A Probabilistic Framework for Improving Dense Object Detection in Underwater Image Data via Annealing-Based Data Augmentation

Object detection models typically perform well on images captured in controlled environments with stable lighting, water clarity, and viewpoint, but their performance degrades substantially in real-world underwater settings characterized by high variability and frequent occlusions. In this work, we address these challenges by introducing a novel data augmentation framework designed to improve robustness in dense and unconstrained underwater scenes. Using the DeepFish dataset, which contains images of fish in natural environments, we first generate bounding box annotations from provided segmentation masks to construct a custom detection dataset. We then propose a pseudo-simulated annealing-based augmentation algorithm, inspired by the copy-paste strategy of Deng et al. [1], to synthesize realistic crowded fish scenarios. Our approach improves spatial diversity and object density during training, enabling better generalization to complex scenes. Experimental results show that our method significantly outperforms a baseline YOLOv10 model, particularly on a challenging test set of manually annotated images collected from live-stream footage in the Florida Keys. These results demonstrate the effectiveness of our augmentation strategy for improving detection performance in dense, real-world underwater environments.

Graph Neural Network-Informed Predictive Flows for Faster Ford-Fulkerson and PAC-Learnability

We propose a learning-augmented framework for accelerating max-flow computation and image segmentation by integrating Graph Neural Networks (GNNs) with the Ford-Fulkerson algorithm. Rather than predicting initial flows, our method learns edge importance probabilities to guide augmenting path selection. We introduce a Message Passing GNN (MPGNN) that jointly learns node and edge embeddings through coupled updates, capturing both global structure and local flow dynamics such as residual capacity and bottlenecks. Given an input image, we propose a method to construct a grid-based flow network with source and sink nodes, extract features, and perform a single GNN inference to assign edge probabilities reflecting their likelihood of belonging to high-capacity cuts. These probabilities are stored in a priority queue and used to guide a modified Ford-Fulkerson procedure, prioritizing augmenting paths via an Edmonds-Karp-style search with bottleneck-aware tie-breaking. This avoids repeated inference over residual graphs while leveraging learned structure throughout optimization. We further introduce a bidirectional path construction strategy centered on high-probability edges and provide a theoretical framework relating prediction quality to efficiency via a weighted permutation distance metric. Our method preserves max-flow/min-cut optimality while reducing the number of augmentations in practice. We also outline a hybrid extension combining flow warm-starting with edge-priority prediction, establishing a foundation for learning-guided combinatorial optimization in image segmentation.

Ollivier-Ricci Curvature of Riemannian Manifolds and Directed Graphs with Applications to Graph Neural Networks

This thesis is an exposition of Ollivier-Ricci Curvature of metric spaces as introduced by Yann Ollivier, which is based upon the 1-Wasserstein Distance and optimal transport theory. We present some of the major results and proofs that connect Ollivier-Ricci curvature with classical Ricci curvature of Riemannian manifolds, including extensions of various theoretical bounds and theorems such as Bonnet-Myers and Levy-Gromov. Then we shift to results introduced by Lin-Lu-Yau on an extension of Ollivier-Ricci curvature on graphs, as well as the work of Jost-Liu on proving various combinatorial bounds for graph Ollivier-Ricci curvature. At the end of this thesis we present novel ideas and proofs regarding extensions of these results to directed graphs, and finally applications of graph-based Ollivier-Ricci curvature to various algorithms in network science and graph machine learning.