Error Metrics for Learning Reliable Manifolds from Streaming Data

Spectral dimensionality reduction is frequently used to identify low-dimensional structure in high-dimensional data. However, learning manifolds, especially from the streaming data, is computationally and memory expensive. In this paper, we argue that a stable manifold can be learned using only a fraction of the stream, and the remaining stream can be mapped to the manifold in a significantly less costly manner. Identifying the transition point at which the manifold is stable is the key step. We present error metrics that allow us to identify the transition point for a given stream by quantitatively assessing the quality of a manifold learned using Isomap. We further propose an efficient mapping algorithm, called S-Isomap, that can be used to map new samples onto the stable manifold. We describe experiments on a variety of data sets that show that the proposed approach is computationally efficient without sacrificing accuracy

Manifold Learning in MR spectroscopy using nonlinear dimensionality reduction and unsupervised clustering

Purpose To investigate whether nonlinear dimensionality reduction improves unsupervised classification of 1H MRS brain tumor data compared with a linear method. Methods In vivo single-voxel 1H magnetic resonance spectroscopy (55 patients) and 1H magnetic resonance spectroscopy imaging (MRSI) (29 patients) data were acquired from histopathologically diagnosed gliomas. Data reduction using Laplacian eigenmaps (LE) or independent component analysis (ICA) was followed by k-means clustering or agglomerative hierarchical clustering (AHC) for unsupervised learning to assess tumor grade and for tissue type segmentation of MRSI data. Results An accuracy of 93% in classification of glioma grade II and grade IV, with 100% accuracy in distinguishing tumor and normal spectra, was obtained by LE with unsupervised clustering, but not with the combination of k-means and ICA. With 1H MRSI data, LE provided a more linear distribution of data for cluster analysis and better cluster stability than ICA. LE combined with k-means or AHC provided 91% accuracy for classifying tumor grade and 100% accuracy for identifying normal tissue voxels. Color-coded visualization of normal brain, tumor core, and infiltration regions was achieved with LE combined with AHC. Conclusion Purpose To investigate whether nonlinear dimensionality reduction improves unsupervised classification of 1H MRS brain tumor data compared with a linear method. Methods In vivo single-voxel 1H magnetic resonance spectroscopy (55 patients) and 1H magnetic resonance spectroscopy imaging (MRSI) (29 patients) data were acquired from histopathologically diagnosed gliomas. Data reduction using Laplacian eigenmaps (LE) or independent component analysis (ICA) was followed by k-means clustering or agglomerative hierarchical clustering (AHC) for unsupervised learning to assess tumor grade and for tissue type segmentation of MRSI data. Results An accuracy of 93% in classification of glioma grade II and grade IV, with 100% accuracy in distinguishing tumor and normal spectra, was obtained by LE with unsupervised clustering, but not with the combination of k-means and ICA. With 1H MRSI data, LE provided a more linear distribution of data for cluster analysis and better cluster stability than ICA. LE combined with k-means or AHC provided 91% accuracy for classifying tumor grade and 100% accuracy for identifying normal tissue voxels. Color-coded visualization of normal brain, tumor core, and infiltration regions was achieved with LE combined with AHC. Conclusion The LE method is promising for unsupervised clustering to separate brain and tumor tissue with automated color-coding for visualization of 1H MRSI data after cluster analysis

Dimensionality Reduction Mappings

A wealth of powerful dimensionality reduction methods has been established which can be used for data visualization and preprocessing. These are accompanied by formal evaluation schemes, which allow a quantitative evaluation along general principles and which even lead to further visualization schemes based on these objectives. Most methods, however, provide a mapping of a priorly given finite set of points only, requiring additional steps for out-of-sample extensions. We propose a general view on dimensionality reduction based on the concept of cost functions, and, based on this general principle, extend dimensionality reduction to explicit mappings of the data manifold. This offers simple out-of-sample extensions. Further, it opens a way towards a theory of data visualization taking the perspective of its generalization ability to new data points. We demonstrate the approach based on a simple global linear mapping as well as prototype-based local linear mappings.