Learning from Examples as an Inverse Problem

By Ernesto De Vito, Lorenzo Rosasco, Andrea Caponnetto, Umberto De Giovannini, Francesca Odone. 2005

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Kernel Mean Estimation via Spectral Filtering (2014, methodology): “Since K may not exist and even if it exists, the computation of it can be numerically unstable, the idea of spectral filtering—this is quite popular in the theory of inverse problems [15] and has been used in kernel least squares [17]—is to replace K by some regularized matrices gλ(K) that approxima”
Kernel Bayes' Rule (2010, methodology): “1+β }, 11 we have bC Y X Cb XX +εnI −1 mbΠ −mQY HY = Op(n −min{2 3α, 2β+1 2β+2α}), (n→ ∞). Theorem 5 shows that the best rate is O(n−2α/3) for β≥ 1/2. The idea of the proof is essentially the same as [7, 10], which uses the standard techniques in regularization theory [8]. The condition π/pX ∈ R(AXC”
Optimal Rates for the Regularized Least-Squares Algorithm (2007, methodology): “of fρ, usually defined for regression in Sobolev spaces (see [11],[30],[17],[5],[9] and references therein).”
Geometry on Probability Spaces (2009, background): “Proposition 1 in various forms can be found in [14] (assuming ρX is nondegenerate), [18, 23] (assuming X ⊆Rn), and [4].”
On Learning with Integral Operators (2010, background): “In De Vito et al. (2005b) and De Vito et al. (2006) the results are stated for real kernels, however the proof does not change ifK is complex valued.”
Finite-Dimensional Projection for Classification and Statistical Learning (2008, unknown): “…if γ is an integer, the RKHS with this kernel is the Sobolev space of order γ with periodic boundary conditions It is known (see e.g. k | | (cid:16)R (cid:17) In this situation, the non-continuity of f ∗ prevents normal convergence of its Fourier series, so that necessarily α ≤ 1 . irregular .”
DISCRETIZATION ERROR ANALYSIS FOR TIKHONOV REGULARIZATION (2006, background): “Our setting appears natural while considering the problem of learning from examples where one has to deal with the stochastic discretization of a linear inverse problem [9].”
Vector Field Learning via Spectral Filtering (2010, background): “The connection between learning and regularization theory of illposed problems [3] motivates considering spectral filtering techniques.”
Multi-output learning via spectral filtering (2012, background): “The rationale behind them is the connection between learning theory and regularization of ill-posed problems (De Vito et al. 2005) and, more generally, the results showing the relation between stability and generalization (Poggio et al. 2004; Bousquet and Elisseeff 2002).”
Spectral Methods for Regularization in Learning Theory (2006, methodology): “In this section we clarify the role of the regularization looking at learning algorithm as an inverse problem as showed in De Vito et al. (2005b).”
Optimal Rates for Regularization Operators in Learning Theory (2006, methodology): “Regularization methods have been recently studied in the context of learning theory in [13, 9, 8, 10, 1].”
Statistical performance of support vector machines (2008, background): “In [27], the SVM is studied from the point of view of inverse problems.”
On Early Stopping in Gradient Descent Learning (2007, methodology): “For a broader discuss on this perspect, see [De Vito, Rosasco, Caponnetto, Giovannini, and Odone 2004].”
Statistical linear estimation with penalized estimators: an application to reinforcement learning (2012, background): “The idea of having learning problems as inverse problems is not new; Rosasco (2006); Vito et al. (2006) study regression in Hilbert spaces as an inverse problem.”
Generalization Properties of Learning with Random Features (2016, unknown): “Remark 4 (from [2, 27]) .”