Multidimensional Scaling on the Sphere

Cox, Trevor F.; Cox, Michael

doi:10.1007/978-3-642-46900-8_44

Cited by 433 publications

(621 citation statements)

References 4 publications

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“…The results of the unit disk experiment are seen in Figure 6. The actual Isomap 2-dimensional embedding was optimally rotated and scaled using a Procrustes rotation [7] to match it to the original parameter space for graphical presentation. Isomap almost perfectly recovers the original spacing and relative positioning of the disk centers.…”

Section: Rigid Translation Of a Single Diskmentioning

confidence: 99%

Image Manifolds which are Isometric to Euclidean Space

2005

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Recently, the Isomap procedure [1] was proposed as a new way to recover a low-dimensional parametrization of data lying on a low-dimensional submanifold in high-dimensional space. The method assumes that the submanifold, viewed as a Riemannian submanifold of the ambient high-dimensional space, is isometric to a convex subset of Euclidean space. This naturally raises the question: what datasets can reasonably be modeled by this condition? In this paper, we consider a special kind of image data: families of images generated by articulation of one or several objects in a scene -for example, images of a black disk on a white background with center placed at a range of locations. The collection of all images in such an articulation family, as the parameters of the articulation vary, makes up an articulation manifold, a submanifold of L 2 . We study the properties of such articulation manifolds, in particular, their lack of differentiability when the images have edges. Under these conditions, we show that there exists a natural renormalization of geodesic distance which yields a well-defined metric. We exhibit a list of articulation models where the corresponding manifold equipped with this new metric is indeed isometric to a convex subset of Euclidean space. Examples include translations of a symmetric object, rotations of a closed set, articulations of a horizon, and expressions of a cartoon face.The theoretical predictions from our study are borne out by empirical experiments with published Isomap code. We also note that in the case where several components of the image articulate independently, isometry may fail; for example, with several disks in an image avoiding contact, the underlying Riemannian manifold is locally isometric to an open, connected, but not convex subset of Euclidean space. Such a situation matches the assumptions of our recently-proposed Hessian Eigenmaps procedure, but not the original Isomap procedure.

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Section: Rigid Translation Of a Single Diskmentioning

confidence: 99%

Image Manifolds which are Isometric to Euclidean Space

2005

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“…The concept of similarity, or metric, is central to many well-known algorithms such as k-means clustering [1], k-nearest neighbors [2], multi-dimensional scaling [3] and semi-supervised clustering [4]. While there are many approaches to metric learning, a large body of work is focussed on learning the Mahalanobis distance, which amounts to learning a feature-space transformation and computing the distance in the transformed space.…”

Section: Introductionmentioning

confidence: 99%

Mirror Descent for Metric Learning: A Unified Approach

Kunapuli

Shavlik

2012

Machine Learning and Knowledge Discovery in Databases

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Abstract. Most metric learning methods are characterized by diverse loss functions and projection methods, which naturally begs the question: is there a wider framework that can generalize many of these methods? In addition, ever persistent issues are those of scalability to large data sets and the question of kernelizability. We propose a unified approach to Mahalanobis metric learning: an online regularized metric learning algorithm based on the ideas of composite objective mirror descent (comid). The metric learning problem is formulated as a regularized positive semidefinite matrix learning problem, whose update rules can be derived using the comid framework. This approach aims to be scalable, kernelizable, and admissible to many different types of Bregman and loss functions, which allows for the tailoring of several different classes of algorithms. The most novel contribution is the use of the trace norm, which yields a sparse metric in its eigenspectrum, thus simultaneously performing feature selection along with metric learning.

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“…In MDS, objects are placed in a Euclidean space while preserving the distance between objects in the space as well as possible. Reviews of the extensive literature on MDS were provided by Davison (1983), Young (1987), Borg and Groenen (2005), and Cox and Cox (2001).…”

Section: Introductionmentioning

confidence: 99%

Model-based Clustering with Dissimilarities: A Bayesian Approach

Oh¹,

Raftery²

2003

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A Bayesian model-based clustering method is proposed for clustering objects on the basis of dissimilarites. This combines two basic ideas. The first is that the objects have latent positions in a Euclidean space, and that the observed dissimilarities are measurements of the Euclidean distances with error. The second idea is that the latent positions are generated from a mixture of multivariate normal distributions, each one corresponding to a cluster. We estimate the resulting model in a Bayesian way using Markov chain Monte Carlo. The method carries out multidimensional scaling and model-based clustering simultaneously, and yields good object configurations and good clustering results with reasonable measures of clustering uncertainties. In the examples we study, the clustering results based on low-dimensional configurations were almost as good as those based on high-dimensional ones. Thus, the method can be used as a tool for dimension reduction when clustering high-dimensional objects, which may be useful especially for visual inspection of clusters.We also propose a Bayesian criterion for choosing the dimension of the object configuration and the number of clusters simultaneously. This is easy to compute and works reasonably well in simulations and real examples.

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Multidimensional Scaling on the Sphere

Cited by 433 publications

References 4 publications

Image Manifolds which are Isometric to Euclidean Space

Image Manifolds which are Isometric to Euclidean Space

Mirror Descent for Metric Learning: A Unified Approach

Model-based Clustering with Dissimilarities: A Bayesian Approach

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