Projects
A summary of my current and past projects during my research activities.
Modeling Quantum States with Tensor Networks and Hypergraphs
The aim of this project is to develop efficient tensor network (TN) algorithms for solving high-dimensional quantum many-body problems on 2D structured graphs. The main goal of this work is to leverage machine learning methods on hypergraphs to accelerate tensor network routines, such as contraction techniques. This is followed by employing the Universal Approximation Theorems of deep learning models to hierarchically enhance tensor network models. These models will simultaneously generate uncorrelated samples for efficient evaluation of Hamiltonian energies.
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Eikonal Equation on Graphs
The Eikonal equation models wavefront propagation as the solution to a nonlinear partial differential equation, often used in fields like physics, geometry, and computer vision. It is closely tied to the Hamilton-Jacobi-Bellman (HJB) equation, which generalizes the Eikonal equation within optimal control theory by incorporating dynamics and cost functions. Both share foundational principles in variational analysis, with the Eikonal equation representing a special case for shortest-path or geodesic problems. This project aims to adapt the continuous Eikonal equation for discrete data on connected graphs, enabling applications such as distance computation, data clustering, image labeling, and vessel or road tracking.
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Data Labeling by gradient descent flows on statistical manifold
This project addresses the image labeling problem, focusing on unique pixel-wise label assignments to simplify images and reduce redundancy. Building on the geometric assignment flow framework, it introduces a novel order-constrained segmentation method for volumetric data, demonstrated with Optical Coherence Tomography (OCT) scans of the retina.
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Local Feature Extraction for Geometric and Deep Image Representation
Local feature extraction is a fundamental task in computer vision and machine learning, enabling the representation of data in compact, discriminative forms for tasks such as classification and segmentation. This project provides two different local feature extractors for image representation geometry by leveraging symmetric positive semidefinite (SPD) manifolds in connection with covariance or similarity matrices. In this context SPD manifolds provide a natural framework for modeling local descriptors, capturing invariant properties such as scale and rotation. Deep neural networks further enhance this process by learning task-specific feature mappings from raw data to the SPD manifold, ensuring robust and adaptive feature extraction. The second feature extractor is baised on representational power of deep learning, paving the way for advancements in diverse applications like medical imaging, biometrics, and object recognition.
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