Unsupervised Feature Extraction and Selection: Feature extraction and selection in the presence of nonlinear dependencies among the data is a fundamental challenge in unsupervised learning. We propose using a Gram-Schmidt (GS) type orthogonalization process over function spaces to detect and map out such dependencies. Specifically, by applying the GS process over some family of functions, we construct a series of covariance matrices that can either be used to identify new large-variance directions, or to remove those dependencies from known directions. In the former case, we provide information-theoretic guarantees in terms of entropy reduction. In the latter, we provide precise conditions by which the chosen function family eliminates existing redundancy in the data. Each approach provides both a feature extraction and a feature selection algorithm. Our feature extraction methods are linear, and can be seen as natural generalization of principal component analysis (PCA). We provide experimental results for synthetic and real-world benchmark datasets which show superior performance over state-of-the-art (linear) feature extraction and selection algorithms. Surprisingly, our linear feature extraction algorithms are comparable and often outperform several important nonlinear feature extraction methods such as autoencoders, kernel PCA, and UMAP. Furthermore, one of our feature selection algorithms strictly generalizes a recent Fourier-based feature selection mechanism, yet at significantly reduced complexity.

Supervised Feature Selection: Supervised feature selection is essential for improving model interpretability and predictive performance, particularly in high-dimensional settings where irrelevant and redundant features can degrade classification accuracy. In this paper, we propose Supervised Gram-Schmidt Feature Selection (SGFS), a novel algorithm that iteratively selects features by leveraging the Gram-Schmidt (GS) orthogonalization process in function spaces. SGFS constructs an orthonormal basis of functions such that each selected feature maximally contributes to the classification task via the Bayes predictor. At each iteration, the algorithm updates the feature space by removing the projections of the remaining features onto the previously selected orthonormal functions, thereby identifying the feature that minimizes the Bayes error rate. By operating in function space, SGFS effectively captures nonlinear dependencies among features. We evaluate the proposed method on real-world datasets and demonstrate its superiority in classification accuracy compared to state-of-the-art feature selection techniques.

The first part of the project introduces the Gram-Schmidt Feature Reduction Class Activation Map (GFR-CAM), a novel framework designed to address the limitations of existing CAM methods in explaining deep learning models. Unlike traditional approaches that generate a single dominant explanation, GFR-CAM leverages Gram-Schmidt orthogonalization to extract a sequence of orthogonal, information-rich components from feature maps. This hierarchical decomposition provides a multi-faceted and holistic view of model reasoning, enabling richer interpretability in complex visual scenes with multiple objects or semantic parts. Experiments on ResNet-50 and Swin Transformer architectures across the ImageNet and PASCAL VOC datasets demonstrate that GFR-CAM produces meaningful, disentangled explanations while maintaining competitive performance with state-of-the-art methods.

The second part of the project develops the Gram-Schmidt Functional Selection Class Activation Map (GFS-CAM), a gradient-free interpretability method that overcomes key drawbacks of existing CAM techniques. By systematically applying Gram-Schmidt Functional Selection (GFS), the method iteratively orthogonalizes and selects feature maps that maximize residual variance, ensuring the extracted features are informative, non-redundant, and mutually orthogonal. This principled selection process allows GFS-CAM to generate disentangled explanations that separate distinct semantic concepts — such as multiple objects or background — into clear, interpretable heatmaps. Evaluations on VGG-16, Inception-v3, and ResNeXt-50 architectures with ILSVRC2012 and PASCAL VOC 2007 benchmarks confirm that GFS-CAM significantly outperforms state-of-the-art methods across faithfulness, localization, and ROAD metrics, establishing it as a powerful tool for trustworthy CNN interpretability.

Conformal Prediction for Time Series: This project introduces Conformal Prediction for Autoregressive Models (CPAM), a computationally efficient and theoretically grounded framework for reliable uncertainty quantification in time-series forecasting. Unlike standard conformal methods that assume independent data, CPAM integrates split conformal inference with classical time-series tools, combining Yule-Walker estimation, AIC-based autoregressive order selection, and residual-based calibration to construct prediction intervals with provable marginal coverage guarantees under weak temporal dependence. The method is analytically tractable, avoids repeated model refitting, and scales efficiently to large datasets. Experiments on both synthetic and real-world data, including wind power and solar irradiance forecasting, show that CPAM achieves coverage close to nominal levels while maintaining practical interval widths, making it a reliable tool for high-stakes forecasting applications.

Optimal Data Splitting for Split Conformal Prediction: This project investigates the problem of optimal data splitting in split conformal prediction, where the choice of dividing data between training and calibration has a direct impact on predictive performance. Standard approaches often use arbitrary or fixed splits, which can yield suboptimal trade-offs between model accuracy and coverage reliability. To address this limitation, the work develops a principled framework for selecting split proportions that balance the competing demands of model training and calibration, thereby improving both the efficiency and robustness of prediction intervals. Theoretical analysis and empirical results across diverse datasets demonstrate that optimal splitting produces narrower and more informative intervals while preserving rigorous coverage guarantees, providing a systematic and practical enhancement to one of the most widely used conformal prediction methods.

Conformal Prediction in Medical Imaging with Vision Transformers This project applies conformal prediction to medical imaging, with a focus on Vision Transformer (ViT) architectures to enable trustworthy AI-assisted decision-making in healthcare. While ViTs have demonstrated state-of-the-art performance in image analysis, their predictions often lack calibrated uncertainty estimates, a critical requirement for safety-sensitive clinical applications. This work integrates conformal calibration into ViT-Base and ViT-Small models, equipping them with the ability to output statistically valid and interpretable prediction sets that adapt to task complexity. The approach remains computationally efficient while delivering reliable uncertainty quantification. Evaluations on medical imaging datasets confirm that conformalized ViTs achieve robust coverage guarantees without sacrificing predictive accuracy, making them well-suited for real-world deployment in clinical decision-support systems.

Security of Cyber-Physical Systems: This project studies replay attack detection in cyber-physical systems (CPSs) through the use of physical watermarking, a well-established defense mechanism. While most watermarking approaches in the literature are designed for discrete-time systems, real-world physical systems generally evolve in continuous time. This work analyzes the effect of watermarking on sampled-data continuous-time systems controlled via a Zero-Order Hold, with a focus on how sampling impacts detection performance. A procedure is proposed to determine a suitable sampling period that balances detectability with acceptable control performance. The effectiveness of the theoretical results is demonstrated through simulations on a quadrotor system.

Safety in Drug Delivery Systems: This project develops an approach for controlling fractional-order semilinear systems subject to linear constraints, motivated by applications in drug delivery. The proposed design procedure consists of two stages: first, a linear state-feedback control law is introduced to prestabilize the system in the absence of constraints, with stability and convergence formally established using Lyapunov theory. Second, a constraint-handling mechanism based on the Explicit Reference Governor (ERG) is employed to ensure constraint satisfaction at all times. The ERG works by translating the linear constraints into bounds on the Lyapunov function and manipulating an auxiliary reference to guarantee that the Lyapunov function remains below a threshold value. The method is applied to the control of drug concentration in a drug delivery system and validated through extensive simulation results.