Semi-Infinite Linear Regression and Its Applications

Shustin, Paz Fink; Avron, Haim

doi:10.1137/21m1411950

Cited by 2 publications

(2 citation statements)

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“…A quasi-matrix can be tall (or wide) in which there is a finite number of columns (or rows) where each one is a functional operator. For more details and a formal definition, see [SA22].…”

Section: Mathematical Preliminariesmentioning

confidence: 99%

Near Optimal Reconstruction of Spherical Harmonic Expansions

Zandieh¹,

Han²,

Avron³

2022

Preprint

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We propose an algorithm for robust recovery of the spherical harmonic expansion of functions defined on the d-dimensional unit sphere S d−1 using a near-optimal number of function evaluations. We show that for any f ∈ L 2 (S d−1 ), the number of evaluations of f needed to recover its degree-q spherical harmonic expansion equals the dimension of the space of spherical harmonics of degree at most q up to a logarithmic factor. Moreover, we develop a simple yet efficient algorithm to recover degree-q expansion of f by only evaluating the function on uniformly sampled points on S d−1 . Our algorithm is based on the connections between spherical harmonics and Gegenbauer polynomials and leverage score sampling methods. Unlike the prior results on fast spherical harmonic transform, our proposed algorithm works efficiently using a nearly optimal number of samples in any dimension d. We further illustrate the empirical performance of our algorithm on numerical examples.

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“…A quasi-matrix can be tall (or wide) in which there is a finite number of columns (or rows) where each one is a functional operator. For more details and a formal definition, see [SA22].…”

Section: Mathematical Preliminariesmentioning

confidence: 99%

Near Optimal Reconstruction of Spherical Harmonic Expansions

Zandieh¹,

Han²,

Avron³

2022

Preprint

Self Cite

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“…However, when the dimensions of the model matrix X become extremely large, then computing by Equation 8 may prove challenging. It is difficult to offer a precise definition of “extremely large.” One theoretical example is a quasi-matrix defined as some matrix having one infinite dimension (Shustin & Avron, 2021). Perhaps more concretely, we can imagine the vector space needed in natural language processing applications to represent words in examinee essay responses as one common example leading to a very large X .…”

Section: Computational Concepts For Linear Systemsmentioning

confidence: 99%

A Collection of Numerical Recipes Useful for Building Scalable Psychometric Applications

Doran

2022

Journal of Educational and Behavioral Statistics

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This article is concerned with a subset of numerically stable and scalable algorithms useful to support computationally complex psychometric models in the era of machine learning and massive data. The subset selected here is a core set of numerical methods that should be familiar to computational psychometricians and considers whitening transforms for dealing with correlated data, computational concepts for linear models, multivariable integration, and optimization techniques.

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Semi-Infinite Linear Regression and Its Applications

Cited by 2 publications

References 25 publications

Near Optimal Reconstruction of Spherical Harmonic Expansions

Near Optimal Reconstruction of Spherical Harmonic Expansions

A Collection of Numerical Recipes Useful for Building Scalable Psychometric Applications

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