Numerical Evaluation of Reproducing Kernel Hilbert Space Inner Products

Oya, Antonia; Navarro-Moreno, J.; Ruiz-Molina, Juan Carlos

doi:10.1109/tsp.2008.2010424

Cited by 5 publications

(2 citation statements)

References 18 publications

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“…Naturally, {(λ n , e n )} N n=1 need to be computed, and are done so by solving the eigenvalue problem Kf = λf. Oya et al [2009] discuss various methods of computing RKHS inner products using this formulation, and suggested using a Ritz-Rayleigh (RR) approach to compute the approximate spectral decomposition of K and inserting the approximate values {( λn , ẽn )} to compute the inner product. To summarize this approach, suppose that A ∈ R n×n is positive semidefinite, and V ∈ R p×n has orthonormal row vectors {v 1 , .…”

Section: Computation Of Rkhs Inner Productmentioning

confidence: 99%

A Kernel-Based Approach for Modelling Gaussian Processes with Functional Information

Nicholson¹,

Kiessler²,

Brown³

2022

Preprint

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Gaussian processes are among the most useful tools in modeling continuous processes in machine learning and statistics. If the value of a process is known at a finite collection of points, one may use Gaussian processes to construct a surface which interpolates these values to be used for prediction and uncertainty quantification in other locations. However, it is not always the case that the available information is in the form of a finite collection of points. For example, boundary value problems contain information on the boundary of a domain, which is an uncountable collection of points that cannot be incorporated into typical Gaussian process techniques. In this paper we construct a Gaussian process model which utilizes reproducing kernel Hilbert spaces to unify the typical finite case with the case of having uncountable information by exploiting the equivalence of conditional expectation and orthogonal projections. We discuss this construction in statistical models, including numerical considerations and a proof of concept.

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Section: Computation Of Rkhs Inner Productmentioning

confidence: 99%

A Kernel-Based Approach for Modelling Gaussian Processes with Functional Information

Nicholson¹,

Kiessler²,

Brown³

2022

Preprint

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“…Oya et al [35] proposed a generalised numerical approach to estimate the inner product in H k . In our approach (Section IV), we use a known analytical form for the inner product, which avoids matrix product or inversion and thus allows to escape the curse of dimensionality.…”

Section: Rkhs Of Gaussian Process and Dpmmentioning

confidence: 99%

High Dimensional Data Clustering by means of Distributed Dirichlet Process Mixture Models

Meguelati

Fontez

Hilgert

et al. 2019

2019 IEEE International Conference on Big Data (Big Data)

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Clustering is a data mining technique intensively used for data analytics, with applications to marketing, security, text/document analysis, or sciences like biology, astronomy, and many more. Dirichlet Process Mixture (DPM) is a model used for multivariate clustering with the advantage of discovering the number of clusters automatically and offering favorable characteristics. However, in the case of high dimensional data, it becomes an important challenge with numerical and theoretical pitfalls. The advantages of DPM come at the price of prohibitive running times, which impair its adoption and makes centralized DPM approaches inefficient, especially with high dimensional data. We propose HD4C (High Dimensional Data Distributed Dirichlet Clustering), a parallel clustering solution that addresses the curse of dimensionality by two means. First it gracefully scales to massive datasets by distributed computing, while remaining DPM-compliant. Second, it performs clustering of high dimensional data such as time series (as a function of time), hyperspectral data (as a function of wavelength) etc. Our experiments, on both synthetic and real world data, illustrate the high performance of our approach.

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