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We provide a theoretical analysis of the representation learning problem aimed at learning the latent variables (design matrix) $\Theta$ of observations $Y$ with the knowledge of the coefficient matrix $X$. The design matrix is learned…
In the (special) smoothing spline problem one considers a variational problem with a quadratic data fidelity penalty and Laplacian regularisation. Higher order regularity can be obtained via replacing the Laplacian regulariser with a…
Most network-based machine learning methods assume that the labels of two adjacent samples in the network are likely to be the same. However, assuming the pairwise relationship between samples is not complete. The information a group of…
As annotations of data can be scarce in large-scale practical problems, leveraging unlabelled examples is one of the most important aspects of machine learning. This is the aim of semi-supervised learning. To benefit from the access to…
Scalings in which the graph Laplacian approaches a differential operator in the large graph limit are used to develop understanding of a number of algorithms for semi-supervised learning; in particular the extension, to this graph setting,…
Tackling semi-supervised learning problems with graph-based methods has become a trend in recent years since graphs can represent all kinds of data and provide a suitable framework for studying continuum limits, e.g., of differential…
We investigate a variational method for ill-posed problems, named $\texttt{graphLa+}\Psi$, which embeds a graph Laplacian operator in the regularization term. The novelty of this method lies in constructing the graph Laplacian based on a…
Recently, manifold regularized semi-supervised learning (MRSSL) received considerable attention because it successfully exploits the geometry of the intrinsic data probability distribution including both labeled and unlabeled samples to…
In this paper, we propose a novel graph-based approach for semi-supervised learning problems, which considers an adaptive adjacency of the examples throughout the unsupervised portion of the training. Adjacency of the examples is inferred…
Motivated by the need to address the degeneracy of canonical Laplace learning algorithms in low label rates, we propose to reformulate graph-based semi-supervised learning as a nonconvex generalization of a \emph{Trust-Region Subproblem}…
We propose graph-dependent implicit regularisation strategies for distributed stochastic subgradient descent (Distributed SGD) for convex problems in multi-agent learning. Under the standard assumptions of convexity, Lipschitz continuity,…
Graph-based semi-supervised learning is the problem of propagating labels from a small number of labelled data points to a larger set of unlabelled data. This paper is concerned with the consistency of optimization-based techniques for such…
In this paper we study Lipschitz regularity of elliptic PDEs on geometric graphs, constructed from random data points. The data points are sampled from a distribution supported on a smooth manifold. The family of equations that we study…
In this paper we give a broad overview of the intersection of partial differential equations (PDEs) and graph-based semi-supervised learning. The overview is focused on a large body of recent work on PDE continuum limits of graph-based…
In machine learning, one must acquire labels to help supervise a model that will be able to generalize to unseen data. However, the labeling process can be tedious, long, costly, and error-prone. It is often the case that most of our data…
In a semi-supervised learning scenario, (possibly noisy) partially observed labels are used as input to train a classifier, in order to assign labels to unclassified samples. In this paper, we study this classifier learning problem from a…
The common graph Laplacian regularizer is well-established in semi-supervised learning and spectral dimensionality reduction. However, as a first-order regularizer, it can lead to degenerate functions in high-dimensional manifolds. The…
Smoothness and low dimensional structures play central roles in improving generalization and stability in learning and statistics. This work combines techniques from semi-infinite constrained learning and manifold regularization to learn…
Semi-supervised learning with manifold regularization is a classical framework for jointly learning from both labeled and unlabeled data, where the key requirement is that the support of the unknown marginal distribution has the geometric…
In this work, we address the solution of both linear and nonlinear ill-posed inverse problems by developing a novel graph-based regularization framework, where the regularization term is formulated through an iteratively updated graph…