Learning linear operators: Infinite-dimensional regression as a well-behaved non-compact inverse problem
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| Format: | Preprint |
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2022
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| _version_ | 1866910520564514816 |
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| author | Mollenhauer, Mattes Mücke, Nicole Sullivan, T. J. |
| author_facet | Mollenhauer, Mattes Mücke, Nicole Sullivan, T. J. |
| contents | We consider the problem of learning a linear operator $θ$ between two Hilbert spaces from empirical observations, which we interpret as least squares regression in infinite dimensions. We show that this goal can be reformulated as an inverse problem for $θ$ with the feature that its forward operator is generally non-compact (even if $θ$ is assumed to be compact or of $p$-Schatten class). However, we prove that, in terms of spectral properties and regularisation theory, this inverse problem is equivalent to the known compact inverse problem associated with scalar response regression.
Our framework allows for the elegant derivation of dimension-free rates for generic learning algorithms under Hölder-type source conditions. The proofs rely on the combination of techniques from kernel regression with recent results on concentration of measure for sub-exponential Hilbertian random variables. The obtained rates hold for a variety of practically-relevant scenarios in functional regression as well as nonlinear regression with operator-valued kernels and match those of classical kernel regression with scalar response. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2211_08875 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Learning linear operators: Infinite-dimensional regression as a well-behaved non-compact inverse problem Mollenhauer, Mattes Mücke, Nicole Sullivan, T. J. Statistics Theory Functional Analysis Probability Machine Learning 62J05, 65J22, 47A52, 47A68 We consider the problem of learning a linear operator $θ$ between two Hilbert spaces from empirical observations, which we interpret as least squares regression in infinite dimensions. We show that this goal can be reformulated as an inverse problem for $θ$ with the feature that its forward operator is generally non-compact (even if $θ$ is assumed to be compact or of $p$-Schatten class). However, we prove that, in terms of spectral properties and regularisation theory, this inverse problem is equivalent to the known compact inverse problem associated with scalar response regression. Our framework allows for the elegant derivation of dimension-free rates for generic learning algorithms under Hölder-type source conditions. The proofs rely on the combination of techniques from kernel regression with recent results on concentration of measure for sub-exponential Hilbertian random variables. The obtained rates hold for a variety of practically-relevant scenarios in functional regression as well as nonlinear regression with operator-valued kernels and match those of classical kernel regression with scalar response. |
| title | Learning linear operators: Infinite-dimensional regression as a well-behaved non-compact inverse problem |
| topic | Statistics Theory Functional Analysis Probability Machine Learning 62J05, 65J22, 47A52, 47A68 |
| url | https://arxiv.org/abs/2211.08875 |