Nonlocal $p$-Laplacian Variational problems on graphs

Hafiene, Yosra; Fadili, Jalal M.; Elmoataz, Abderrahim

doi:10.48550/arxiv.1810.12817

Cited by 2 publications

(2 citation statements)

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“…Several works have since suggested using the p-Laplacian instead [AL11, BZ13,ZB11] with large p, and have established its consistency and effectiveness for graph based SSL with small amounts of data [ACR + 16, Cal17, RCL19, ST17, KRSS15]. Recently, p-Laplacians have also been used for data clustering and learning problems [ETT15,EDT17,HFE18]. Minimizing the p-Laplacian can be easily seen as an p -regression problem.…”

Section: Introductionmentioning

confidence: 99%

Fast, Provably convergent IRLS Algorithm for p-norm Linear Regression

Adil,

Peng,

Sachdeva

2019

Preprint

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Linear regression in p -norm is a canonical optimization problem that arises in several applications, including sparse recovery, semi-supervised learning, and signal processing. Generic convex optimization algorithms for solving p -regression are slow in practice. Iteratively Reweighted Least Squares (IRLS) is an easy to implement family of algorithms for solving these problems that has been studied for over 50 years. However, these algorithms often diverge for p > 3, and since the work of Osborne (1985), it has been an open problem whether there is an IRLS algorithm that is guaranteed to converge rapidly for p > 3. We propose p-IRLS, the first IRLS algorithm that provably converges geometrically for any p ∈ [2, ∞). Our algorithm is simple to implement and is guaranteed to find a (1 + ε)-approximate solution in O(p 3.5 m p−2Our experiments demonstrate that it performs even better than our theoretical bounds, beats the standard Matlab/CVX implementation for solving these problems by 10-50x, and is the fastest among available implementations in the high-accuracy regime.

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Section: Introductionmentioning

confidence: 99%

Fast, Provably convergent IRLS Algorithm for p-norm Linear Regression

Adil,

Peng,

Sachdeva

2019

Preprint

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“…εn,con (f ; ρ) was considered in [61] as an intermediary functional to provide convergence between E (p) n,con (f ) and E (p) ∞,con and has appeared in [30][31][32] as the continuum limit of E (p) n,con when ε n = ε is fixed (although not using the hard constraint). We also develop a numerical method for computing minimisers of the above functional that are efficient for large n.…”

Section: Introductionmentioning

confidence: 99%

PDE-Inspired Algorithms for Semi-Supervised Learning on Point Clouds

Crook¹,

Hurst²,

Schönlieb³

et al. 2019

Preprint

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Given a data set and a subset of labels the problem of semi-supervised learning on point clouds is to extend the labels to the entire data set. In this paper we extend the labels by minimising the constrained discrete p-Dirichlet energy. Under suitable conditions the discrete problem can be connected, in the large data limit, with the minimiser of a weighted continuum p-Dirichlet energy with the same constraints. We take advantage of this connection by designing numerical schemes that first estimate the density of the data and then apply PDE methods, such as pseudo-spectral methods, to solve the corresponding Euler-Lagrange equation. We prove that our scheme is consistent in the large data limit for two methods of density estimation: kernel density estimation and spline kernel density estimation.

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Nonlocal $p$-Laplacian Variational problems on graphs

Cited by 2 publications

References 0 publications

Fast, Provably convergent IRLS Algorithm for p-norm Linear Regression

Fast, Provably convergent IRLS Algorithm for p-norm Linear Regression

PDE-Inspired Algorithms for Semi-Supervised Learning on Point Clouds

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