Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Analysis and algorithms for $\ell_p$-based semi-supervised learning on
  graphs   This paper addresses theory and applications of $\ell_p$-based Laplacian
regularization in semi-supervised learning. The graph $p$-Laplacian for $p>2$
has been proposed recently as a replacement for the standard ($p=2$) graph
Laplacian in semi-supervised learning problems with very few labels, where
Laplacian learning is degenerate.
  In the first part of the paper we prove new discrete to continuum convergence
results for $p$-Laplace problems on $k$-nearest neighbor ($k$-NN) graphs, which
are more commonly used in practice than random geometric graphs. Our analysis
shows that, on $k$-NN graphs, the $p$-Laplacian retains information about the
data distribution as $p\to \infty$ and Lipschitz learning ($p=\infty$) is
sensitive to the data distribution. This situation can be contrasted with
random geometric graphs, where the $p$-Laplacian forgets the data distribution
as $p\to \infty$. We also present a general framework for proving discrete to
continuum convergence results in graph-based learning that only requires
pointwise consistency and monotonicity.
  In the second part of the paper, we develop fast algorithms for solving the
variational and game-theoretic $p$-Laplace equations on weighted graphs for
$p>2$. We present several efficient and scalable algorithms for both
formulations, and present numerical results on synthetic data indicating their
convergence properties. Finally, we conduct extensive numerical experiments on
the MNIST, FashionMNIST and EMNIST datasets that illustrate the effectiveness
of the $p$-Laplacian formulation for semi-supervised learning with few labels.
In particular, we find that Lipschitz learning ($p=\infty$) performs well with
very few labels on $k$-NN graphs, which experimentally validates our
theoretical findings that Lipschitz learning retains information about the data
distribution (the unlabeled data) on $k$-NN graphs.

23
The categories of this paper are: solv-int
34
The categories of this paper are: chem-ph
0
The categories of this paper are: hep-ph
16
The categories of this paper are: stat
17
The categories of this paper are: q-fin
24
The categories of this paper are: dg-ga
12
The categories of this paper are: nucl-th
7
The categories of this paper are: hep-th
33
The categories of this paper are: plasm-ph
25
The categories of this paper are: patt-sol
3
The categories of this paper are: physics
18
The categories of this paper are: econ
30
The categories of this paper are: q-alg
35
The categories of this paper are: mtrl-th
37
The categories of this paper are: supr-con
20
The categories of this paper are: acc-phys