Linear discriminant analysis (LDA) is a well-known dimensionality reduction method which can be easily extended for data classification. Traditional LDA aims to preserve the separability of different classes and the compactness of the same class in the output space by maximizing the between-class covariance and simultaneously minimizing the within-class covariance. However, the performance of L...

Semi-supervised learning plays an important role in the recent literature on machine learning and data mining and the developed semisupervised learning techniques have led to many data mining applications in recent years. This paper addresses the semi-supervised learning problem by developing a semiparametric regularization based approach, which attempts to discover the marginal distribution of...

In semi-supervised learning, at the limit of infinite unlabeled points while fixing labeled ones, the solutions of several graph Laplacian regularization based algorithms were shown by Nadler et al. (2009) to degenerate to constant functions with “spikes” at labeled points in R for d ≥ 2. These optimization problems all use the graph Laplacian regularizer as a common penalty term. In this paper...

Abstract We demonstrate that strong global asymptotic stability (GAS) of the origin for an upper semicontinuous difference inclusion is equivalent to the existence of a smooth Lyapunov function. This result is of interest in discrete-time because the robustness of the stability property is dependent on the existence of such a smooth Lyapunov function. We also propose a regularization that allow...

We address in this paper the question of how the knowledge of the marginal distribution P (x) can be incorporated in a learning algorithm. We suggest three theoretical methods for taking into account this distribution for regularization and provide links to existing graph-based semi-supervised learning algorithms. We also propose practical implementations.

Semi-supervised learning gets estimated marginal distribution X P with a large number of unlabeled examples and then constrains the conditional probability ) | ( x y p with a few labeled examples. In this paper, we focus on a regularization approach for semi-supervised classification. The label information graph is first defined to keep the pairwise label relationship and can be incorporated wi...

We consider a novel “online semi-supervised learning” setting where (mostly unlabeled) data arrives sequentially in large volume, and it is impractical to store it all before learning. We propose an online manifold regularization algorithm. It differs from standard online learning in that it learns even when the input point is unlabeled. Our algorithm is based on convex programming in kernel sp...

We address the minimization of the sum of a proper, convex and lower semicontinuous with a (possibly nonconvex) smooth function from the perspective of an implicit dynamical system of forward-backward type. The latter is formulated by means of the gradient of the smooth function and of the proximal point operator of the nonsmooth one. The trajectory generated by the dynamical system is proved t...

Given a directed graph in which some of the nodes are labeled, we investigate the question of how to exploit the link structure of the graph to infer the labels of the remaining unlabeled nodes. To that extent we propose a regularization framework for functions defined over nodes of a directed graph that forces the classification function to change slowly on densely linked subgraphs. A powerful...

A family of regularization functionals is said to admit a linear representer theorem if every member of the family admits minimizers that lie in a fixed finite dimensional subspace. A recent characterization states that a general class of regularization functionals with differentiable regularizer admits a linear representer theorem if and only if the regularization term is a non-decreasing func...