| Lee, J.A., Lendasse, A., Donckers, N., Verleysen, M. A robust nonlinear projection method 2000 :13-20 [html] |
| This paper describes a new nonlinear projection method. The aim is to design a user-friendly method, tentativ ely as easy to use as the linear PCA (Principal Component Analysis). The method is based on CCA (Curvilinear Component Analysis). This paper presen ts tw o improvements with respect to the original CCA: a better beha vior in the projection of highly nonlinear databases (like spirals) and a complete automation in the choice of the parameters value. |
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| Lee, D.D., Seung, H.S. Algorithm for Non-negative Matrix Factorization 2001 (13) [pdf] |
| Non-negative matrix factorization (NMF) has previously been shown to be a useful decomposition for multivariate data. Two different multiplicative algorithms for NMF are analyzed. They differ only slightly in the multiplicative factor used in the update rules. One algorithm can be shown to minimize the conventional least squares error while the other minimizes the generalized Kullback-Leibler divergence. The monotonic convergence of both algorithms can be proven using an auxiliary function analogous to that used for proving convergence of the Expectation- Maximization algorithm. The algorithms can also be interpreted as diagonally rescaled gradient descent, where the rescaling factor is optimally chosen to ensure convergence. |
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| Olver, J.P., Sapiro, G., Tannenbaum, A. Differential Invariant Signatures and Flows in Computer Vision: A Symmetry Group Approach 1993 [html] |
| Computer vision deals with image understanding at various levels. At the low level, it addresses issues such us planar shape recognition and analysis. Some classical results on differential invariants associated to planar curves are relevant to planar object recognition under different views and partial occlusion, and recent results concerning the evolution of planar shapes under curvature controlled diffusion have found applications in geometric shape decomposition, smoothing, and analysis, as well as in other image processing applications. In this work we first give a modern approach to the theory of differential invariants, describing concepts like Lie theory, jets, and prolongations. Based on this and the theory of symmetry groups, we present a high level way of defining invariant geometric flows for a given Lie group. We then analyze in detail different subgroups of the projective group, which are of special interest for computer vision. We classify the corresponding invariant flows and show that the geometric heat flow is the simplest possible one. This uniqueness result, together with previously reported results which we review in this paper, confirms the importance of this class of flows. |
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| Lee, D.D., Seung, H.S. Learning the Parts of Objects by Non-negative Matrix Factorization Nature 1999 (401):788 [pdf] |
| Is perception of the whole based on perception of its parts? There is psychological and physiological evidence for parts-based representations in the brain, and certain computational theories of object recognition rely on such representations. But little is known about how brains or computers might learn the parts of objects. Here we demonstrate an algorithm for non-negative matrix factorization that is able to learn parts of faces and semantic features of text. This is in contrast to other methods, such as principal components analysis and vector quantization, that learn holistic, not parts-based, representations. Non-negative matrix factorization is distinguished from the other methods by its use of non-negativity constraints. These constraints lead to a parts-based representation because they allow only additive, not subtractive, combinations. When non-negative matrix factorization is implemented as a neural network, parts-based representations emerge by virtue of two properties: the firing rates of neurons are never negative and synaptic strengths do not change sign. |
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