In this paper we introduce the pure exploration transductive linear bandit problem: given a set of measurement vectors $\mathcal{X}\subset \mathbb{R}^d$, a set of items $\mathcal{Z}\subset \mathbb{R}^d$, a fixed confidence $\delta$, and an unknown vector $\theta^{\ast}\in \mathbb{R}^d$, the goal is to infer $\arg\max_{z\in \mathcal{Z}} z^\top\theta^\ast$ with probability $1-\delta$ by making as few sequentially chosen noisy measurements of the form $x^\top\theta^{\ast}$ as possible. When $\mathcal{X}=\mathcal{Z}$, this setting generalizes linear bandits, and when $\mathcal{X}$ is the standard basis vectors and $\mathcal{Z}\subset \{0,1\}^d$, combinatorial bandits. The transductive setting naturally arises when the set of measurement vectors is limited due to factors such as availability or cost. As an example, in drug discovery the compounds and dosages $\mathcal{X}$ a practitioner may be willing to evaluate in the lab in vitro due to cost or safety reasons may differ vastly from those compounds and dosages $\mathcal{Z}$ that can be safely administered to patients in vivo. Alternatively, in recommender systems for books, the set of books $\mathcal{X}$ a user is queried about may be restricted to known best-sellers even though the goal might be to recommend more esoteric titles $\mathcal{Z}$. In this paper, we provide instance-dependent lower bounds for the transductive setting, an algorithm that matches these up to logarithmic factors, and an evaluation. In particular, we present the first non-asymptotic algorithm for linear bandits that nearly achieves the information-theoretic lower bound.

We make progress on both computational and statistical aspects of the problem.

The sustainability of modern cities highly depends on efficient water distribution management, including effective pressure control and leak detection and localization. Accurate information about the network hydraulic state is therefore essential. This article presents a comparison between two data-driven state estimation methods based on the Unscented Kalman Filter (UKF), fusing pressure, demand and flow data for head and flow estimation. One approach uses a joint state vector with a single estimator, while the other uses a dual-estimator scheme. We analyse their main characteristics, discussing differences, advantages and limitations, and compare them theoretically in terms of accuracy and complexity. Finally, we show several estimation results for the L-TOWN benchmark, allowing to discuss their properties in a real implementation.

Uncertainty quantification plays an important role in achieving trustworthy and reliable learning-based computational imaging. Recent advances in generative modeling and Bayesian neural networks have enabled the development of uncertainty-aware image reconstruction methods. Current generative model-based methods seek to quantify the inherent (aleatoric) uncertainty on the underlying image for given measurements by learning to sample from the posterior distribution of the underlying image. On the other hand, Bayesian neural network-based approaches aim to quantify the model (epistemic) uncertainty on the parameters of a deep neural network-based reconstruction method by approximating the posterior distribution of those parameters. Unfortunately, an ongoing need for an inversion method that can jointly quantify complex aleatoric uncertainty and epistemic uncertainty patterns still persists. In this paper, we present a scalable framework that can quantify both aleatoric and epistemic uncertainties. The proposed framework accepts an existing generative model-based posterior sampling method as an input and introduces an epistemic uncertainty quantification capability through Bayesian neural networks with latent variables and deep ensembling. Furthermore, by leveraging the conformal prediction methodology, the proposed framework can be easily calibrated to ensure rigorous uncertainty quantification. We evaluated the proposed framework on magnetic resonance imaging, computed tomography, and image inpainting problems and showed that the epistemic and aleatoric uncertainty estimates produced by the proposed framework display the characteristic features of true epistemic and aleatoric uncertainties. Furthermore, our results demonstrated that the use of conformal prediction on top of the proposed framework enables marginal coverage guarantees consistent with frequentist principles.

We call (1) the process model and (2) the measurement model.

Choose a random initial value for p as well as h, and construct H = diag( h) and P = diag( p) .

A proposition that connects randomness and compression is put forward via Gibbs entropy over set of measurement vectors associated with a compression process. The proposition states that a lossy compression process is equivalent to {\it directed randomness} that preserves information content. The proposition originated from the observed behaviour in newly proposed {\it Dual Tomographic Compression} (DTC) compress-train framework. This is akin to tomographic reconstruction of layer weight matrices via building compressed sensed projections, via so-called {\it weight rays}. This tomographic approach is applied to previous and next layers in a dual fashion, that triggers neuronal-level pruning. This novel model compress-train scheme appears in iterative fashion and acts as a smart neural architecture search, The experiments demonstrated the utility of this dual-tomography producing state-of-the-art performance with efficient compression during training, accelerating and supporting lottery ticket hypothesis. However, random compress-train iterations having similar performance demonstrated the connection between randomness and compression from statistical physics perspective, we formulated the so-called {\it Gibbs randomness-compression proposition}, signifying randomness-compression relationship via Gibbs entropy. Practically, the DTC framework provides a promising approach for massively energy- and resource-efficient deep learning training.

State estimation or filtering serves as a fundamental task to enable intelligent decision-making in applications such as autonomous vehicles, robotics, healthcare monitoring, smart grids, intelligent transportation, and predictive maintenance. Standard filtering assumes prior knowledge of noise statistics to extract latent system states from noisy sensor data. However, real-world scenarios involve abnormalities like outliers, biases, drifts, and missing observations with unknown or partially known statistics, limiting conventional approaches. This thesis presents novel robust nonlinear filtering methods to mitigate these challenges. Based on insights from our filtering proposals, we extend the formulations to offline estimation/learning setups and propose smoothing extensions. Our methods leverage Bayesian inference frameworks, employing both deterministic and stochastic approximation techniques including Variational Inference (VI) and Particle Filters/Sequential Monte Carlo (SMC). We also study theoretical estimation limits using Bayesian Cramér-Rao bounds (BCRBs) in the context of measurement abnormalities. To validate the performance gains of the proposed methods, we perform simulations and experiments in scenarios including target tracking, indoor localization, 3D point cloud registration, mesh registration, and pose graph optimization. The fundamental nature of the work makes it useful in diverse applications, with possible future extensions toward developing outlier-robust machine learning pipelines, learning system dynamics from anomalous data, and addressing challenges in generative AI where standard diffusion models struggle with outliers, imbalanced datasets, and mode collapse.