On Learning Vector − Valued Functions

By Charles A. Micchelli, Massimiliano Pontil. 2003

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In Statistical Learning Theory

Citations in the corpus

Kernel Mean Embedding of Distributions: A Review and Beyonds (2016, methodology): “Given a sample (x1, z1), . . . , (xn, zn) ∈ X × G , a vector-valued regression problem can be formulated as Êλ(f) = n∑ i=1 ‖zi − f(xi)‖2G + λ‖f‖2HΓ (4.9) where G is a Hilbert space and HΓ denotes a RKHS of vector-valued functions from X to G (see Micchelli and Pontil (2005) for more details).”
Learning Theory for Distribution Regression (2014, methodology): “The kernel (say KG) is defined on M + 1 (X), and the regressor is then modelled by an element in the RKHS G = G(KG) of functions mapping from M + 1 (X) to Y .”
Modelling transition dynamics in MDPs with RKHS embeddings (2012, background): “It can be shown that the conditional embeddings 5 We thank the authors for providing code. solve, where H ⊆ ( H L ) ( X×A ) , recovering the vector-valued regression setting of Micchelli & Pontil (2005) (see Sec.”
Conditional mean embeddings as regressors (2012, background): “(Micchelli & Pontil, 2005, Th. 4) If f ∗ min-imises (cid:98) E λ in H Γ then it is unique and has the form, where the coefﬁcients { c i } i ≤ n , c i ∈ V are the unique solu-tion of the system of linear equations In the scalar case we have that | f ( x ) | ≤ K ( x, x ) (cid:107) f (cid:107) K .”
Optimal Rates for the Regularized Least-Squares Algorithm (2007, background): “We now introduce some basic concepts of statistical learning theory in the regression setting for vector-valued outputs (for details see [32], [15], [24], [6], [20] and references therein).”
Calibration tests in multi-class classification: A unifying framework (2019, background): “A more detailed general introduction to RKHSs of vector-valued functions can be found in the publications by Caponnetto et al. (2008); Carmeli et al. (2010); Micchelli and Pontil (2005).”
Learning Multiple Tasks with Kernel Methods (2005, methodology): “…is currently an open questions that can be approached, for example, using convex optimization techniques, see, for example, (Lanckriet et al., 2004; Micchelli and Pontil, 2005b) To explore the second point further, we show in Figure 1 the change in performance for the identity matrix based…”
Vector valued reproducing kernel Hilbert spaces and universality (2008, methodology): “In the case Y is itself a Hilbert space, the theory can be simplified as shown in [21, 6, 5].”
Kernel Bayes' rule: Bayesian inference with positive definite kernels (2013, background): “Note that more complex vector-valued RKHS are possible; see, for example, Micchelli and Pontil (2005).”
A Generalized Kernel Approach to Structured Output Learning (2012, methodology): “We now provide a few definitions related to operatorvalued kernels and their associated RKHSs that are used in the paper (see (Micchelli & Pontil, 2005; Caponnetto et al., 2008; Álvarez et al., 2012) for more details).”
Semi-Supervised Multi-Label Learning with Incomplete Labels (2015, background): “The solution of (1) takes the form of f * = i=1 K xi a i with {a i , i = 1, · · ·, } ⊆ Y [Micchelli and Pontil, 2005] .”
Learning the coordinate gradients (2011, methodology): “The above contributions rely on very recent developments from machine learning: multi-task learning [5, 20, 21] and multiple kernel learning with sparse regularization (aka nonparametric group lasso) [3, 16, 30].”
Bayesian Online Multitask Learning of Gaussian Processes (2010, methodology): “Recently, vector-valued Reproducing Kernel Hilbert Spaces (RKHSs) [20] were used to derive multitask-regular-ized kernel methods [21].”
Cross-Domain Object Recognition Via Input-Output Kernel Analysis (2013, methodology): “According to the representer theorem [15], the optimal solution for the inner minimization has the form g * (x) = l i=1 H (x, x i )c i = L l i=1 c i K (x, x i ), (3) where K is the input kernel function.”
Vector Field Learning via Spectral Filtering (2010, methodology): “The development of the theory of RKHS in the vector case is essentially the same as in the scalar case and we refer to [1,19] for further details and references.”
Kernel Exponential Family Estimation via Doubly Dual Embedding (2018, methodology): “We consider f : Ω x Denoting T ∈ H Ω x : Ω x → H y such that T x ( y ) = f ( x, y ), we can derive its kernel function following Micchelli and Pontil (2005); Arbel and Gretton (2017).”