Topological Learning for Brain Networks

Abstract

This paper proposes a novel topological learning framework that can integrate networks of different sizes and topology through persistent homology. This is possible through the introduction of a new topological loss function that enables such challenging task. The use of the proposed loss function bypasses the intrinsic computational bottleneck associated with matching networks. We validate the method in extensive statistical simulations with ground truth to assess the effectiveness of the topological loss in discriminating networks with different topology. The method is further applied to a twin brain imaging study in determining if the brain network is genetically heritable. The challenge is in overlaying the topologically different functional brain networks obtained from the resting-state functional magnetic resonance imaging (fMRI) onto the template structural brain network obtained through the diffusion tensor imaging (DTI).

Discussion of ‘Event history and topological data analysis’

“…These types of persistent diagrams are difficult to analyse since the locations of the scatter points and the number of scatter points do not correspond across different persistent diagrams. For a 1-skeleton, there exists a more efficient 1D filtration called the graph filtration, which filters edge weights varying from −∞ to ∞ ( Chung et al, 2019 ; Songdechakraiwut & Chung, 2020b ).…”

“…For convenience, we set the death value of 0-cycles to some fixed number c > w ( q −1) . Then the persistence diagram of the graph filtration is simply ( w (1) , c ), ( w (2) , c ), …, ( w ( q −1) , c ) forming 1D scatter points along the horizontal line y = c , and making various analysis and operations, including matching, significantly simplified ( Songdechakraiwut & Chung, 2020b ). Figure 1 illustrates the graph filtration and corresponding 1D scatter points in persistence diagrams on the binary tree used in Garside et al (2021) .…”

“…For general graphs beyond trees, there will be 1-cycles and β 1 is monotonically decreasing over the graph filtration ( Chung et al, 2019 ). Similarly, the persistence diagram is given as 1D scatter points along the vertical line ( Songdechakraiwut & Chung, 2020b ). Subsequently, statistical analysis on β 0 , β 1 curves as well as their persistence diagram can be performed using existing tools in the order statistics.…”

Topological Data Analysis of Human Brain Networks Through Order Statistics

“…Since the real brain networks are often affected by heterogeneity and intrinsic randomness [30, 31], it is challenging to build a coherent statistical framework to transform these topological features as quantitative measures to compare across different brain networks by averaging or matching [32]. The brain networks are inherently noisy which makes it even harder to establish similarity across networks.…”

“…We consider a random complete graph, where all the nodes are connected with its edge weights randomly drawn from a continuous distribution. The consideration of a complete graph model simplifies building graph filtration straightforward [22, 32]. We then compute the expected 0D and 1D barcodes through the order statistics [38–43].…”