AI summaryⓘ
The authors address a problem in creating spherical coordinates that reflect the shape of data when data points are unevenly spaced. They build on existing work for circular coordinates by using a subsampling method that picks more even subsets of data, then combine these subsets into one global coordinate system. Their main challenge was figuring out how to align separate spherical maps, which they solved using a mathematical technique called the spherical Procrustes problem. This method is both resistant to uneven data sampling and can handle larger datasets more efficiently than older approaches. They tested it on simulated data and showed it works well even with strong sampling biases and large numbers of points.
persistent cohomologynon-uniform samplingspherical coordinatescircular coordinatessubsamplingrejection samplingspherical Procrustes problemvariational problemtopological data analysiscoordinate optimization
Abstract
Persistent cohomology provides a principled framework for constructing nonlinear coordinates that reflect the topology of data. However, these topological coordinates can be severely distorted by non-uniform sampling density, limiting their applicability to real-world data. While density-robust circular coordinates have recently been developed, the extension to spherical coordinates remains an open challenge: unlike the circular case, spherical coordinates are obtained through a nonlinear variational problem for sphere-valued maps, to which existing density-correction mechanisms are not directly applicable. In this paper, we introduce the first density-robust construction of spherical coordinates from persistent cohomology. Rather than modifying the coordinate optimization itself, we extend a subsampling-and-alignment framework for circular coordinates to $S^2$, which first computes spherical coordinates on approximately uniform subsamples obtained by rejection sampling and then combines them into a global consensus map. The principal mathematical difficulty is the alignment of independently computed sphere-valued coordinates. We formulate this challenge as a spherical Procrustes problem and establish approximation guarantees for a computationally tractable Euclidean relaxation. Our resulting construction is robust to non-uniform sampling and retains the accuracy of classical spherical coordinates under uniform sampling. Moreover, by computing persistent cohomology only on fixed-size subsamples, our approach avoids the quartic memory bottleneck of the classical spherical coordinate pipeline and scales to substantially larger datasets. We conduct experiments on synthetic data and demonstrate accurate coordinate recovery under severe sampling bias and scalability to datasets of 10,000 points.