Accompanying code for paper Mapper Comparison with Wasserstein Metrics.
This repo is intended for the evaluation of different candidate metrics for comparing Mapper graphs. Mapper is an unsupervised learning algorithm based in Topological Data Analysis that generates a simplified nerve complex representation from a data set that captures topological features of a metric space.
One challenge using Mapper graphs is that without an indicator of goodness of fit, it can be challenging to determine when a Mapper model (or nerve complex representations in general) no longer fits that data. However, if we had a metric over Mapper graphs that captured all of the properties we're interested in, then we could evaluate the degree to which Mapper graphs would be expected to change over different samples of the same population, which could be used to identify when new samples have significantly deviated from the previous model.
This repository contains code from the paper Mapper Comparison with Wasserstein Metrics, which attempts to address this issue by comparing Mapper graphs as metric-measure spaces and introduces a Wasserstein distance variant, the Network Augmented Wasserstein (NAW) distance, which intuitively captures differences between the topological, metric, and density information represented by Mapper graphs.
This repo comes equipped with the actual distance measures and with a lightweight mapper implementation that the distance measures are built around. The mapper builder objects automatically create a networkx object that can be used for visualization. The basic procedure for generating a network and using a network metric is:
from lightweight_mapper.clustering import SingleLinkageHistogram
from lightweight_mapper.networks import Network, Cover, Partition
from lightweight_mapper.partitioners import UniformPartitioner
from distances import naw_distance
metric_space = pairwise_distances(X) # Create finite metric space
partition = UniformPartitioner() # Create partitioner
cover1 = partition.fit_transform(X[:, 0], 130, .7, 'PCA1') # Define cover parameters
cover2 = partition.fit_transform(X[:, 1], 130, .7, 'PCA2') # Define cover parameters
cover = [cover1, cover2] # Create cross product cover
cluster = SingleLinkageHistogram(metric='precomputed', threshold = 'histogram') # Any sklearn API compatible clusterer
network1 = Network(metric_space, cover, cluster, prune=True)
...
distance = naw_distance(network1, network2, metric_space)Examples can be found in examples (Eventually)
Michael McCabe. "Mapper Comparison with Wasserstein Metrics." arXiv preprint arXiv:1812.06232, 2018. [arxiv]