aiqbench analysis
This paper presents the Functional Spectral-Newton Method (FSNM), a novel algorithm for learning the spectral structure of conditional expectation operators without relying on fixed bases or specific kernel Hilbert spaces. By leveraging vector-valued regression trees within a boosting framework, the method achieves global optimality at local minima and efficiently answers multiple conditional queries using a single learned kernel. The approach advances theoretical machine learning by establishing rigorous convergence rates for low-rank density ratio approximations in high-dimensional settings.
Learning Conditional Expectation Operators via Functional Newton Updates
arXiv:2609.35598v1 Announce Type: cross Abstract: We introduce the Functional Spectral-Newton Method (FSNM) for learning the leading singular structure of a conditional expectation operator without fixing a basis or reproducing kernel Hilbert space. FSNM fits a low-rank representation of the centered joint-to-product density ratio kernel by alternating functional Newton updates. Each update reduces to a preconditioned regression, which we approximate with vector-valued regression trees in a stagewise boosting procedure. At the population level, we establish descent and an $O(1/T)$ best-iterate block-stationarity rate under a relative weak-learner accuracy condition, and show that every nondegenerate local minimum over the full centered $L^2$ spaces is a globally optimal rank-$d$ approximation. Synthetic experiments show that FSNM recovers a low-rank density ratio and its leading spectral structure, and that the same learned kernel can answer multiple conditional queries without refitting.
Abstract:We introduce the Functional Spectral-Newton Method (FSNM) for learning the leading singular structure of a conditional expectation operator without fixing a basis or reproducing kernel Hilbert space. FSNM fits a low-rank representation of the centered joint-to-product density ratio kernel by alternating functional Newton updates. Each update reduces to a preconditioned regression, which we approximate with vector-valued regression trees in a stagewise boosting procedure. At the population level, we establish descent and an $O(1/T)$ best-iterate block-stationarity rate under a relative weak-learner accuracy condition, and show that every nondegenerate local minimum over the full centered $L^2$ spaces is a globally optimal rank-$d$ approximation. Synthetic experiments show that FSNM recovers a low-rank density ratio and its leading spectral structure, and that the same learned kernel can answer multiple conditional queries without refitting.
Machine Learning (stat.ML); Machine Learning (cs.LG)
arXiv:2609.35598 [stat.ML]
(or arXiv:2609.35598v1 [stat.ML] for this version)
https://doi.org/10.48550/arXiv.2609.35598 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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Full text under the publisher's open licence · 226 words
