On Procrustes Analysis in Hyperbolic Space

Tabaghi, Puoya

doi:10.1109/lsp.2021.3081379

Cited by 8 publications

(4 citation statements)

References 23 publications

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“…R U represents rotation or reflection in space dimension (without changing the time dimension), while R b denotes a hyperbolic rotation across the time dimension and each space dimension. [33] established an equivalence between Lorentz boost and Möbius addition (or hyperbolic translation). Hence, HyboNet inherently models each relation as a combination of a rotation/reflection and a hyperbolic translation.…”

Section: Connections With Hyperbolic Methodsmentioning

confidence: 99%

Ultrahyperbolic Knowledge Graph Embeddings

Xiong,

Zhu,

Nayyeri

et al. 2022

Preprint

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Recent knowledge graph (KG) embeddings have been advanced by hyperbolic geometry due to its superior capability for representing hierarchies. The topological structures of real-world KGs, however, are rather heterogeneous, i.e., a KG is composed of multiple distinct hierarchies and non-hierarchical graph structures. Therefore, a homogeneous (either Euclidean or hyperbolic) geometry is not sufficient for fairly representing such heterogeneous structures. To capture the topological heterogeneity of KGs, we present an ultrahyperbolic KG embedding (UltraE) in an ultrahyperbolic (or pseudo-Riemannian) manifold that seamlessly interleaves hyperbolic and spherical manifolds. In particular, we model each relation as a pseudo-orthogonal transformation that preserves the pseudo-Riemannian bilinear form. The pseudo-orthogonal transformation is decomposed into various operators (i.e., circular rotations, reflections and hyperbolic rotations), allowing for simultaneously modeling heterogeneous structures as well as complex relational patterns. Experimental results on three standard KGs show that UltraE outperforms previous Euclidean-and hyperbolic-based approaches. CCS CONCEPTS• Computing methodologies → Knowledge representation and reasoning; Semantic networks.

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Section: Connections With Hyperbolic Methodsmentioning

confidence: 99%

Ultrahyperbolic Knowledge Graph Embeddings

Xiong,

Zhu,

Nayyeri

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Connections to hyperbolic models: Existing hyperbolic models such as AttH, RotH and RefH (Chami et al, 2020) represent each relation as the combination of a rotation (or reflection) and a Möbius addition in the Poincaré ball. Since the Poincaré ball is isometric to the hyperboloid (Nickel & Kiela, 2017), and the Möbius addition is equivalent to the hyperbolic boost (Tabaghi & Dokmanic, 2021), all these models can be viewed as the special cases of our proposed hyperbolic parameterization. Note that these models parameterize rotations/reflections using the 2-dimensional Givens transformations, while our approach breaks the dimensional restriction without loss of degrees of freedom, thereby generalizing them and achieving superior modeling capacity.…”

Section: Hyperbolic Orthogonal Parameterizationmentioning

confidence: 99%

“…On the one hand, the orthogonal relation transformations in the elliptic component spaces are naturally suitable for capturing the cyclical structures (Wilson et al, 2014;Liu et al, 2017). On the other hand, the hyperbolic orthogonal transformation in Equation ( 12) inherently encodes the hierarchical structures-the Euclidean orthogonal matrix models the transformation between entities at the same level of hierarchies, and the hyperbolic boost matrix models the transformation between entities at different levels of hierarchies (Tabaghi & Dokmanic, 2021).…”

Section: It Follows Thatmentioning

confidence: 99%

Enhancing Knowledge Graph Embedding with Relational Constraints

Sun

Zhang

et al. 2020

2020 IEEE International Conference on Knowledge Graph (ICKG)

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Recent advances in knowledge graph embedding (KGE) rely on Euclidean/hyperbolic orthogonal relation transformations to model intrinsic logical patterns and topological structures. However, existing approaches are confined to rigid relational orthogonalization with restricted dimension and homogeneous geometry, leading to deficient modeling capability. In this work, we move beyond these approaches in terms of both dimension and geometry by introducing a powerful framework named GoldE, which features a universal orthogonal parameterization based on a generalized form of Householder reflection. Such parameterization can naturally achieve dimensional extension and geometric unification with theoretical guarantees, enabling our framework to simultaneously capture crucial logical patterns and inherent topological heterogeneity of knowledge graphs. Empirically, GoldE achieves state-of-the-art performance on three standard benchmarks. Codes are available at https://github.com/xxrep/GoldE.

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“…Extending this idea from shapes to high-dimensional point clouds facilitated an appealing data alignment approach, which is simple, efficient, and mathematically tractable, and it does not require any rigid a-priori model assumptions or estimates of the whole distribution of the data. Indeed, data alignment using PA has been successfully applied to various fields, including Brain-Computer Interface (BCI) [2], genetics and bioinformatics [3], [4], [5], indoor navigation [6], face recognition [7], and hierarchical representation [5], [8], to name but a few.…”

mentioning

confidence: 99%

Procrustes Analysis on the Manifold of SPSD Matrices for Data Sets Alignment

Lahav

Talmon

2023

IEEE Trans. Signal Process.

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In contemporary high-dimensional data analysis, intrinsically similar and related data sets are often significantly different due to various undesired factors that could arise from different acquisition equipment, calibration, environmental conditions, and many other sources of batch effects. Therefore, the task of aligning such data sets has become ubiquitous. In this work, we present a method for the alignment of different, but related, sets of Symmetric Positive Semidefinite (SPSD) matrices, which constitute a commonly-used family of features, e.g., covariance and correlation matrices, various kernels, and prototypical graph and network representations. Our method does not require any a-priori correspondence, and it is based on non-Euclidean Procrustes Analysis (PA) using a particular Riemannian geometry of SPSD matrices. While the derivation is focused on the manifold of SPSD matrices, we show that our alignment method can be applied directly in the original high-dimensional data space, when considering SPSD features that are sample covariance matrices. We demonstrate the advantage of our approach over competing methods in simulations and in an application to Brain-Computer Interface (BCI) with electroencephalographic (EEG) recordings.

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On Procrustes Analysis in Hyperbolic Space

Cited by 8 publications

References 23 publications

Ultrahyperbolic Knowledge Graph Embeddings

Ultrahyperbolic Knowledge Graph Embeddings

Enhancing Knowledge Graph Embedding with Relational Constraints

Procrustes Analysis on the Manifold of SPSD Matrices for Data Sets Alignment

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