🍩 Database of Original & NonTheoretical Uses of Topology
(found 5 matches in 0.001265s)


Reviews: Topological Distances and Losses for Brain Networks (2021)
Moo K. Chung, Alexander Smith, Gary ShiuAbstract
Almost all statistical and machine learning methods in analyzing brain networks rely on distances and loss functions, which are mostly Euclidean or matrix norms. The Euclidean or matrix distances may fail to capture underlying subtle topological differences in brain networks. Further, Euclidean distances are sensitive to outliers. A few extreme edge weights may severely affect the distance. Thus it is necessary to use distances and loss functions that recognize topology of data. In this review paper, we survey various topological distance and loss functions from topological data analysis (TDA) and persistent homology that can be used in brain network analysis more effectively. Although there are many recent brain imaging studies that are based on TDA methods, possibly due to the lack of method awareness, TDA has not taken as the mainstream tool in brain imaging field yet. The main purpose of this paper is provide the relevant technical survey of these powerful tools that are immediately applicable to brain network data. 
Quantitative and Interpretable Order Parameters for Phase Transitions From Persistent Homology (2020)
Alex Cole, Gregory J. Loges, Gary ShiuAbstract
We apply modern methods in computational topology to the task of discovering and characterizing phase transitions. As illustrations, we apply our method to four twodimensional lattice spin models: the Ising, square ice, XY, and fullyfrustrated XY models. In particular, we use persistent homology, which computes the births and deaths of individual topological features as a coarsegraining scale or sublevel threshold is increased, to summarize multiscale and highpoint correlations in a spin configuration. We employ vector representations of this information called persistence images to formulate and perform the statistical task of distinguishing phases. For the models we consider, a simple logistic regression on these images is sufficient to identify the phase transition. Interpretable order parameters are then read from the weights of the regression. This method suffices to identify magnetization, frustration, and vortexantivortex structure as relevant features for phase transitions in our models. We also define "persistence" critical exponents and study how they are related to those critical exponents usually considered. 
Topological Echoes of Primordial Physics in the Universe at Large Scales (2020)
Alex Cole, Matteo Biagetti, Gary ShiuAbstract
We present a pipeline for characterizing and constraining initial conditions in cosmology via persistent homology. The cosmological observable of interest is the cosmic web of large scale structure, and the initial conditions in question are nonGaussianities (NG) of primordial density perturbations. We compute persistence diagrams and derived statistics for simulations of dark matter halos with Gaussian and nonGaussian initial conditions. For computational reasons and to make contact with experimental observations, our pipeline computes persistence in subboxes of full simulations and simulations are subsampled to uniform halo number. We use simulations with large NG (\$f_\\rm NL\\textasciicircum\\rm loc\=250\$) as templates for identifying data with mild NG (\$f_\\rm NL\\textasciicircum\\rm loc\=10\$), and running the pipeline on several cubic volumes of size \$40~(\textrm\Gpc/h\)\textasciicircum\3\\$, we detect \$f_\\rm NL\\textasciicircum\\rm loc\=10\$ at \$97.5\%\$ confidence on \$\sim 85\%\$ of the volumes for our best single statistic. Throughout we benefit from the interpretability of topological features as input for statistical inference, which allows us to make contact with previous firstprinciples calculations and make new predictions. 
The Persistence of Large Scale Structures I: Primordial NonGaussianity (2020)
Matteo Biagetti, Alex Cole, Gary ShiuAbstract
We develop an analysis pipeline for characterizing the topology of large scale structure and extracting cosmological constraints based on persistent homology. Persistent homology is a technique from topological data analysis that quantifies the multiscale topology of a data set, in our context unifying the contributions of clusters, filament loops, and cosmic voids to cosmological constraints. We describe how this method captures the imprint of primordial local nonGaussianity on the latetime distribution of dark matter halos, using a set of Nbody simulations as a proxy for real data analysis. For our best single statistic, running the pipeline on several cubic volumes of size \$40~(\rm\Gpc/h\)\textasciicircum\3\\$, we detect \$f_\\rm NL\\textasciicircum\\rm loc\=10\$ at \$97.5\%\$ confidence on \$\sim 85\%\$ of the volumes. Additionally we test our ability to resolve degeneracies between the topological signature of \$f_\\rm NL\\textasciicircum\\rm loc\\$ and variation of \$\sigma_8\$ and argue that correctly identifying nonzero \$f_\\rm NL\\textasciicircum\\rm loc\\$ in this case is possible via an optimal template method. Our method relies on information living at \$\mathcal\O\(10)\$ Mpc/h, a complementary scale with respect to commonly used methods such as the scaledependent bias in the halo/galaxy power spectrum. Therefore, while still requiring a large volume, our method does not require sampling longwavelength modes to constrain primordial nonGaussianity. Moreover, our statistics are interpretable: we are able to reproduce previous results in certain limits and we make new predictions for unexplored observables, such as filament loops formed by dark matter halos in a simulation box.