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### MREC: a fast and versatile framework for aligning and matching point clouds with applications to single cell molecular data

Comparing and aligning large datasets is a pervasive problem occurring across many different knowledge domains. We introduce and study MREC, a recursive decomposition algorithm for computing matchings between data sets. The basic idea is to partition the data, match the partitions, and then recursively match the points within each pair of identified partitions. The matching itself is done using black box matching procedures that are too expensive to run on the entire data set. Using an absolute measure of the quality of a matching, the framework supports optimization over parameters including partitioning procedures and matching algorithms. By design, MREC can be applied to extremely large data sets. We analyze the procedure to describe when we can expect it to work well and demonstrate its flexibility and power by applying it to a number of alignment problems arising in the analysis of single cell molecular data.

### Intrinsic persistent homology via density-based metric learning

We address the problem of estimating intrinsic distances in a manifold from a finite sample. We prove that the metric space defined by the sample endowed with a computable metric known as sample Fermat distance converges a.s. in the sense of Gromov-Hausdorff. The limiting object is the manifold itself endowed with the population Fermat distance, an intrinsic metric that accounts for both the geometry of the manifold and the density that produces the sample. This result is applied to obtain sample persistence diagrams that converge towards an intrinsic persistence diagram. We show that this method outperforms more standard approaches based on Euclidean norm with theoretical results and computational experiments.

### Non-Aligned Distribution Distance using Metric Measure Embedding and Optimal Transport

We propose a novel approach for comparing distributions whose supports do not necessarily lie on the same metric space. Unlike Gromov-Wasserstein (GW) distance that compares pairwise distance of elements from each distribution, we consider a method that embeds the metric measure spaces in a common Euclidean space and computes an optimal transport (OT) on the embedded distributions. This leads to what we call a sub-embedding robust Wasserstein(SERW). Under some conditions, SERW is a distance that considers an OT distance of the (low-distorted) embedded distributions using a common metric. In addition to this novel proposal that generalizes several recent OT works, our contributions stand on several theoretical analyses: i) we characterize the embedding spaces to define SERW distance for distribution alignment; ii) we prove that SERW mimics almost the same properties of GW distance, and we give a cost relation between GW and SERW. The paper also provides some numerical experiments illustrating how SERW behaves on matching problems in real-world.

### Recovering metric from full ordinal information

Given a geodesic space (E, d), we show that full ordinal knowledge on the metric d-i.e. knowledge of the function D d : (w, x, y, z) $\rightarrow$ 1 d(w,x)$\le$d(y,z) , determines uniquely-up to a constant factor-the metric d. For a subspace En of n points of E, converging in Hausdorff distance to E, we construct a metric dn on En, based only on the knowledge of D d on En and establish a sharp upper bound of the Gromov-Hausdorff distance between (En, dn) and (E, d).

### Aligning Time Series on Incomparable Spaces

Dynamic time warping (DTW) is a useful method for aligning, comparing and combining time series, but it requires them to live in comparable spaces. In this work, we consider a setting in which time series live on different spaces without a sensible ground metric, causing DTW to become ill-defined. To alleviate this, we propose Gromov dynamic time warping (GDTW), a distance between time series on potentially incomparable spaces that avoids the comparability requirement by instead considering intra-relational geometry. We derive a Frank-Wolfe algorithm for computing it and demonstrate its effectiveness at aligning, combining and comparing time series living on incomparable spaces. We further propose a smoothed version of GDTW as a differentiable loss and assess its properties in a variety of settings, including barycentric averaging, generative modeling and imitation learning.