Nonlinear dimensionality reduction techniques for manifold learning, e.g., Isomap, may become exceedingly expensive to carry out for large data sets. This paper explores a multilevel framework with the goal of reducing the cost of unsupervised manifold learning. In addition to savings in computational time, the proposed multilevel technique essentially preserves the geodesic information, and so...

In this paper we introduce Maximum Variance Correction (MVC), which finds largescale feasible solutions to Maximum Variance Unfolding (MVU) by post-processing embeddings from any manifold learning algorithm. It increases the scale of MVU embeddings by several orders of magnitude and is naturally parallel. This unprecedented scalability opens up new avenues of applications for manifold learning,...

We consider the performance of local tangent space alignment (Zhang and Zha, 2004), one of several manifold learning algorithms which has been proposed as a dimension reduction method, when errors are present in the observations. Matrix perturbation theory is applied to obtain a worst-case upper bound on the deviation of the solution, which is an invariant subspace. Although we only prove this ...

For decades, bearing factory quality evaluation has been a key problem and the methods used are always static tests. This paper investigates the use of piezoelectric ultrasonic transducers (PUT) as dynamic diagnostic tools and a relevant signal classification technique, wavelet packet entropy (WPEntropy) flow manifold learning, for the evaluation of bearing factory quality. The data were analyz...

Manifold learning is an important theme in machine learning. This paper proposes a new visualization approach to manifold clustering. The method is based on pie charts in order to obtain meaningful visualizations of the clustering results when applying a manifold technique. In addition to this, the proposed approach extracts all the existing relationships among the attributes of the different c...

In many multi-dimensional tracking problems, the quantities of interest are restricted to a manifold in observation space. Learning the manifold shape is a necessary step for dimensionality reduction, which in turn allows faster and more robust tracking performance. For manifolds with arbitrary topology, learning the shape from noisy scattered data is not trivial. This paper presents a geometri...

This paper presents varifold learning, a learning framework based on the mathematical concept of varifolds. Different from manifold based methods, our varifold learning framework does not treat data as being sampled from a manifold; but rather, we presume a weaker varifold structure, based upon which we utilize a Grassmannian manifold at each data point, and convert Grassmannian Laplacians to f...

This paper addresses the numerical estimation of the principal curvature of pattern transformation manifolds. When a visual pattern undergoes a geometric transformation, it forms a (sub)manifold in the ambient space, which is usually called the transformation manifold. The manifold curvature is an important property characterizing the manifold geometry, with several applications in manifold lea...

Abstract. Manifold learning has been successfully used for finding dominant factors (low-dimensional manifold) in a high-dimensional data set. However, most existing manifold learning algorithms only consider one manifold based on one dissimilarity matrix. For utilizing multiple manifolds, a key question is how different pieces of information can be integrated when multiple measurements are ava...