Low-rank approximations of hyperbolic embeddings

Jawanpuria, Pratik; Meghwanshi, Mayank; Mishra, Bamdev

doi:10.1109/cdc40024.2019.9029297

Cited by 8 publications

(10 citation statements)

References 15 publications

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“…Proof. Using (13), the usual first order characterization of convex functions implies that, for all minimal geodesic segments γ : I → C, the composition f • γ : I → R is convex if and only if JDf (γ(t 2 )), γ ′ (t 2 ) − JDf (γ(t 1 )), γ ′ (t 1 ) (t 2 − t 1 ) ≥ 0, ∀ t 2 , t 1 ∈ I.…”

Section: Hyperbolically Convex Quadratic Functionsmentioning

confidence: 99%

“…For instance, several problems in machine learning, artificial intelligence, financial networks, as well as procrustes problems and many other practical questions can be modeled in this setting. Papers dealing with these subjects include, machine learning [20], artificial intelligence [19], neural circuits [24], low-rank approximations of hyperbolic embeddings [13,26], procrustes problems [25], financial networks [14], complex networks [15,18], embeddings of data [30] and there references therein. We also mention that there are many related papers on strain analysis, see for example, [28,31].…”

Section: Introductionmentioning

confidence: 99%

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Convexity of sets and quadratic functions on the hyperbolic space

Ferreira¹,

Németh²,

Zhu³

2021

Preprint

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In this paper some concepts of convex analysis on hyperbolic space are studied. We first study properties of the intrinsic distance, for instance, we present the spectral decomposition of its Hessian. Then, we study the concept of convex sets and the intrinsic projection onto these sets. We also study the concept of convex function and present the first and second order characterizations for these functions. An extensive study of the hyperbolic convex quadratic function is also presented.

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Section: Hyperbolically Convex Quadratic Functionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Convexity of sets and quadratic functions on the hyperbolic space

Ferreira¹,

Németh²,

Zhu³

2021

Preprint

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“…In Algorithm Algorithm 3 we propose a simple but suboptimal procedure to solve this low-rank approximation problem. Unlike iterative refinement algorithms based on optimization on manifolds [25], our proposed method is one-shot. It is based on the spectral factorization of the the estimated hyperbolic Gramian and involves the following steps:…”

Section: Low-rank Approximation Of H-gramiansmentioning

confidence: 99%

Hyperbolic Distance Matrices

Tabaghi

2020

Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery &Amp; Data Mining

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Hyperbolic space is a natural setting for mining and visualizing data with hierarchical structure. In order to compute a hyperbolic embedding from comparison or similarity information, one has to solve a hyperbolic distance geometry problem. In this paper, we propose a unified framework to compute hyperbolic embeddings from an arbitrary mix of noisy metric and non-metric data. Our algorithms are based on semidefinite programming and the notion of a hyperbolic distance matrix, in many ways parallel to its famous Euclidean counterpart. A central ingredient we put forward is a semidefinite characterization of the hyperbolic Gramian-a matrix of Lorentzian inner products. This characterization allows us to formulate a semidefinite relaxation to efficiently compute hyperbolic embeddings in two stages: first, we complete and denoise the observed hyperbolic distance matrix; second, we propose a spectral factorization method to estimate the embedded points from the hyperbolic distance matrix. We show through numerical experiments how the flexibility to mix metric and non-metric constraints allows us to efficiently compute embeddings from arbitrary data. CCS CONCEPTS • Human-centered computing → Hyperbolic trees; • Computing methodologies → Machine learning; • Networks;

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“…, i K ) ∈ Ω. We convert the problem (2) into a minimax problem by constructing a partial dual similar to [15]. This leads to a factorization of the tensor W in a form with separate factors for the nonnegative and low-rank constraints.…”

Section: Introductionmentioning

confidence: 99%

Nonnegative Low-Rank Tensor Completion via Dual Formulation with Applications to Image and Video Completion

Sinha

Naram

Kumar

2022

2022 IEEE/CVF Winter Conference on Applications of Computer Vision (WACV)

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Recent approaches to the tensor completion problem have often overlooked the nonnegative structure of the data. We consider the problem of learning a nonnegative lowrank tensor, and using duality theory, we propose a novel factorization of such tensors. The factorization decouples the nonnegative constraints from the low-rank constraints. The resulting problem is an optimization problem on manifolds, and we propose a variant of Riemannian conjugate gradients to solve it. We test the proposed algorithm across various tasks such as colour image inpainting, video completion, and hyperspectral image completion. Experimental results show that the proposed method outperforms many state-of-the-art tensor completion algorithms.

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Low-rank approximations of hyperbolic embeddings

Cited by 8 publications

References 15 publications

Convexity of sets and quadratic functions on the hyperbolic space

Convexity of sets and quadratic functions on the hyperbolic space

Hyperbolic Distance Matrices

Nonnegative Low-Rank Tensor Completion via Dual Formulation with Applications to Image and Video Completion

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