Hyperbolic Distance Matrices

Tabaghi, Puoya

doi:10.1145/3394486.3403224

Cited by 19 publications

(21 citation statements)

References 43 publications

(42 reference statements)

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“…Note that LR embedding states that n − 1 dimensions are sufficient, but it does not state that n − 1 dimensions are necessary. Future work should explore recent advances in hyperbolic neural networks (Ganea et al, 2018) and hyperbolic distances (Tabaghi and Dokmanić, 2020) to overcome limitations of Euclidean distances.…”

Section: Discussionmentioning

confidence: 99%

DEPP: Deep Learning Enables Extending Species Trees using Single Genes

Jiang

Balaban

Zhu

et al. 2021

Preprint

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Identifying samples in an evolutionary context is a fundamental step in the study of microbiome, and more broadly, biodiversity. Extending a reference phylogeny by placing new query sequences onto it has been increasingly used for sample identification and other applications. Existing phylogenetic placement methods have assumed that the query sequence is homologous to the data used to infer the reference phylogeny. Thus, they are designed to place data from a single gene onto a gene tree (e.g., they can place 16S sequences onto a 16S gene tree). While this assumption is reasonable, ultimately, sample identification is a question of identifying the species not individual genes. The placement of single gene data on a gene tree is therefore used as a proxy for a more ambitious goal: extending a species tree given sequence data from one or more gene. This goal poses difficult algorithmic questions. Nevertheless, a sufficiently accurate solution would not only improve sample identification using marker genes, it would also help achieving the long-standing goal of combining 16S and metagenomic data. We approach this problem using deep neural networks (DNN) and introduce a method called DEPP. Given a reference species tree and sequence data from one (or a handful of) genes, DEPP learns how to extend the species tree to include new species. DEPP does not rely on pre-specified models of sequence evolution or gene tree discordance; instead, it uses highly parameterized DNNs to learn both aspects from the data. We test DEPP both in simulations and on real microbial data and show high accuracy.

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Section: Discussionmentioning

confidence: 99%

DEPP: Deep Learning Enables Extending Species Trees using Single Genes

Jiang

Balaban

Zhu

et al. 2021

Preprint

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“…This Euclidean space's growth speed is significantly slower than embedding hierarchical data such as an r-ary tree (r ≥ 2) requires, which is exponential. To overcome this limitation, a few recent papers (Suzuki et al, 2019;Tabaghi & Dokmanic, 2020) have proposed ordinal embedding methods using hyperbolic space for hierarchical data, which we call hyperbolic ordinal embedding (HOE) in this paper. In contrast to Euclidean space's polynomial growth property, hyperbolic space has the exponential growth property, that is, the volume of any ball in hyperbolic space grows exponentially with respect to its radius (Lamping & Rao, 1994;Ritter, 1999;Nickel & Kiela, 2017).…”

Section: Proceedings Of the 38 Th International Conference On Machinementioning

confidence: 99%

Generalization Error Bound for Hyperbolic Ordinal Embedding

Suzuki,

Nitanda,

Wang

et al. 2021

Preprint

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Hyperbolic ordinal embedding (HOE) represents entities as points in hyperbolic space so that they agree as well as possible with given constraints in the form of entity i is more similar to entity j than to entity k. It has been experimentally shown that HOE can obtain representations of hierarchical data such as a knowledge base and a citation network effectively, owing to hyperbolic space's exponential growth property. However, its theoretical analysis has been limited to ideal noiseless settings, and its generalization error in compensation for hyperbolic space's exponential representation ability has not been guaranteed. The difficulty is that existing generalization error bound derivations for ordinal embedding based on the Gramian matrix do not work in HOE, since hyperbolic space is not inner-product space. In this paper, through our novel characterization of HOE with decomposed Lorentz Gramian matrices, we provide a generalization error bound of HOE for the first time, which is at most exponential with respect to the embedding space's radius. Our comparison between the bounds of HOE and Euclidean ordinal embedding shows that HOE's generalization error is reasonable as a cost for its exponential representation ability.

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“…Procrustes analysis can be performed in any metric space. In particular, hyperbolic Procrustes analysis is of great relevance due to the recent surge of interest in hyperbolic embeddings and machine learning [9][10][11][12][13][14][15][16]. Furthermore, hyperbolic embeddings are closely connected to the study of hierarchical or tree-like data structures and hyperbolic Procrustes problem solutions may be used to align hierarchical data, e.g., ontologies and phylogenies [17][18][19].…”

Section: Introductionmentioning

confidence: 99%

On Procrustes Analysis in Hyperbolic Space

Tabaghi

2021

IEEE Signal Process. Lett.

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Congruent Procrustes analysis aims to find the best matching between two point sets through rotation, reflection and translation. We formulate the Procrustes problem for hyperbolic spaces, review the canonical definition of the center mass for a point set, and give a closed-form solution for the optimal isometry between noise-free point sets. Our algorithm is analogous to the Euclidean Procrustes analysis, with centering and rotation replaced by their hyperbolic counterparts. When the data is corrupted with noise, our algorithm computes a sub-optimal alignment. We thus propose a gradient-based fine-tuning method to improve the matching accuracy.

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Hyperbolic Distance Matrices

Cited by 19 publications

References 43 publications

DEPP: Deep Learning Enables Extending Species Trees using Single Genes

DEPP: Deep Learning Enables Extending Species Trees using Single Genes

Generalization Error Bound for Hyperbolic Ordinal Embedding

On Procrustes Analysis in Hyperbolic Space

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