The covering number in learning theory

By Ding-Xuan Zhou. 2002

378 citations recorded by the source corpus.

Subfields: Applied Mathematics, Mathematical Physics

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Citations in the corpus

Interpretable Machine Learning: Fundamental Principles and 10 Grand Challenges (2021, background): “Ways that complexity of function classes, algorithms, or learning problems are typically measured in statistical learning theory include the VC dimension, covering numbers, algorithmic stability, Rademacher complexity, or flat minima (see, for instance, Srebro et al., 2010; Kakade et al., 2009; Zhou”
Generative and Latent Mean Map Kernels (2010, background): “For convolutional kernels, Zhou [31] showed that when F [k] (i.”
Quantile Regression in Reproducing Kernel Hilbert Spaces (2007, methodology): “Using the property of RKHS with a radial basis kernel ( Zhou 2002 ), we have HB(u, F(k)) = O(log2(k/u)) for any given k. Because F1 ⊂ F(b) ,w e have H B(u, F1) = O(log2(1/u)) .L et”
Consistency of spectral clustering (2008, methodology): “To this, end we use bounds on the ‖ ·‖∞-covering numbers of K obtained in Proposition 1 in [51].”
Regularization in kernel learning (2010, background): “Zhou, [37], however, argues against this assumption and provides an example of a C∞ kernel without uniformly bounded eigenfunctions.”
Are Loss Functions All the Same? (2004, methodology): “The covering number has been estimated from the upper bounds found in Zhou (2002) for the Gaussian kernel.”
Reproducing Kernel Hilbert Spaces Associated with Analytic Translation-Invariant Mercer Kernels (2008, methodology): “In the above second case, we restrict c> 4 + 2n log 4 because our approach in [ 20 ] requires the Fourier transform of ϕ to decay exponentially fast with a large exponent.”
Application of integral operator for regularized least-square regression (2009, background): “For example, the covering numbers of RKHS is only well estimated for certain very smooth kernels such as polynomial and Gaussians and the underlying input space X is finite dimensional compact set [12,16,17].”
Capacity of reproducing kernel spaces in learning theory (2003, background): “The example of kernel presented in [25] tells us that the regularity of the kernel function (or the decay of the eigenvalues of the integral operator ) does not ensure this assumption to be valid.”
Regularized Policy Iteration with Nonparametric Function Spaces (2016, background): “Assumption A4 is suitable for large function spaces and is indeed satisfied for the Sobolev spaces and various RKHS. Refer to van de Geer (2000); Zhou (2002, 2003); Steinwart and Christmann (2008) for many examples.”
Multi-View Intact Space Learning (2015, methodology): “We proceed to bound the Rademacher complexity of F by ﬁrst bounding its covering number [50], [51].”
Bridging Theory and Algorithm for Domain Adaptation (2019, methodology): “To this end, we describe the notion of covering number from Zhou (2002); Anthony & Bartlett (2009); Talagrand (2014).”