Optimality of the rescaled pure greedy learning algorithms

Zhang, Wenhui; Ye, Peixin; Xing, Shuo

doi:10.1142/s0219691322500485

Cited by 3 publications

(2 citation statements)

References 33 publications

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“…• • • and N = 1, then the VWRPGA degenerates into the RPGA. In [5], the authors applied the RPGA to a kernel-based regression. They defined the Rescaled Pure Greedy Learning Algorithm (RPGLA) and studied its efficiency.…”

Section: Discussionmentioning

confidence: 99%

“…Approximation using a sparse linear combination of elements from a fixed redundant family is actively used because of its concise representations and increased computational efficiency. It has been applied widely to signal processing, image compression, machine learning and PDE solvers (see [1][2][3][4][5][6][7][8][9][10]). Among others, simultaneous sparse approximation has been utilized in signal vector processing and multi-task learning (see [11][12][13][14]).…”

Section: Introductionmentioning

confidence: 99%

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Approximation Properties of the Vector Weak Rescaled Pure Greedy Algorithm

Guo

et al. 2023

Mathematics

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We first study the error performances of the Vector Weak Rescaled Pure Greedy Algorithm for simultaneous approximation with respect to a dictionary D in a Hilbert space. We show that the convergence rate of the Vector Weak Rescaled Pure Greedy Algorithm on A1(D) and the closure of the convex hull of the dictionary D is optimal. The Vector Weak Rescaled Pure Greedy Algorithm has some advantages. It has a weaker convergence condition and a better convergence rate than the Vector Weak Pure Greedy Algorithm and is simpler than the Vector Weak Orthogonal Greedy Algorithm. Then, we design a Vector Weak Rescaled Pure Greedy Algorithm in a uniformly smooth Banach space setting. We obtain the convergence properties and error bound of the Vector Weak Rescaled Pure Greedy Algorithm in this case. The results show that the convergence rate of the VWRPGA on A1(D) is sharp. Similarly, the Vector Weak Rescaled Pure Greedy Algorithm is simpler than the Vector Weak Chebyshev Greedy Algorithm and the Vector Weak Relaxed Greedy Algorithm.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Approximation Properties of the Vector Weak Rescaled Pure Greedy Algorithm

Guo

et al. 2023

Mathematics

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Reproducing property of bounded linear operators and kernel regularized least square regressions

Sheng

2024

Int. J. Wavelets Multiresolut Inf. Process.

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We consider bounded linear operators from the view of functional reproducing property. For some bounded linear operators associated with orthogonal polynomials we define an inner product space associated with a kernel constructed with orthogonal polynomials, show that it is a functional reproducing kernel Hilbert space (FRKHS) associated with these bounded linear operators and give decay rate for the best FRKHS approximation with a [Formula: see text]-functional associated with the FRKHS. On this basis, we provide a learning rate for kernel regularized regression whose hypothesis space is the defined FRKHS. As applications, we define some concrete FRKHSs associated with polynomial operators such as the Bernstein–Durrmeyer operators, the de la Vallée Poussin operators on both the unit sphere [Formula: see text] and the unit ball [Formula: see text]. We show that these polynomial operators have reproducing property with respect to the corresponding concrete FRKHSs and show the learning rate for the kernel regularized regression. In short, we provide a way of constructing FRKHS operators with Fourier multipliers and show a learning framework from the view of operator approximation.

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Sparse Signal Recovery via Rescaled Matching Pursuit

Li,

2024

Axioms

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We propose the Rescaled Matching Pursuit (RMP) algorithm to recover sparse signals in high-dimensional Euclidean spaces. The RMP algorithm has less computational complexity than other greedy-type algorithms, such as Orthogonal Matching Pursuit (OMP). We show that if the restricted isometry property is satisfied, then the upper bound of the error between the original signal and its approximation can be derived. Furthermore, we prove that the RMP algorithm can find the correct support of sparse signals from random measurements with a high probability. Our numerical experiments also verify this conclusion and show that RMP is stable with the noise. So, the RMP algorithm is a suitable method for recovering sparse signals.

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Optimality of the rescaled pure greedy learning algorithms

Cited by 3 publications

References 33 publications

Approximation Properties of the Vector Weak Rescaled Pure Greedy Algorithm

Approximation Properties of the Vector Weak Rescaled Pure Greedy Algorithm

Reproducing property of bounded linear operators and kernel regularized least square regressions

Sparse Signal Recovery via Rescaled Matching Pursuit

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