Nonlocal Estimation of Manifold Structure

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Generalized Denoising Auto-Encoders as Generative Models (2013, background): “This idea is already behind the non-local manifold Parzen windows (Bengio et al. , 2006a) and non-local manifold tangent learning (Bengio et al. , 2006b) algorithms: the local density around a point ˜ X can be approximated by a multivariate Gaussian whose covariance matrix has leading eigenvectors…”
A Generative Process for Contractive Auto-Encoders (2012, methodology): “A ﬁrst non-linear manifold learning algorithm with non-local generalization was proposed by Bengio et al. (2006a), stimulating the work presented here, which also follows up on more recent work in the area of Deep Learning (Hinton et al., 2006; Bengio, 2009), more speciﬁcally the Con-tractive…”
Deep Learning of Representations: Looking Forward (2013, background): “, 2006a) and non-local manifold tangent learning (Bengio et al., 2006b) algorithms: the local density around a point X̃ can be approximated by a multivariate Gaussian whose covariance matrix has leading eigenvectors that span the local tangent of the manifold near which the data concentrates (if it ”
Learning Deep Architectures for AI (2007, background): “In that case one can show (Bengio et al., 2006) that the number of examples necessary with a Gaussian kernel machine is exponential in the input dimension: Theorem 3.2.”