7,906 research outputs found
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.
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
Masking Strategies for Image Manifolds
We consider the problem of selecting an optimal mask for an image manifold,
i.e., choosing a subset of the pixels of the image that preserves the
manifold's geometric structure present in the original data. Such masking
implements a form of compressive sensing through emerging imaging sensor
platforms for which the power expense grows with the number of pixels acquired.
Our goal is for the manifold learned from masked images to resemble its full
image counterpart as closely as possible. More precisely, we show that one can
indeed accurately learn an image manifold without having to consider a large
majority of the image pixels. In doing so, we consider two masking methods that
preserve the local and global geometric structure of the manifold,
respectively. In each case, the process of finding the optimal masking pattern
can be cast as a binary integer program, which is computationally expensive but
can be approximated by a fast greedy algorithm. Numerical experiments show that
the relevant manifold structure is preserved through the data-dependent masking
process, even for modest mask sizes
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