TDA and Deep Learning: A Practical Study on Arrhythmia Detection

Healthcare, as a field, offers an exciting spectrum of possibilities. Every incremental improvement in performance could translate into saved lives or enhanced quality of life. Therefore, achieving satisfactory generalization is a non-negotiable goal.

Topological Data Analysis, or TDA, is a relatively young discipline that sprouted from the fertile grounds of applied topology and computational geometry. Its goal is to devise robust mathematical, statistical, and algorithmic techniques to leverage the topological and geometric structures buried within data. While TDA is primarily adept at handling three-dimensional data, it has demonstrated its versatility in other areas too, such as time-series data.

There are many ways to improve generalization; some augment the size of their dataset, others the quality. Then there are those who explore new models and a few pioneers who create their own unique techniques. The advent of deep learning is perhaps the best testament to this innovation. Today, I invite you to walk a different path—innovate by modifying the description of your data. Our guide on this journey will be Topological Data Analysis (TDA). A recently developed field, TDA merges applied topology and computational geometry to exploit the topological and geometric structures in data.

A deeper dive into TDA and its integration with deep learning can be found in an article I previously penned here.

This article is a fragment of the work I accomplished at Fujitsu’s AI laboratories in Tokyo, in collaboration with the Datashape team from INRIA (French Research Institute). Although I cannot disclose the entirety of our work, I assure you, there's ample information available via the referenced Github and paper to help you comprehend and apply TDA.

Did you know heart attacks and strokes are among the top five causes of death in the US? It is little wonder, then, that tech giants like Apple are developing smart heart monitors. Your heart, that indefatigable muscle, is truly miraculous but even it falters occasionally, be it due to alcohol, stress, intense exercise, or just a scary movie. Such failures can sometimes prove lethal. Arrhythmias, conditions that disrupt your heartbeat's rhythm or frequency, fall into this category. Detecting these events and tracking their frequency can pave the way to timely health interventions. But how do we achieve smart monitoring?

Welcome to the realm of machine learning—our tool for smart monitoring. However, possessing an impressive model isn’t enough. Thankfully, researchers have compiled a series of open-source datasets, available on the Physionet platform, to facilitate further work. The annotation of these datasets alleviates the issue of heartbeat detection, which, though not a major hurdle, still requires an efficient solution such as wavelet transform or 1D-CNN.

Given that ECGs are one-dimensional time series, how do we describe their shape and time relationships? This challenge isn’t exclusive to ECGs and finds relevance in numerous domains. And this is where topology lends a helping hand.

By analyzing a single heartbeat, we can gather data regarding PQRST events, while the examination of an ECG can yield the RR-intervals—the delay between consecutive R peaks. These measurements are useful, but not quite enough.

The main challenges for arrhythmia classification include individual differences, bradycardia, and tachycardia. TDA, specifically persistent homology theory, addresses these challenges by uniquely characterizing the shape of ECG signals, while being robust against the deformations of the patterns.

The theory of persistent homology allows us to represent a filtered simplicial complex (in our case, the ECG signal) with a persistence barcode. A barcode diagram is not directly exploitable by machine learning due to its non-uniform dimensions. Hence, we convert these diagrams into Betti Curves.

With two uniformly-sized Betti Curves for each heartbeat, we can now apply previously trained models to similar problems that take similarly distributed curves as input.

Given the rich shape information contained in the Betti Curves, our next task is to find the best model to leverage this information. Options include kNN with Euclidean or Dynamic Time Warping metrics, Boosted Trees, or CNN networks.

So, does TDA really improve performance? After running numerous experiments, we found that TDA not only aids generalization but also has a more significant impact on classification. TDA, being a specialized measure of topology and shape, offers differentiation between abnormal heartbeats—vital for proper classification.

Finally, while this introduction to applied Topological Data Analysis hopefully piqued your interest, it's crucial to remember that no matter your level of innovation, the quality and availability of data remain the biggest limiting factors. As we dream of a future with centralized high-quality data sources, we might just change the game!