From Geometry to Topology: Inverse Theorems for Distributed Persistence

Solomon, Elchanan; Wagner, A.; Bendich, Paul

doi:10.48550/arxiv.2101.12288

Cited by 6 publications

(9 citation statements)

References 14 publications

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“…DR methods that do not incorporate the information of the global geometry or topology well enough would lead to the problematic results (Figure 1.1). Although topology-based statistics has been proposed to describe this aspect of the dimension reduction problem (Solomon et al, 2021), it is also not clear at the moment how geometry and topology can interplay with each other in the context of dimension reduction Luo and Strait, 2019). In the current paper, the loss function is a summation of functions in the form of u(x i , θ) with the parameters θ being the center c, radius r and the selected subspace.…”

Section: Discussion and Future Workmentioning

confidence: 99%

“…That is, we have identical ordering of these pairwise distances in the original space and the dimension-reduced space. Coranking matrix is a finer analysis tool than the so-called ijk rank test (See, e.g., Solomon et al (2021)).…”

Section: Coranking Matrixmentioning

confidence: 99%

“…Statistical dimension reduction with a topologically induced loss function, where the distributed persistence serves as a statistics that aids dimension reduction (section VI inSolomon et al (2021)). …”

mentioning

confidence: 99%

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Spherical Rotation Dimension Reduction with Geometric Loss Functions

Luo¹,

Purvis²,

Li³

2022

Preprint

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Modern datasets witness high-dimensionality and nontrivial geometries of spaces they live in. It would be helpful in data analysis to reduce the dimensionality while retaining the geometric structure of the dataset. Motivated by this observation, we propose a general dimension reduction method by incorporating geometric information. Our Spherical Rotation Component Analysis (SRCA) is a dimension reduction method that uses spheres or ellipsoids, to approximate a low-dimensional manifold.This method not only generalizes the Spherical Component Analysis (SPCA) method in terms of theories and algorithms and presents a comprehensive comparison of our method, as an optimization problem with theoretical guarantee and also as a structural preserving low-rank representation of data. Results relative to state-of-the-art competitors show considerable gains in ability to accurately approximate the subspace with fewer components and better structural preserving. In addition, we have pointed out that this method is a specific incarnation of a grander idea of using a geometrically induced loss function in dimension reduction tasks.

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Section: Discussion and Future Workmentioning

confidence: 99%

Section: Coranking Matrixmentioning

confidence: 99%

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Spherical Rotation Dimension Reduction with Geometric Loss Functions

Luo¹,

Purvis²,

Li³

2022

Preprint

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“…Notably, have studied the behavior of the average persistence landscape-a vector representation of persistence diagrams -computed from subsamples and found that the empirical average landscape accurately approximates the true mean landscape. More recently, Solomon et al (2021) propose the notion of distributed persistence to describe the topology of a dataset, which relies on subsampling. Distributed persistence produces a collection of persistence diagrams of smaller subsets, rather than a single one computed on the large dataset; this collection has been shown to be stable to outliers and possess desirable inverse properties.…”

Section: Introductionmentioning

confidence: 99%

Approximating Persistent Homology for Large Datasets

Cao¹,

Monod²

2022

Preprint

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Persistent homology is an important methodology from topological data analysis which adapts theory from algebraic topology to data settings and has been successfully implemented in many applications. It produces a statistical summary in the form of a persistence diagram, which captures the shape and size of the data. Despite its widespread use, persistent homology is simply impossible to implement when a dataset is very large. In this paper we address the problem of finding a representative persistence diagram for prohibitively large datasets. We adapt the classical statistical method of bootstrapping, namely, drawing and studying smaller multiple subsamples from the large dataset. We show that the mean of the persistence diagrams of subsamples-taken as a mean persistence measure computed from the subsamples-is a valid approximation of the true persistent homology of the larger dataset. We give the rate of convergence of the mean persistence diagram to the true persistence diagram in terms of the number of subsamples and size of each subsample. Given the complex algebraic and geometric nature of persistent homology, we adapt the convexity and stability properties in the space of persistence diagrams together with random set theory to achieve our theoretical results for the general setting of point cloud data. We demonstrate our approach on simulated and real data, including an application of shape clustering on complex large-scale point cloud data.

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“…We note that the study of the (non-) injectivity of certain topological transforms is also an aspect of topological inverse problems, see [23,13,6,21,25] for a sampling of these articles and [24] for a recent survey. Better understanding the precise failure of injectivity of certain TDA invariants led to the development of enriched topological summaries (ETS) that remediate these failures, opening a promising line of research; see [2] and [5] for some examples of these ETS.…”

Section: Introductionmentioning

confidence: 99%

From Trees to Barcodes and Back Again II: Combinatorial and Probabilistic Aspects of a Topological Inverse Problem

Curry¹,

DeSha²,

Garin³

et al. 2021

Preprint

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In this paper we consider two aspects of the inverse problem of how to construct merge trees realizing a given barcode. Much of our investigation exploits a recently discovered connection between the symmetric group and barcodes in general position, based on the simple observation that death order is a permutation of birth order. The first important outcome of our study is a clear combinatorial distinction between the space of phylogenetic trees (as defined by Billera, Holmes and Vogtmann) and the space of merge trees. Generic BHV trees on n + 1 leaf nodes fall into (2n − 1)!! distinct strata, but the analogous number for merge trees is equal to the number of maximal chains in the lattice of partitions, i.e., (n+1)!n!2 −n . The second aspect of our study is the derivation of precise formulas for the distribution of tree realization numbers (the number of merge trees realizing a given barcode) when we assume that barcodes are sampled using a uniform distribution on the symmetric group. We are able to characterize some of the higher moments of this distribution, thanks in part to a reformulation in terms of Dirichlet convolution. This characterization provides a type of null hypothesis, apparently different from the distributions observed in real neuron data and opens the door to doing more precise science.

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From Geometry to Topology: Inverse Theorems for Distributed Persistence

Cited by 6 publications

References 14 publications

Spherical Rotation Dimension Reduction with Geometric Loss Functions

Spherical Rotation Dimension Reduction with Geometric Loss Functions

Approximating Persistent Homology for Large Datasets

From Trees to Barcodes and Back Again II: Combinatorial and Probabilistic Aspects of a Topological Inverse Problem

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