Hodge Laplacians on graphs

Lim, Lek-Heng

doi:10.48550/arxiv.1507.05379

Cited by 6 publications

(11 citation statements)

References 41 publications

(66 reference statements)

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“…Let f ∈ L 2 (V) and F ∈ L 2 (E) be functions on the vertices and edges of the graphs, respectively. We can define differential operators acting on such functions analogously to differential operators on manifolds [72]. The graph gradient is an operator ∇ : L 2 (V) → L 2 (E) mapping functions defined on vertices to functions defined on edges,…”

Section: Redmentioning

confidence: 99%

Geometric Deep Learning: Going beyond Euclidean data

Bronstein

Bruna

LeCun

et al. 2017

IEEE Signal Process. Mag.

2,668

1,629

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Many scientific fields study data with an underlying structure that is a non-Euclidean space. Some examples include social networks in computational social sciences, sensor networks in communications, functional networks in brain imaging, regulatory networks in genetics, and meshed surfaces in computer graphics. In many applications, such geometric data are large and complex (in the case of social networks, on the scale of billions), and are natural targets for machine learning techniques. In particular, we would like to use deep neural networks, which have recently proven to be powerful tools for a broad range of problems from computer vision, natural language processing, and audio analysis. However, these tools have been most successful on data with an underlying Euclidean or grid-like structure, and in cases where the invariances of these structures are built into networks used to model them.Geometric deep learning is an umbrella term for emerging techniques attempting to generalize (structured) deep neural models to non-Euclidean domains such as graphs and manifolds. The purpose of this paper is to overview different examples of geometric deep learning problems and present available solutions, key difficulties, applications, and future research directions in this nascent field.

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Section: Redmentioning

confidence: 99%

Geometric Deep Learning: Going beyond Euclidean data

Bronstein

Bruna

LeCun

et al. 2017

IEEE Signal Process. Mag.

2,668

1,629

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“…The classical algorithms for persistent homology of [ZC05] work for field coefficients such as Z2. The Betti numbers of homology groups over R coefficients do not capture the torsion [Lim15]. Nonetheless, this seems to be enough for many of the situations in TDA.…”

Section: Preliminaries On Persistent Homologymentioning

confidence: 99%

Quantum algorithm for persistent Betti numbers and topological data analysis

Hayakawa¹

2021

Preprint

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Topological data analysis (TDA) based on persistent homology is an emergent field of data analysis. The critical step of TDA is computing the persistent Betti numbers. Existing classical algorithms for TDA have been limited because the number of simplices can grow exponentially in the size of data for higher-dimensional analysis. In the context of quantum computation, it has been previously shown that there exists an efficient quantum algorithm for estimating the Betti numbers even in high dimensions. However, the Betti numbers are less general than the persistent Betti numbers, and there have been no quantum algorithms that can estimate the persistent Betti numbers of arbitrary dimensions. This paper shows the first quantum algorithm that can estimate the persistent Betti numbers of arbitrary dimensions. Our algorithm is efficient for the construction of simplicial complexes such as the Vietoris-Rips complex and demonstrates exponential speedup over the known classical algorithms.

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“…In this subsection, we define harmonic cycles, a main object of study of this paper, via combinatorial Laplacians, and recall combinatorial Hodge theory relating harmonic cycles and homology groups. For readers who search for a basic material introducing harmonic spaces, refer to [18]. For an application of this theory to the ranking system of items, can be found in [12].…”

Section: Harmonic Space and Combinatorial Hodge Theorymentioning

confidence: 99%

“…A harmonic cycle is energy minimizing among its homologous cycles, and as we shall see, allows intriguing combinatorial interpretations. For a basic material about harmonic cycles, check [18]. See [21] and [3] for previous studies concerning harmonic spaces as a projection of cycle spaces.…”

Section: Introductionmentioning

confidence: 99%

Winding number and Cutting number of Harmonic cycle

Kim,

Kook

2018

Preprint

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A harmonic cycle λ, also called a discrete harmonic form, is a solution of the Laplace's equation with the combinatorial Laplace operator obtained from the boundary operators of a chain complex. By the combinatorial Hodge theory, harmonic spaces are isomorphic to the homology groups with real coefficients. In particular, if a cell complex has a one dimensional reduced homology, it has a unique harmonic cycle up to scalar, which we call the standard harmonic cycle. In this paper, we will present a formula for the standard harmonic cycle λ of a cell complex based on a high-dimensional generalization of cycletrees. Moreover, by using duality, we will define the standard harmonic cocycle λ * , and show intriguing combinatorial properties of λ and λ * in relation to (dual) spanning trees, (dual) cycletrees, winding numbers w(•) and cutting numbers c(•) in high dimensions.

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Hodge Laplacians on graphs

Cited by 6 publications

References 41 publications

Geometric Deep Learning: Going beyond Euclidean data

Geometric Deep Learning: Going beyond Euclidean data

Quantum algorithm for persistent Betti numbers and topological data analysis

Winding number and Cutting number of Harmonic cycle

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