imperceptible to the human eye, but can lead the model to misclassify the
output (Chakraborty et al., 2018). It has been speculated that models based on
topological features are more robust against adversarial attacks (Gabrielsson
and Carlsson, 2019; Carlsson and Gabrielsson, 2018). A topological layer can
also be placed in neural networks to prevent from adversarial attacks and
enhance topological fidelity. In Brüel-Gabrielsson et al. (2019), it was shown
that the success rate of adversarial attacks is much smaller on networks with
a topological layer. Even thought the concept of adversarial attacks have not
been popular in brain network modeling, building deep learning models on brain
networks that are robust against adversarial attacks would be highly useful.

##  9 Conclusion & Disucssion

Figure 19: Top: Sulcal (blue) and gyral (red) pattern of two different
subjects. The sucal pattern can be represented as a forest, a collection of
trees (Huang et al., 2020a). Since the nodes of the forests are the surface
mesh vertices, the forest can be represented as about 300000 $\times$ 300000
sparse adjacency matrices. Computing similarity measures between the sulcal
forest is not trivial and may require topological distances. Middle:
enlargement of the temporal lobe. Bottom: The weighted spherical harmonic
representation with degree 70 and heat kernel bandwidth $0.001$ (Chung et al.,
2008) smoothing the node value where sulci are assigned value -1 and gyri are
assigned value 1. The smoothing obliviates the need for matching heterogeneous
data structure and comparisons of the different sulcal pattern across subject
can be done at the mesh vertex level. Such local geometric solution is not
readily available for brain networks.

In this review paper, we surveyed various topological distance and loss
functions based on persistent homology. We showed how such distance and loss
functions can be used in wide variety of learning applications including
regression, clustering and classification. ManyTDA approaches can successfully
differentiate the topological network differences. However, often it is
unclear where the difference is localized within the networks. In diagnostics
related problems such as determining if a subject belong to particular
clinical population, it is important to determine the topological differences;
however, more important questions is localizing the source of differences.
Since there is no one-to-one correspondence between the topological features
and the original data, it is often not possible to localize the signals, which
has been the biggest limitation of the TDA methods in brain imaging
applications. The future development TDA should be toward this important but
very difficult question.

Figure 20: Top: Dynamically changing correlation matrices computed from rs-
fMRI using the sliding window of size 60 for a subject (Huang et al., 2019a).
Bottom: The constructed correlation matrices are superimposed on top of the
white matter fibers, which are colored based on correlations between
parcellations. The structural brain network is a more or less a tree like
structure without many loops or cycles while the functional bran networks can
have many loops and cycles. Thus, aligning functional brain networks on top of
structural brain network directly will not work. It requires a new topological
approach for aligning topologically different networks.

Compared to TDA methods, geometric methods are more adapt at detecting
localized signals in brain imaging. Figure 19 displays the sucal and gyral
trees obtained from the brain surface mesh (Huang et al., 2019b), where trees
are teated as heat source with value +1 on gyral trees and heat sink with
value -1 on sulcal trees. Then Isotropic diffusion is applied to produce the
smooth map of sulcal and gyral trees. Such smooth maps can be easily compared
across different subjects. For instance, two-sample $t$-statistic at each mesh
vertex is performed localizing group differences. Such localized signal
detection is difficult if not impossible with many existing TDA methods.

Persistent homology features are by definition global summary measures and
they might be more useful for tasks that do not involves identifying the
source of signal differences. It might be more useful in discrete decision
making tasks such as clustering and classifications. In fact TDA has begin to
be more useful in deep learning (Chen et al., 2019).

The paper reviewed how TDA can be used in statistic brain networks that do not
change over time. For dynamically changing brain networks over time, it is
unclear how TDA can be applied yet. Consider the dynamically changing
correlating matrices of resting-state fMRI (Figure 20). At each each
correlation matrix, we can apply persistent homology and obtain PD, which is
the discrete representation of connectivity at one particular time point. It
is unclear how to integrate such discrete topological representation over
multiple time points in a continuous fashion. Dynamic-TDA as introduced in
Songdechakraiwut and Chung (2020a) encodes rsfMRI as a time-ordered sequence
of Rips complexes and their corresponding barcodes in studying dynamically
changing topological patterns over time. Dyanmic-TDA may provide some
suggestion for extending TDA to time varying brain networks. This is left as
promising future research direction.

## Acknowledgements

This study was supported by NIH grants R01 EB022856 and R01 EB028753 and NSF
grant MDS-2010778. We would like to thank Andery Gritsenko of Northeastern
University for the computation of the largest cycle in a graph. We also like
to thank Alex Cole of University of Amsterdam for the discussion of the
witness complex. We thank Hyekyung Lee of Seoul National University Hospital
for discussion on sparse filtrations and kernel distances. We would like to
thank Robin Henderson of Newcastle University, U.K. for providing the
coordinates and connectivity information of nodes of the binary tree used in
Garside et al. (2020). We would like to thank Hill Goldsmith of University of
Wisconsin for providing one subject rsfMRI data used in the figure, which came
from the study Huang et al. (2019a). We thank Taniguchi Masanobu of Waseda
University for discussion on canonical correlations.

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