2023/03/14 by Adrien Weihs, Matthew Thorpe, Weihs, Adrien +1 · 2 citations
Biochemistry, Genetics and Molecular Biology · Mathematics · #Circular RNAs in diseases #FOS: Mathematics #Mathematical Approximation and Integration #MicroRNA in disease regulation #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2303.07818
openalex publication_date 2023/03/14 · openalex created_date 2023/03/17 · openalex updated_date 2026/08/01
Laplace learning is a popular machine learning algorithm for finding missing labels from a small number of labelled feature vectors using the geometry of a graph. More precisely, Laplace learning is based on minimising a graph-Dirichlet energy, equivalently a discrete Sobolev \Wkp21 semi-norm, constrained to taking the values of known labels on a given subset. The variational problem is asymptotically ill-posed as the number of unlabeled feature vectors goes to infinity for finite given labels due to a lack of regularity in minimisers of the continuum Dirichlet energy in any dimension higher than one. In particular, continuum minimisers are not continuous. One solution is to consider higher-order regularisation, which is the analogue of minimising Sobolev \Wkps2 semi-norms. In this paper we consider the asymptotics of minimising a graph variant of the Sobolev \Wkps2 semi-norm with pointwise constraints. We show that, as expected, one needs s>d/2 where d is the dimension of the data manifold. We also show that there must be an upper bound on the connectivity of the graph; that is, highly connected graphs lead to degenerate behaviour of the minimiser even when s>d/2.