Probability error bounds for approximation of functions in reproducing kernel Hilbert spaces
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arXiv
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| Natura: | Preprint |
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2020
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| _version_ | 1866911782189137920 |
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| author | Aydin, Ata Deniz Gheondea, Aurelian |
| author_facet | Aydin, Ata Deniz Gheondea, Aurelian |
| contents | We find probability error bounds for approximations of functions $f$ in a separable reproducing kernel Hilbert space $\mathcal{H}$ with reproducing kernel $K$ on a base space $X$, firstly in terms of finite linear combinations of functions of type $K_{x_i}$ and then in terms of the projection $π^n_x$ on $\mathrm{Span}\{K_{x_i}\}^n_{i=1}$, for random sequences of points $x=(x_i)_i$ in $X$. Given a probability measure $P$, letting $P_K$ be the measure defined by $\mathrm{d} P_K(x)=K(x,x)\mathrm{d} P(x)$, $x\in X$, our approach is based on the nonexpansive operator \[L^2(X;P_K)\niλ\mapsto L_{P,K}λ:=\int_X λ(x)K_x\mathrm{d} P(x)\in \mathcal{H},\] where the integral exists in the Bochner sense. Using this operator, we then define a new reproducing kernel Hilbert space, denoted by $\mathcal{H}_P$, that is the operator range of $L_{P,K}$. Our main result establishes bounds, in terms of the operator $L_{P,K}$, on the probability that the Hilbert space distance between an arbitrary function $f\in\mathcal{H}$ and linear combinations of functions of type $K_{x_i}$, for $(x_i)_i$ sampled independently from $P$, falls below a given threshold. For sequences of points $(x_i)_{i=1}^\infty$ constituting a so-called uniqueness set, the orthogonal projections $π^n_x$ to $\mathrm{Span}\{K_{x_i}\}^n_{i=1}$ converge in the strong operator topology to the identity operator. We prove that, under the assumption that $\mathcal{H}_P$ is dense in $\mathcal{H}$, any sequence of iid samples from $P$ yields a uniqueness set with probability $1$. This result improves on previous error bounds in weaker norms, such as uniform or $L^p$ norms, which yield only convergence in probability and not a.c. convergence. Two examples that show the applicability of this result to a uniform distribution on a compact interval and to the Hardy space $H^2(\mathbb{D})$ are presented as well. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2003_12801 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Probability error bounds for approximation of functions in reproducing kernel Hilbert spaces Aydin, Ata Deniz Gheondea, Aurelian Numerical Analysis Information Theory Functional Analysis Primary 41A65, Secondary 68T05, 46E22 We find probability error bounds for approximations of functions $f$ in a separable reproducing kernel Hilbert space $\mathcal{H}$ with reproducing kernel $K$ on a base space $X$, firstly in terms of finite linear combinations of functions of type $K_{x_i}$ and then in terms of the projection $π^n_x$ on $\mathrm{Span}\{K_{x_i}\}^n_{i=1}$, for random sequences of points $x=(x_i)_i$ in $X$. Given a probability measure $P$, letting $P_K$ be the measure defined by $\mathrm{d} P_K(x)=K(x,x)\mathrm{d} P(x)$, $x\in X$, our approach is based on the nonexpansive operator \[L^2(X;P_K)\niλ\mapsto L_{P,K}λ:=\int_X λ(x)K_x\mathrm{d} P(x)\in \mathcal{H},\] where the integral exists in the Bochner sense. Using this operator, we then define a new reproducing kernel Hilbert space, denoted by $\mathcal{H}_P$, that is the operator range of $L_{P,K}$. Our main result establishes bounds, in terms of the operator $L_{P,K}$, on the probability that the Hilbert space distance between an arbitrary function $f\in\mathcal{H}$ and linear combinations of functions of type $K_{x_i}$, for $(x_i)_i$ sampled independently from $P$, falls below a given threshold. For sequences of points $(x_i)_{i=1}^\infty$ constituting a so-called uniqueness set, the orthogonal projections $π^n_x$ to $\mathrm{Span}\{K_{x_i}\}^n_{i=1}$ converge in the strong operator topology to the identity operator. We prove that, under the assumption that $\mathcal{H}_P$ is dense in $\mathcal{H}$, any sequence of iid samples from $P$ yields a uniqueness set with probability $1$. This result improves on previous error bounds in weaker norms, such as uniform or $L^p$ norms, which yield only convergence in probability and not a.c. convergence. Two examples that show the applicability of this result to a uniform distribution on a compact interval and to the Hardy space $H^2(\mathbb{D})$ are presented as well. |
| title | Probability error bounds for approximation of functions in reproducing kernel Hilbert spaces |
| topic | Numerical Analysis Information Theory Functional Analysis Primary 41A65, Secondary 68T05, 46E22 |
| url | https://arxiv.org/abs/2003.12801 |