Noncommutative Model Selection and the Data-Driven Estimation of Real Cohomology Groups

We propose three completely data-driven methods for estimating the real cohomology groups $H^k (X ; \mathbb{R})$ of a compact metric-measure space $(X, d_X, \mu_X)$ embedded in a metric-measure space $(Y,d_Y,\mu_Y)$, given a finite set of points $S$ sampled from a uniform distrbution $\mu_X$ on $X$, possibly corrupted with noise from $Y$. We present the results of several computational experiments in the case that $X$ is embedded in $\mathbb{R}^n$, where two of the three algorithms performed well.

Noncommutative Model Selection for Data Clustering and Dimension Reduction Using Relative von Neumann Entropy

We propose a pair of completely data-driven algorithms for unsupervised classification and dimension reduction, and we empirically study their performance on a number of data sets, both simulated data in three-dimensions and images from the COIL-20 data set. The algorithms take as input a set of points sampled from a uniform distribution supported on a metric space, the latter embedded in an ambient metric space, and they output a clustering or reduction of dimension of the data. They work by constructing a natural family of graphs from the data and selecting the graph which maximizes the relative von Neumann entropy of certain normalized heat operators constructed from the graphs. Once the appropriate graph is selected, the eigenvectors of the graph Laplacian may be used to reduce the dimension of the data, and clusters in the data may be identified with the kernel of the associated graph Laplacian. Notably, these algorithms do not require information about the size of a neighborhood or the desired number of clusters as input, in contrast to popular algorithms such as $k$-means, and even more modern spectral methods such as Laplacian eigenmaps, among others. In our computational experiments, our clustering algorithm outperforms $k$-means clustering on data sets with non-trivial geometry and topology, in particular data whose clusters are not concentrated around a specific point, and our dimension reduction algorithm is shown to work well in several simple examples.

A Topological Approach to Spectral Clustering

We propose a clustering algorithm which, for input, takes data assumed to be sampled from a uniform distribution supported on a metric space $X$, and outputs a clustering of the data based on a topological estimate of the connected components of $X$. The algorithm works by choosing a weighted graph on the samples from a natural one-parameter family of graphs using an error based on the heat operator on the graphs. The estimated connected components of $X$ are identified as the support of the eigenfunctions of the heat operator with eigenvalue $1$, which allows the algorithm to work without requiring the number of expected clusters as input.