RESEARCH

Graph representation learning

Graph representation learning is a field of machine learning focused on developing techniques to learn meaningful and effective representations of graph-structured data. Graphs are flexible data structures that can be used to summarize interactions between individual units in complex systems and are commonly used in various domains such as social networks, molecular structures, financial networks and transportation systems. However, while the flexibility of their structure makes them easy to use, graphs can become complex objects when used as an organized information repository from a machine learning (ML) perspective. While traditional ML methods have been successful with data in vector form in affine spaces, graph-structured data has been a challenging area for ML and deep learning (DL). 

Over the past decade, topological data analysis (TDA) has emerged as a potent approach for capturing shape-pattern developments within graph-structured data across varying resolutions. These topological characteristics have exhibited considerable prowess in fundamental ML tasks within graph representation learning, including node classification, link prediction, and graph classification.

Within our research group, we introduce innovative methodologies that harness these topological methods and fuse them with cutting-edge deep learning models, notably graph neural networks. We deploy these pioneering machine learning algorithms across diverse domains such as computer-aided drug discovery, the resilience of power grids, anomaly detection, blockchain analysis, and numerous others.

medical image analysis

Much like graph representation learning, TDA has emerged as a potent tool for extracting features in image analysis. By harnessing the inherent shape patterns within images, TDA has proven particularly effective. Over the last decade, TDA techniques have found successful applications within medical image analysis across various domains such as neurology, radiology, ophthalmology, and histopathology, among others.

Within our research group, we've utilized topological methods to delve into diverse areas. This includes cancer diagnosis through the examination of histopathological images, the detection of thoracic diseases via chest X-rays, and the grading of tumors using MRI images. Remarkably positive outcomes have been achieved in these endeavors. Presently, we're taking a step further by synergizing these crucial topological features with state-of-the-art deep learning models. Our overarching goal is to craft robust and precise clinical decision support systems.

THEORY OF TOPOLOGICAL DATA ANALYSIS

Over the past twenty years, Topological Data Analysis (TDA) has emerged as a powerful feature extraction method for complex data, including point clouds, graphs, and images. Its application across various domains introduces a fresh perspective that complements existing approaches. To enhance its capabilities and glean more insights, a deeper comprehension of its mathematical underpinnings is necessary, along with the refinement of current techniques. Our research group is dedicated to pursuing two primary avenues of exploration in this regard.

The first focus lies in devising methods to adapt multiparameter persistence to real-world data, a step that holds the potential to significantly enhance the efficacy of single parameter persistence. Within TDA, persistent homology plays a pivotal role by capturing the evolution of topological features in relation to a filtering function. However, the existing framework restricts the use of a single domain function, despite the fact that many datasets can be characterized by multiple domain functions that yield crucial insights. Our group endeavors to facilitate the implementation of a multiparameter persistence approach across diverse domains, effectively harnessing the capabilities of the multiparameter persistence module.

The second aspect of our research involves establishing a connection between two distinct yet interconnected mathematical domains: applied topology and metric geometry. Gaining a geometric understanding of persistent homology holds the promise of enhancing its utility, with metric geometry providing an array of sophisticated tools to aid in this endeavor. Our group is actively immersed in this pursuit, striving to refine persistent homology's computational efficiency and unlock its potential as a robust dimension reduction tool.