Sample Complexity of Testing the Manifold Hypothesis

By Hariharan Narayanan, Sanjoy K. Mitter. 2010

262 citations recorded by the source corpus.

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Scaling & Learning Theory

Citations in the corpus

Better Mixing via Deep Representations (2012, background): “…a fundamental problem of mixing : it is difﬁcult for the Markov chain to jump from one mode of the distribution to another, when these are separated by large low-density regions, a common situation in real-world data, and under the manifold hypothesis (Cayton, 2005; Narayanan and Mitter, 2010).”
A Generative Process for Contractive Auto-Encoders (2012, methodology): “This means that if the true density tends to concentrate near low-dimensional manifolds (this is called the manifold hypothesis (Cay-ton, 2005; Narayanan and Mitter, 2010; Rifai et al., 2011a)), then a lot of densely packed training examples will be required to achieve a high model density near the…”
Representation Learning: A Review and New Perspectives (2012, background): “PRESENTATION LEARNING AS MANIFOLD LEARNING Another important perspective on representation learning is based on the geometric notion of manifold. Its premise is the manifold hypothesis (Cayton, 2005; Narayanan and Mitter, 2010), according to which real-world data presented in high dimensional spaces”
Towards Principled Methods for Training Generative Adversarial Networks (2017, background): “There is strong empirical and theoretical evidence to believe that P r is indeed extremely concentrated on a low dimensional manifold (Narayanan & Mitter, 2010).”
Mitigating Evasion Attacks to Deep Neural Networks via Region-based Classification (2017, background): “It would be an interesting work to explore other types of regions, e.g., lp-norm ball (i.e., Bp„x;r”= fyjyj 2»0;1…and jjy xjjp rg), hypersphere, and intersection between a lp-norm ball and manifolds [27] formed by the natural examples. Hypercube is essentially a l1-norm ball. Moreover, we essentiall”
MagNet: A Two-Pronged Defense against Adversarial Examples (2017, background): “Researchers believe thatNt constitute amanifold that is of much lower dimension than S [23].”
What regularized auto-encoders learn from the data-generating distribution (2012, background): “ata-generating distribution being associated with an unknown target density function, denoted p. Manifold learning algorithms assume that pis concentrated in regions of lower dimension (Cayton, 2005; Narayanan and Mitter, 2010), i.e., the training examples are by deﬁnition located very close to thes”
Stabilizing Training of Generative Adversarial Networks through Regularization (2017, methodology): “This may occur, whenever the model and/or data are confined to low-dimensional manifolds [3, 23].”
Deep Learning of Representations: Looking Forward (2013, methodology): “…for the Markov chain of MCMC methods to jump from one mode of the distribution to another, when these are separated by large low-density regions embedded in a high-dimensional space, a common situation in real-world data, and under the manifold hypothesis (Cayton, 2005; Narayanan and Mitter, 2010).”
Laplacian Auto-Encoders: An explicit learning of nonlinear data manifold (2015, background): “In many areas of machine learning and computer vision, a general assumption about sensory data is that they lie on or near a lowdimensional nonlinear manifold embedded in a high-dimensional space: when representing these data as points in a high-dimensional space, their probability density may be re”
Unsupervised Feature Learning and Deep Learning: A Review and New Perspectives (2012, background): “Its premise is the manifold hypothesis (Cayton, 2005; Narayanan and Mitter, 2010) according to which real-world data presented in high dimensional spaces are likely to concentrate in the vicinity of a non-linear manifold M of much lower dimensionality d M , embedded in high dimensional input space…”
Text to Image Synthesis Using Generative Adversarial Networks (2018, background): “In the case of natural images, there is substantial evidence that the support of Pr lies on a low dimensional manifold [24].”
Regularized Auto-Encoders Estimate Local Statistics (2012, background): “…being associated with an unknown target density function, denoted p. Manifold learning algorithms assume that p is concentrated in regions of lower dimension (Cayton, 2005; Narayanan and Mitter, 2010), i.e., the training examples are by definition located very close to these highdensity manifolds.”
Machine Learning in Aerodynamic Shape Optimization (2022, background): “The manifold hypothesis that high dimensional data tends to lie in the vicinity of a low dimensional manifold enables the local distance measurement in high-dimensional space [210, 211].”
Improved Autoregressive Modeling with Distribution Smoothing (2021, methodology): “Several existing methods (Roweis & Saul, 2000; Tenenbaum et al., 2000) rely on the manifold hypothesis, i.e . that real-world high-dimensional data tends to lie on a low-dimensional manifold (Narayanan & Mitter, 2010).”
The Manifold Tangent Classifier (2011, background): “The (unsupervised) manifold hypothesis, according to which real world data presented in high dimensional spaces is likely to concentrate in the vicinity of non-linear sub-manifolds of much lower dimensionality (Cayton, 2005; Narayanan and Mitter, 2010).”
Lipschitz Continuity in Model-based Reinforcement Learning (2018, unknown): “(Narayanan & Mitter, 2010)”
On the Privacy Risks of Model Explanations (2019, background): “[28] H Narayanan and S Mitter.”