Gideon Klaila | Searcharxiv

arXiv2024

Stability of the persistence transformation

In this paper, we introduce the persistence transformation, a novel methodology in Topological Data Analysis (TDA) for applications in time series data which can be obtained in various areas such as science, politics, economy, healthcare, engineering, and beyond. This approach captures the enduring presence or `persistence' of signal peaks in time series data arising from Morse functions while preserving their positional information. Through rigorous analysis, we demonstrate that the proposed persistence transformation exhibits stability and outperforms the persistent diagram of Morse functions (with respect to filtration, e.g., the upper levelset filtration). Moreover, we present a modified version of the persistence transformation, termed the reduced persistence transformation, which retains stability while enjoying dimensionality reduction in the data. Consequently, the reduced persistence transformation yields faster computational results for subsequent tasks, such as classification, albeit at the cost of reduced overall accuracy compared to the persistence transformation. However, the reduced persistence transformation finds relevance in specific domains, e.g., MALDI-Imaging, where positional information is of greater significance than the overall signal height. Finally, we provide a conceptual outline for extending the persistence diagram to accommodate higher-dimensional input while assessing its stability under these modifications.

math.AT↗

arXiv2023

Supervised topological data analysis for MALDI mass spectrometry imaging applications

Background: Matrix-assisted laser desorption/ionization mass spectrometry imaging (MALDI MSI) displays significant potential for applications in cancer research, especially in tumor typing and subtyping. Lung cancer is the primary cause of tumor-related deaths, where the most lethal entities are adenocarcinoma (ADC) and squamous cell carcinoma (SqCC). Distinguishing between these two common subtypes is crucial for therapy decisions and successful patient management. Results: We propose a new algebraic topological framework, which obtains intrinsic information from MALDI data and transforms it to reflect topological persistence. Our framework offers two main advantages. Firstly, topological persistence aids in distinguishing the signal from noise. Secondly, it compresses the MALDI data, saving storage space and optimizes computational time for subsequent classification tasks. We present an algorithm that efficiently implements our topological framework, relying on a single tuning parameter. Afterwards, logistic regression and random forest classifiers are employed on the extracted persistence features, thereby accomplishing an automated tumor (sub-)typing process. To demonstrate the competitiveness of our proposed framework, we conduct experiments on a real-world MALDI dataset using cross-validation. Furthermore, we showcase the effectiveness of the single denoising parameter by evaluating its performance on synthetic MALDI images with varying levels of noise. Conclusion: Our empirical experiments demonstrate that the proposed algebraic topological framework successfully captures and leverages the intrinsic spectral information from MALDI data, leading to competitive results in classifying lung cancer subtypes. Moreover, the frameworks ability to be fine-tuned for denoising highlights its versatility and potential for enhancing data analysis in MALDI applications.

stat.ML↗