm efficiency, and market power plays a critical role in shaping welfare outcomes in privatization decisions. While the first three chapters evaluate privatization using consumer surplus as a welfare metric under standard oligopoly assumptions, the fourth chapter introduces a new dimension: congestion. In many markets such as healthcare, education, telecommunications, transportation etc., congestion disutility arises as firms serve more consumers, reducing individual utility. We model a Cournot oligopoly with congestion under both mixed and fully private regimes and characterize equilibrium outcomes in the presence of congestion. We conduct comparative static analysis with respect to market size and competition and determine a consumer surplus threshold: a fully private regime yields a higher consumer surplus if the relative cost inefficiency of the public firm exceeds this threshold. We show that congestion and increases in market size both lower this threshold, making privatization more favorable. However, the effect of competition is more nuanced. An increase in competition facilitates privatization only when the initial level of competition is low. Beyond a certain point, additional competition could in fact facilitate public provision. These results highlight how market structures and conditions influence optimal ownership structure in the presence of congestion. Collectively, the four chapters of this thesis underscore the importance of evaluating privatization and ownership design through the lens of consumer welfare, particularly in the context of developing economies. The results challenge simplistic assumptions about public versus private provisions and offer a nuanced framework for understanding when mixed markets can be not just a compromise, but an optimal institutional structure.
Design of reliability acceptance sampling plans
(Indian Statistical Institute, 2026-07-27) Das, Rathin
A reliability acceptance sampling plan (RASP) is used for sampling and decision-making in the acceptance or rejection of a lot of products based on lifetime data obtained from a life test. In practice, censored life tests are employed due to limitations in cost, time, and other testing resources for the collection of lifetime data. This thesis develops the design of optimal RASPs under various censoring schemes and testing environments. Design of optimal Bayesian RASPs (BRASPs) are considered under interval censoring schemes (ICS) and hybrid censoring schemes using Bayesian decision-theoretic approaches. These models incorporate the adversarial relationship between manufacturers and consumers, who differ in prior beliefs and utility functions. Global market competitiveness and rapid technological advancement have pushed manufacturers to produce products with very high reliability. For such products, the mean time to failure under normal operating conditions is often prohibitively long. To address this issue, a BRASP based on a novel adaptive simple step-stress partial accelerated life test (ASSSPALT) framework is proposed under Type-I censoring using common prior and utility functions. The adaptive scheme dynamically adjusts stress levels based on observed failures, providing a general framework that accommodates both accelerated and non-accelerated testing under Type-I censoring. For complex products, failure may occur due to multiple causes. The work considers the design of RASP for competing risk data under progressive Type-I interval censoring using the producer's and consumer's risk approaches. The asymptotic properties of maximum likelihood estimators are derived to develop optimal plans. A frailty-based model is employed to capture dependence among competing risks and evaluate its influence on sampling plan performance. Subsequently, RASP is extended to a Bayesian framework under interval censoring. Further, for complex products with very high reliability, the design of BRASP is considered based on ASSSPALT under Type-II censoring for competing risk data. This framework unifies both accelerated and non-accelerated testing scenarios for competing risk data under Type-II censoring. The proposed methodologies for designing RASPs are illustrated using real-life data.
Distributed Computation of Graph Structures by Mobile Agents
(2026-07-15) Chand, Prabhat Kumar
This thesis investigates how mobile agents with no centralised control can be employed in anonymous networks to perform efficient distributed graph computations. The network is modelled as a simple, undirected, anonymous graph with n nodes and m edges, where nodes are memoryless and indistinguishable, and edges represent bidirectional communication links or traversal paths for the agents. The mobile agents are uniquely identifiable, possess limited local memory, and operate under a local communication model, in which communication is restricted to agents colocated at the same node. Under this computational model, we explore how mobile agents can collaborate effectively to solve global network problems—including dispersion in the presence of crashes, the construction of spanning trees, the identification of dominating sets, and the analysis of sub-graph hierarchies. We study the time complexity and memory usage per agent required to solve the above problems. The first contributory chapter addresses the fault-tolerant dispersion problem, aiming to evenly spread mobile agents across an anonymous graph in the presence of crash faults, from two initial configurations: rooted, where all the agents start at a single node, and arbitrary, where agents are initially scattered across the graph in multiple clusters. This chapter explores how dispersion can be achieved under both settings, ensuring that each node eventually hosts at most one functional agent despite crashes. The next chapter presents the problem of computing dominating sets using mobile agents, where two different algorithms are introduced: one for computing minimal dominating sets in O(m) time when the agents are gathered at a single node, and another for scenarios where agents start from multiple clusters. Additionally, an ln ∆ approximation algorithm for the minimum dominating set problem is provided, where ∆ is the maximum degree of the graph. The subsequent chapter focuses on subgraph analytics using mobile agents dispersed across the nodes of a graph. We present algorithms for triangle counting (3-cycles) in general graphs and butterfly counting (4-cycles) in bipartite graphs. The triangle counting framework extends to related problems such as truss decomposition, triangle centrality, and local clustering coefficient. These methods enable the distributed identification of cohesive structures and dense subgraphs, with butterfly counting being of relevance to bipartite graphs commonly found in social network analysis and recommendation systems. The final chapter focuses on constructing tree structures, specifically BFS trees and minimum spanning trees, using mobile agents. These algorithms, which assume minimal prior knowledge, improve upon existing methods by achieving better time complexity and optimal memory usage. Throughout the thesis, these graph problems are explored through the lens of the mobile agent framework, focusing on minimising the time complexity of the algorithms and memory usage per agent.
A Regression Tree Framework for Denoising and Monitoring of Image Data
(Indian Statistical Institute, 2026-07-02) Basak, Subhasish
The proliferation of advanced image acquisition technologies has led to the routine collection of large-scale image data across numerous scientific domains. This widespread reliance on image data accentuates the imperative to develop robust and efficient imaging techniques, which are essential for supporting modern applications across various scientific and industrial domains. This dissertation focuses on the development and analysis of methods for image denoising and image monitoring, two fundamental tasks in modern image analysis. A wide array of image denoising techniques exists in the literature, each tailored to handle specific types of noise or structural characteristics. However, no single method proves universally optimal, as each comes with its own advantages and trade-offs. In the first part of the dissertation, different configurations of local neighbourhoods are investigated, and an adaptive framework is proposed that combines these with local clustering-based smoothing to effectively harness the advantages of both methodologies. The dissertation then introduces a regression tree-based framework utilizing Oblique-axis Regression Trees (ORT) to estimate discontinuous regression functions in finite-dimensional spaces and applies this methodology to achieve effective image denoising. Due to an alternative set of assumptions on the underlying regression function, the overall structure of the proofs is substantially simpler than those typically found in the existing literature on regression trees. Finally, leveraging the ORT framework, the dissertation introduces an original approach to monitor drift patterns within an image sequence. Even though gradual temporal variations, known as drifts, are frequently observed in image sequences, drift monitoring remains an underexplored research area. This dissertation thus makes an effort to address that gap. Theoretical analysis and numerical studies, conducted on both simulated and real-world data, demonstrate the broad applicability and effectiveness of the proposed methods.
Dynamic Property Ordering for Efficient Multi-Property Bounded Model Checking
(Indian Statistical Institute, 2026-06-16) Kumar, Vivek
Formal verification plays a critical role in ensuring the correctness of modern hardware designs. As the complexity of digital systems increases, designs are often associated with a large number of verification properties that must be analyzed within limited computational resources. In conventional multi-property bounded model checking (BMC), all properties are verified simultaneously. While this approach enables parallel analysis, difficult properties can consume a disproportionate amount of resources, causing simpler properties to be delayed and reducing the overall efficiency of bug detection. This thesis presents dynamic property ordering techniques for efficient multi-property verification using SAT-based bounded model checking in the ABC verification framework. The central idea is to verify properties individually and dynamically prioritize them based on their observed verification progress, allowing computational resources to be directed toward properties that are more likely to yield results within a given time budget.Two dynamic property ordering algorithms are proposed. The first algorithm, ALG1, employs a round-robin style strategy in which unsolved properties are periodically reordered according to the maximum verification depth (frame) reached, prioritizing properties that demonstrate greater progress. The second algorithm, ALG2, adopts a priority-based scheduling approach where each property’s priority is determined by its verification rate, measured as frames explored per second. Properties with higher progress rates are allocated greater verification resources. The proposed approaches are evaluated on benchmark suites from the Hardware Model Checking Competition (HWMCC) 2012 and 2013 and compared against two baselines: the conventional ABC multi-property verification method and an Equal Time Bounding (ETB) strategy that distributes the available verification time equally among all properties. Experimental results demonstrate that dynamic property ordering significantly improves verification efficiency. Both ALG1 and ALG2 solve more properties and achieve greater verification depth within the same time budget, while also accelerating bug discovery. Across the benchmark set, the proposed methods provide improvements exceeding 40% over the baseline approaches in key performance metrics. The results demonstrate that dynamic property scheduling is an effective technique for improving the scalability and effectiveness of multi-property bounded model checking, offering a practical solution for faster bug detection and enhanced utilization of verification resources.

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