Eigenvalues and Spectral Dimension of Random Geometric Graphs in Thermodynamic Regime

Avrachenkov, Konstantin; Cottatellucci, Laura; Hamidouche, Mounia

doi:10.1007/978-3-030-36687-2_80

Cited by 2 publications

(5 citation statements)

References 14 publications

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“…This result is proved at the end of this section. Similar results were proved for supercritical percolation bonds in [22] and for the random geometric graphs in [3].…”

Section: Decay Rate Of the Averaging Processsupporting

confidence: 79%

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Tight Nonparametric Convergence Rates for Stochastic Gradient Descent under the Noiseless Linear Model

Berthier¹,

Bach²,

Gaillard³

2020

Preprint

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In the context of statistical supervised learning, the noiseless linear model assumes that there exists a deterministic linear relation Y = θ * , Φ(U ) between the random output Y and the random feature vector Φ(U ), a potentially non-linear transformation of the inputs U . We analyze the convergence of single-pass, fixed step-size stochastic gradient descent on the least-square risk under this model. The convergence of the iterates to the optimum θ * and the decay of the generalization error follow polynomial convergence rates with exponents that both depend on the regularities of the optimum θ * and of the feature vectors Φ(U ). We interpret our result in the reproducing kernel Hilbert space framework; as a special case, we analyze an online algorithm for estimating a real function on the unit interval from the noiseless observation of its value at randomly sampled points. The convergence depends on the Sobolev smoothness of the function and of a chosen kernel. Finally, we apply our analysis beyond the supervised learning setting to obtain convergence rates for the averaging process (a.k.a. gossip algorithm) on a graph depending on its spectral dimension.

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“…This result is proved at the end of this section. Similar results were proved for supercritical percolation bonds in [22] and for the random geometric graphs in [3].…”

Section: Decay Rate Of the Averaging Processsupporting

confidence: 79%

“…In this section, we introduce a quantitive version of the notion of spectral dimension of a graph (see [3] and references therein for other definitions). We use this quantity to build polynomial convergence rates for the expected squared 2…”

Section: Decay Rate Of the Averaging Processmentioning

confidence: 99%

Tight Nonparametric Convergence Rates for Stochastic Gradient Descent under the Noiseless Linear Model

Berthier¹,

Bach²,

Gaillard³

2020

Preprint

View full text Add to dashboard Cite

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“…close to the relevant xed point, where λ > 1 is the largest eigenvalue of the linearized RG equations close to the relevant xed point. Using equation (36), the spectral density ρ(µ) can be expressed as…”

Section: The Free-energy Density and The Spectral Dimensionmentioning

confidence: 99%

“…with G(µ 1 ) given by equation (60). Using equation (36), we can deduce that the spectral density ρ(µ) is given by…”

Section: The Free-energy Density and The Spectral Dimensionmentioning

confidence: 99%

“…The network Laplacian [23][24][25][26] is fundamental to understand the interplay between topology and dynamics and its spectral properties are known to affect diffusion and synchronization on network structures. In particular the spectral dimension [27][28][29][30][31][32][33][34][35][36] characterizes the spectral properties of networks with distinct geometrical features and determines the late time behaviour of diffusion and more general dynamical processes on networks [37][38][39][40][41]. The spectral dimension can also be de ned on simplicial complexes [28] by focussing on their skeleton (the network obtained from a simplicial complex by retaining only its nodes and links).…”

Section: Introductionmentioning

confidence: 99%

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The higher-order spectrum of simplicial complexes: a renormalization group approach

Reitz¹,

Bianconi²

2020

J. Phys. A: Math. Theor.

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Network topology is a flourishing interdisciplinary subject that is relevant for different disciplines including quantum gravity and brain research. The discrete topological objects that are investigated in network topology are simplicial complexes. Simplicial complexes generalize networks by not only taking pairwise interactions into account, but also taking into account many-body interactions between more than two nodes. Higher-order Laplacians are topological operators that describe higher-order diffusion on simplicial complexes and constitute the natural mathematical objects that capture the interplay between network topology and dynamics. We show that higher-order up and down Laplacians can have a finite spectral dimension, characterizing the long time behaviour of the diffusion process on simplicial complexes that depends on their order m. We provide a renormalization group theory for the calculation of the higher-order spectral dimension of two deterministic models of simplicial complexes: the Apollonian and the pseudo-fractal simplicial complexes. We show that the RG flow is affected by the fixed point at zero mass, which determines the higher-order spectral dimension d S of the up-Laplacians of order m with m ⩾ 0.

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Eigenvalues and Spectral Dimension of Random Geometric Graphs in Thermodynamic Regime

Cited by 2 publications

References 14 publications

Tight Nonparametric Convergence Rates for Stochastic Gradient Descent under the Noiseless Linear Model

Tight Nonparametric Convergence Rates for Stochastic Gradient Descent under the Noiseless Linear Model

The higher-order spectrum of simplicial complexes: a renormalization group approach

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