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Finite-sample representer theorem when a minimizer exists

Statement

Let $H$ be a real reproducing kernel Hilbert space with kernel $k$ , let $n \geq 0$ , let $x_{1}, \dots, x_{n}$ be sample points, let $L : R^{n} \to R$ be finite-valued, and let $Ω : [0, \infty) \to R$ be nondecreasing. Define $J (f) : = L (f (x_{1}), \dots, f (x_{n})) + Ω (∥ f ∥_{H})$ . If $J$ has a minimizer, then it has a minimizer in $span {k (\cdot, x_{i}) : 1 \leq i \leq n}$ ; equivalently, at least one minimizer has the finite expansion $f (\cdot) = \sum_{i = 1}^{n} α_{i} k (\cdot, x_{i})$ . This is an existence-conditional statement and asserts no minimizer exists when none is separately guaranteed.

Statement references

Hastie, Tibshirani, and Friedman, The Elements of Statistical Learning, §5.8

Rasmussen and Williams, Gaussian Processes for Machine Learning, §6.2

Facts & Assumptions

Given: A real RKHS $H$ with kernel $k$ , finitely many sample points, a finite-valued loss $L$ , a nondecreasing penalty $Ω$ , and a minimizer $f$ of $J$ .

[F1]

For each $x$ , the section $k_{x} = k (\cdot, x)$ lies in the RKHS and $⟨ h, k_{x} ⟩ = h (x)$ (Positive semidefinite kernels and reproducing kernel Hilbert spaces).

[F2]

Every finite-dimensional subspace of a real inner-product space admits a unique orthogonal projection, the residual is orthogonal to the subspace, and the projection does not increase norm (Projection onto a finite-dimensional subspace of an arbitrary real inner-product space).

Proof

1.1givenF2construct

Let $M : = span {k_{x_{i}} : 1 \leq i \leq n}$ . It is finite-dimensional. Apply [F2] to $f$ and $M$ , and write $f = p + h$ with $p \in M$ and $h ⊥ M$ .

2.1step 1.1F1algebra

For every $i$ , [F1] gives $h (x_{i}) = ⟨ h, k_{x_{i}} ⟩ = 0$ , so $p (x_{i}) = f (x_{i})$ . Hence the loss terms in $J (p)$ and $J (f)$ agree.

3.1step 1.1step 2.1F2

The norm-contraction clause of [F2] gives $∥ p ∥_{H} \leq ∥ f ∥_{H}$ . Monotonicity of $Ω$ therefore gives $J (p) \leq J (f)$ .

4.1givenstep 3.1∎

Since $f$ is a minimizer, $J (f) \leq J (p)$ . Thus $J (p) = J (f)$ and $p$ is also a minimizer; by its definition $p$ is a finite linear combination of the sample kernel sections.

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Sources

Hastie, Tibshirani, and Friedman, The Elements of Statistical Learning, 2nd ed., corrected 12th printing (standard reference, not scraped)
Rasmussen and Williams, Gaussian Processes for Machine Learning (standard reference, not scraped)