On Learning with Integral Operators

By Lorenzo Rosasco, Mikhail Belkin, Ernesto De Vito. 2010

263 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Mathematical Physics, Statistical Learning Theory

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Citations in the corpus

Kernel Mean Embedding of Distributions: A Review and Beyonds (2016, background): “see, e.g., Rosasco et al. (2010) for further details.”
Kernel Mean Estimation via Spectral Filtering (2014, background): “It is known that the operator C k is of ﬁnite rank, self-adjoint, and positive. b b If ˜ γ i is a nonzero eigenvalue and ˜ v i is the corresponding eigenfunction of C k , then the following decomposition holds Moreover, its spectrum has only ﬁnitely many nonzero elements [28].”
Towards Understanding the Spectral Bias of Deep Learning (2019, methodology): “The theory of integral operator with respect to kernel function has been well studied in literature [Smale and Zhou, 2007; Rosasco et al., 2010] thus we only give a brief introduction here.”
Exponential convergence of testing error for stochastic gradient methods (2017, unknown): “To bound v n we apply again the Pinelis inequality (see also Rosasco et al., 2010) considering that the space of Hilbert-Schmidt operators is again a Hilbert space and that (cid:98) Σ = In particular we apply it with D = 1 and b 2 ∗ = 4 R 4 n , so with probability 1 − 2 e − τ .”
Nonparametric Estimation of Multi-View Latent Variable Models (2013, methodology): “Similarly for the second term in (19), we have Proof We will use similar arguments as in Rosasco et al. (2010) which deals with symmetric operator.”
A Spectral Approach to Gradient Estimation for Implicit Distributions (2018, background): “Note that when this assumption does not hold, we could proceed as in Rosasco et al. (2010) (Theorem 12) and derive the error bound in a similar way.”
Learning Sets with Separating Kernels (2012, background): “This last result is the key towards a new computational approach to learn the support from data, since the integral operator can be approximated with high probability from random samples [53, 60].”
Anisotropic diffusion on sub-manifolds with application to Earth structure classification (2012, unknown): “[8–12].”
Identification of the Diffusion Parameter in Nonlocal Steady Diffusion Problems (2013, methodology): “Among those applications we have image analyses [14, 24, 25, 32], machine learning [36], nonlocal Dirichlet forms [6], kinetic equations [8, 31], phase transitions [11, 23], and nonlocal heat conduction [12].”
Analysis and Approximation of Nonlocal Diffusion Problems with Volume Constraints (2012, background): “The operator L and its generalizations arise in many applications such as image analyses [19, 32, 33, 39], machine learning [47], nonlocal Dirichlet forms [5, Sec.”
On Some Extensions of Bernstein's Inequality for Self-adjoint Operators (2011, background): “rator norm? Note that the spectrum of L K;n can be easily evaluated in practice since the sets of non-zero eigenvalues of L K;n and a matrix K = 1 n K(X i;X j) n i;j=1 are equal, see Proposition 9 in [10]. Theorem2.1gives an answer to this question. To apply the theorem, dene operator-valued random ”
Multivariate nonparametric test of independence (2017, background): “Proposition 10 in [26] is precisely tailored for such continuous and uniformly bounded kernels and their empirical counterparts.”
Reproducing kernel Hilbert spaces (2019, background): “I’m not sure what the immediate benefits of this representation are, but I have found work [4] that discusses reconstructing a continuous integral operator from finite-dimensional data, i.”