Approximations of the Reproducing Kernel Hilbert Space (RKHS) Embedding Method over Manifolds

Guo, Jia; Paruchuri, Sai Tej; Kurdila, Andrew J.

doi:10.1109/cdc42340.2020.9304331

Cited by 11 publications

(12 citation statements)

References 17 publications

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“…This paper extends the results in the recent papers [23], [41], [24], [25], [42] in several fundamental ways. The first result states sufficient conditions that guarantee the PE condition for finite-dimensional RKHS over a manifold M that is defined in terms of a finite number of kernel basis functions.…”

Section: B Summary Of New Resultssupporting

confidence: 86%

“…As in [29], we see that the RKHS is PE if the trajectory repeatedly visits any (geodesic) neighborhoods of the kernel basis centers, and the time of visitation is bounded below in some sense. This result has direct applicability to finite-dimensional cases of the RKHS embedding methods discussed in [23], [41], [24], [25], [42]. It serves as a foundation for practical choices of PE subsets and spaces, which is not addressed in these references.…”

Section: B Summary Of New Resultsmentioning

confidence: 99%

“…[23], [41] The corresponding PE conditions are given in [24], [25]. Guo et al study the rate of convergence of the finite-dimensional approximations of reproducing kernel Hilbert spaces defined over manifolds in [42]. From an adaptive estimation perspective, this is equivalent to studying the rate at which the finite-dimensional function estimate fn converges to the infinite-dimensional function estimate f .…”

Section: A Pe Conditions and Convergencementioning

confidence: 99%

“…We next study in the following corollary a new error bound that is an immediate result of the sufficient condition derived in Theorem 4 and Theorem 5. Following the derivation of the corollary, we compare and contrast the nature of the new convergence rate with the results in [42].…”

Section: Implications Of the Sufficient Condition In Infinite Dimensionsmentioning

confidence: 99%

“…It is shown in [42] that for certain kernels, the rate of convergence of the finite-dimensional function estimate fn to the infinite-dimensional function estimate f depends on the fill distance. We can think of the infinite-dimensional estimate as the one that makes the error P Ω f (t) → 0 as t → ∞, where Ω is the PE set (that consists of infinite number of elements).…”

Section: Remarks On Corollarymentioning

confidence: 99%

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Sufficient Conditions for Parameter Convergence Over Embedded Manifolds Using Kernel Techniques

Paruchuri

Guo

Kurdila

2023

IEEE Trans. Automat. Contr.

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The persistence of excitation (PE) condition is sufficient to ensure parameter convergence in adaptive estimation problems. Recent results on adaptive estimation in reproducing kernel Hilbert spaces (RKHS) introduce PE conditions for RKHS. This paper presents sufficient conditions for PE for the particular class of uniformly embedded reproducing kernel Hilbert spaces (RKHS) defined over smooth Riemannian manifolds. This paper also studies the implications of the sufficient condition in the case when the RKHS is finite or infinite-dimensional. When the RKHS is finite-dimensional, the sufficient condition implies parameter convergence as in the conventional analysis. On the other hand, when the RKHS is infinite-dimensional, the same condition implies that the function estimate error is ultimately bounded by a constant that depends on the approximation error in the infinite-dimensional RKHS. We illustrate the effectiveness of the sufficient condition in a practical example.

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Section: B Summary Of New Resultssupporting

confidence: 86%

Section: B Summary Of New Resultsmentioning

confidence: 99%

Section: A Pe Conditions and Convergencementioning

confidence: 99%

Section: Implications Of the Sufficient Condition In Infinite Dimensionsmentioning

confidence: 99%

Section: Remarks On Corollarymentioning

confidence: 99%

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Sufficient Conditions for Parameter Convergence Over Embedded Manifolds Using Kernel Techniques

Paruchuri

Guo

Kurdila

2023

IEEE Trans. Automat. Contr.

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Kernel center adaptation in the reproducing kernel Hilbert space embedding method

Paruchuri

Guo

Kurdila

2022

Adaptive Control & Signal

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SummaryThe performance of adaptive estimators that employ embedding in reproducing kernel Hilbert spaces (RKHS) depends on the choice of the location of basis kernel centers. Parameter convergence and error approximation rates depend on where and how the kernel centers are distributed in the state‐space. In this article, we develop the theory that relates parameter convergence and approximation rates to the position of kernel centers. We develop criteria for choosing kernel centers in a specific class of systems by exploiting the fact that the state trajectory regularly visits the neighborhood of the positive limit set. Two algorithms, based on centroidal Voronoi tessellations and Kohonen self‐organizing maps, are derived to choose kernel centers in the RKHS embedding method. Finally, we implement these methods on two practical examples and test their effectiveness.

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Kernel methods for regression in continuous time over subsets and manifolds

et al. 2023

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Approximations of the Reproducing Kernel Hilbert Space (RKHS) Embedding Method over Manifolds

Cited by 11 publications

References 17 publications

Sufficient Conditions for Parameter Convergence Over Embedded Manifolds Using Kernel Techniques

Sufficient Conditions for Parameter Convergence Over Embedded Manifolds Using Kernel Techniques

Kernel center adaptation in the reproducing kernel Hilbert space embedding method

Kernel methods for regression in continuous time over subsets and manifolds

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