Reproducing Kernel (Quasi-)Banach Spaces with the $\ell^p$ Coefficient Norms

Tianfang Wang; Guohui Song; Haizhang Zhang

cs.LGNov 5, 2026

Reproducing Kernel (Quasi-)Banach Spaces with the $\ell^p$ Coefficient Norms

Authors: Tianfang Wang, Guohui Song, Haizhang Zhang

Organizations: School of Mathematics, Sun Yat-sen University, Zhuhai, P. R. China. · Department of Mathematics and Statistics, Old Dominion University, Norfolk, VA 23529, USA.

Abstract

This paper advances the theory of reproducing kernel Banach spaces (RKBSs) by investigating the general $\ell^p$ regularization framework, with a particular focus on the non-convex regime $0<p<1$ . While our preceding work established the foundation for RKBSs with the $\ell^1$ norm, the pursuit of stronger sparsity---closer to the $\ell^0$ pseudo-norm---naturally motivates this exploration into $\ell^p$ quasi-norms. For $0<p<1$ , we construct reproducing kernel quasi-Banach spaces (RKQBSs) with $\ell^p$ quasi-norms and establish existence and sparsity results for interpolation and regularized learning under appropriate assumptions. We characterize the linear and relaxed representer theorems in terms of a generalized Lebesgue constant and derive bounds under kernel decay, sample spacing, and stability assumptions. For $p>1$ , we construct RKBSs using a fixed countable dictionary, establish minimizer existence, and identify the optimal relaxed representer factor through an off-sample coefficient operator. These results reveal fundamental differences between the regimes $0<p<1$ , $p=1$ , and $p>1$ . Numerical experiments compare the accuracy and sparsity of solutions for these three cases.

Figures & tables

	Gaussian noise		Uniform noise		Pepper sauce noise
	Error	Sparsity (Max)	Error	Sparsity (Max)	Error	Sparsity (Max)
$\ell^{\frac{1}{2}}$ -Regularized Model	7.38E-4	13 (16)	8.18E-4	13.08 (15)	9.00E-4	13.1 (16)
$\ell^{\frac{2}{3}}$ -Regularized Model	6.90E-4	13.66 (16)	7.77E-4	13.66 (16)	8.63E-4	13.52 (16)
$\ell^{1}$ -Regularized Model	6.32E-4	23.04 (33)	7.19E-4	23.14(40)	8.20E-4	21.92(28)
$\ell^{\frac{3}{2}}$ -Regularized Model	6.21E-4	199.98(200)	7.11E-4	200 (200)	8.23E-4	200 (200)
$\ell^{2}$ -Regularized Model	6.52E-4	200 (200)	7.43E-4	200 (200)	8.64E-4	200 (200)

Table 6.1 : Comparison of Least-Squares Regularization in RKHS and with

\ell^{p}

-Regularized Models Using the Gaussian Kernel for the GMO function.

	Gaussian noise		Uniform noise		Pepper sauce noise
	Error	Sparsity (Max)	Error	Sparsity (Max)	Error	Sparsity (Max)
$\ell^{\frac{1}{2}}$ -Regularized Model	6.88E-04	11.3 (14)	7.52E-04	11.5 (14)	8.27E-04	11.38 (14)
$\ell^{\frac{2}{3}}$ -Regularized Model	5.73E-04	12.32 (15)	6.37E-04	12.04 (14)	7.12E-04	12.26 (15)
$\ell^{1}$ -Regularized Model	5.51E-04	20.42 (25)	6.20E-04	20.04 (24)	7.10E-04	20.18 (24)
$\ell^{\frac{3}{2}}$ -Regularized Model	5.86E-04	200 (200)	6.64E-04	200 (200)	7.72E-04	200 (200)
$\ell^{2}$ -Regularized Model	6.37E-04	200 (200)	7.20E-04	200 (200)	8.39E-04	200 (200)

Table 6.2 : Comparison of Least-Squares Regularization in RKHS and with

\ell^{p}

-Regularized Models Using the Gaussian Kernel for the HRG function.

	Gaussian noise		Uniform noise		Pepper sauce noise
	Error	Sparsity (Max)	Error	Sparsity (Max)	Error	Sparsity (Max)
$\ell^{\frac{1}{2}}$ -Regularized Model	6.71E-04	11.84 (14)	7.40E-04	12.32 (14)	8.58E-04	11.86 (14)
$\ell^{\frac{2}{3}}$ -Regularized Model	6.35E-04	12.36 (15)	7.30E-04	12.4 (16)	8.12E-04	12.44 (14)
$\ell^{1}$ -Regularized Model	6.08E-04	20.52 (25)	6.97E-04	20.4 (30)	8.06E-04	20.5 (24)
$\ell^{\frac{3}{2}}$ -Regularized Model	6.49E-04	199.98 (200)	7.46E-04	200 (200)	8.60E-04	200 (200)
$\ell^{2}$ -Regularized Model	6.84E-04	200 (200)	7.81E-04	200 (200)	9.05E-04	200 (200)

Table 6.3 : Comparison of Least-Squares Regularization in RKHS and with

\ell^{p}

-Regularized Models Using the Gaussian Kernel for the SPF function.

	Gaussian noise		Uniform noise		Pepper sauce noise
	Error	Sparsity (Max)	Error	Sparsity (Max)	Error	Sparsity (Max)
$\ell^{\frac{1}{2}}$ -Regularized Model	6.99E-04	12.02 (15)	7.33E-04	12 (14)	8.52E-04	12.16 (14)
$\ell^{\frac{2}{3}}$ -Regularized Model	6.66E-04	12.38 (15)	7.24E-04	12.44 (15)	8.50E-04	12.66 (15)
$\ell^{1}$ -Regularized Model	6.38E-04	21.46 (29)	7.18E-04	20.96 (26)	8.29E-04	20.9 (27)
$\ell^{\frac{3}{2}}$ -Regularized Model	6.59E-04	200 (200)	7.47E-04	199.98 (200)	8.67E-04	199.98 (200)
$\ell^{2}$ -Regularized Model	6.92E-04	200 (200)	7.80E-04	200 (200)	9.09E-04	200 (200)

Table 6.4 : Comparison of Least-Squares Regularization in RKHS and with

\ell^{p}

-Regularized Models Using the Gaussian Kernel for the MTH function.

	Gaussian noise		Uniform noise		Pepper sauce noise
	Error	Sparsity (Max)	Error	Sparsity (Max)	Error	Sparsity (Max)
$\ell^{\frac{1}{2}}$ -Regularized Model	1.38E-03	18.9 (22)	1.49E-03	18.94 (23)	1.71E-03	18.94 (24)
$\ell^{\frac{2}{3}}$ -Regularized Model	1.32E-03	19.98 (23)	1.46E-03	19.92 (22)	1.66E-03	19.62 (23)
$\ell^{1}$ -Regularized Model	1.12E-03	62.36 (93)	1.25E-03	58.16 (81)	1.43E-03	59.36 (78)
$\ell^{\frac{3}{2}}$ -Regularized Model	9.92E-04	200 (200)	1.11E-03	200 (200)	1.26E-03	200 (200)
$\ell^{2}$ -Regularized Model	9.92E-04	200 (200)	1.10E-03	200 (200)	1.26E-03	200 (200)

Table 6.5 : Comparison of Least-Squares Regularization in RKHS and with

\ell^{p}

-Regularized Models Using the Laplacian Kernel for the GMO function.

	Gaussian noise		Uniform noise		Pepper sauce noise
	Error	Sparsity (Max)	Error	Sparsity (Max)	Error	Sparsity (Max)
$\ell^{\frac{1}{2}}$ -Regularized Model	5.33E-04	9.24 (11)	5.78E-04	9.64 (12)	6.61E-04	9.52 (12)
$\ell^{\frac{2}{3}}$ -Regularized Model	4.99E-04	10.24 (12)	5.57E-04	10.36 (14)	6.45E-04	10.24 (13)
$\ell^{1}$ -Regularized Model	4.75E-04	25.78 (46)	5.42E-04	25.6 (37)	6.56E-04	24.9 (38)
$\ell^{\frac{3}{2}}$ -Regularized Model	5.62E-04	200 (200)	6.40E-04	200 (200)	7.66E-04	200 (200)
$\ell^{2}$ -Regularized Model	6.74E-04	200 (200)	7.68E-04	200 (200)	9.05E-04	200 (200)

Table 6.6 : Comparison of Least-Squares Regularization in RKHS and with

\ell^{p}

-Regularized Models Using the Laplacian Kernel for the HRG function.

	Gaussian noise		Uniform noise		Pepper sauce noise
	Error	Sparsity (Max)	Error	Sparsity (Max)	Error	Sparsity (Max)
$\ell^{\frac{1}{2}}$ -Regularized Model	3.43E-04	4.02 (5)	4.32E-04	4.04 (5)	4.78E-04	4.04 (5)
$\ell^{\frac{2}{3}}$ -Regularized Model	3.43E-04	4.24 (6)	4.08E-04	4.14 (5)	4.82E-04	4.18 (6)
$\ell^{1}$ -Regularized Model	2.69E-04	11.42 (19)	3.05E-04	11.16 (20)	3.61E-04	11.02 (17)
$\ell^{\frac{3}{2}}$ -Regularized Model	3.30E-04	200 (200)	3.63E-04	200 (200)	4.33E-04	200 (200)
$\ell^{2}$ -Regularized Model	4.15E-04	200 (200)	4.67E-04	200 (200)	5.48E-04	200 (200)

Table 6.7 : Comparison of Least-Squares Regularization in RKHS and with

\ell^{p}

-Regularized Models Using the Laplacian Kernel for the SPF function.

	Gaussian noise		Uniform noise		Pepper sauce noise
	Error	Sparsity (Max)	Error	Sparsity (Max)	Error	Sparsity (Max)
$\ell^{\frac{1}{2}}$ -Regularized Model	1.09E-03	15.36 (18)	1.20E-03	15.54 (21)	1.37E-03	15.5 (19)
$\ell^{\frac{2}{3}}$ -Regularized Model	1.04E-03	16.16 (19)	1.14E-03	16.42 (21)	1.28E-03	16.32 (20)
$\ell^{1}$ -Regularized Model	8.69E-04	50.7 (85)	9.63E-04	48.36 (71)	1.11E-03	47.6 (75)
$\ell^{\frac{3}{2}}$ -Regularized Model	8.40E-04	200 (200)	9.15E-04	200 (200)	1.06E-03	200 (200)
$\ell^{2}$ -Regularized Model	9.15E-04	200 (200)	9.98E-04	200 (200)	1.15E-03	200 (200)

Table 6.8 : Comparison of Least-Squares Regularization in RKHS and with

\ell^{p}

-Regularized Models Using the Laplacian Kernel for the MTH function.

	Gaussian noise		Uniform noise		Pepper sauce noise
	Error	Sparsity (Max)	Error	Sparsity (Max)	Error	Sparsity (Max)
$\ell^{\frac{1}{2}}$ -Regularized Model	5.32E-04	13.04 (17)	5.88E-04	13.14 (16)	6.94E-04	13.24 (17)
$\ell^{\frac{2}{3}}$ -Regularized Model	5.15E-04	18.62 (25)	5.82E-04	18.94 (23)	6.76E-04	18.46 (23)
$\ell^{1}$ -Regularized Model	5.16E-04	129.48 (156)	5.84E-04	126.34 (150)	6.81E-04	125.78 (149)
$\ell^{\frac{3}{2}}$ -Regularized Model	5.57E-04	200 (200)	6.34E-04	200 (200)	7.37E-04	200 (200)
$\ell^{2}$ -Regularized Model	6.00E-04	200 (200)	6.87E-04	200 (200)	7.98E-04	200 (200)

Table 6.9 : Comparison of Least-Squares Regularization in RKHS and with

\ell^{p}

-Regularized Models Using the IMQ Kernel for the GMO function.

	Gaussian noise		Uniform noise		Pepper sauce noise
	Error	Sparsity (Max)	Error	Sparsity (Max)	Error	Sparsity (Max)
$\ell^{\frac{1}{2}}$ -Regularized Model	3.80E-04	9.22 (12)	4.19E-04	9.36 (11)	4.95E-04	9.54 (12)
$\ell^{\frac{2}{3}}$ -Regularized Model	3.90E-04	10.28 (12)	4.40E-04	10.46 (13)	5.21E-04	10.62 (13)
$\ell^{1}$ -Regularized Model	4.53E-04	66.54 (109)	5.02E-04	70.14 (118)	5.97E-04	75.22 (112)
$\ell^{\frac{3}{2}}$ -Regularized Model	5.34E-04	200 (200)	6.00E-04	200 (200)	7.00E-04	200 (200)
$\ell^{2}$ -Regularized Model	5.89E-04	200 (200)	6.68E-04	200 (200)	7.77E-04	200 (200)

Table 6.10 : Comparison of Least-Squares Regularization in RKHS and with

\ell^{p}

-Regularized Models Using the IMQ Kernel for the HRG function.

	Gaussian noise		Uniform noise		Pepper sauce noise
	Error	Sparsity (Max)	Error	Sparsity (Max)	Error	Sparsity (Max)
$\ell^{\frac{1}{2}}$ -Regularized Model	4.01E-04	6.34 (8)	4.19E-04	6.22 (8)	5.15E-04	6.16 (8)
$\ell^{\frac{2}{3}}$ -Regularized Model	3.83E-04	6.22 (8)	3.94E-04	6.26 (7)	4.65E-04	6.22 (8)
$\ell^{1}$ -Regularized Model	3.50E-04	14.02 (23)	3.85E-04	14.38 (22)	4.59E-04	14.92 (29)
$\ell^{\frac{3}{2}}$ -Regularized Model	4.27E-04	200 (200)	4.82E-04	200 (200)	5.71E-04	200 (200)
$\ell^{2}$ -Regularized Model	5.03E-04	200 (200)	5.62E-04	200 (200)	6.56E-04	200 (200)

Table 6.11 : Comparison of Least-Squares Regularization in RKHS and with

\ell^{p}

-Regularized Models Using the IMQ Kernel for the SPF function.

	Gaussian noise		Uniform noise		Pepper sauce noise
	Error	Sparsity (Max)	Error	Sparsity (Max)	Error	Sparsity (Max)
$\ell^{\frac{1}{2}}$ -Regularized Model	4.93E-04	11.04 (14)	5.58E-04	10.82 (13)	6.53E-04	10.7 (13)
$\ell^{\frac{2}{3}}$ -Regularized Model	4.86E-04	12.92 (15)	5.57E-04	12.46 (15)	6.47E-04	12.42 (14)
$\ell^{1}$ -Regularized Model	4.91E-04	104.22 (140)	5.57E-04	102.1 (131)	6.51E-04	101.7 (131)
$\ell^{\frac{3}{2}}$ -Regularized Model	5.49E-04	200 (200)	6.24E-04	200 (200)	7.25E-04	200 (200)
$\ell^{2}$ -Regularized Model	5.96E-04	200 (200)	6.83E-04	200 (200)	7.93E-04	200 (200)

Table 6.12 : Comparison of Least-Squares Regularization in RKHS and with

\ell^{p}

-Regularized Models Using the IMQ Kernel for the MTH function.

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In this paper, we establish a novel connection between the metric entropy growth and the embeddability of function spaces into reproducing kernel Hilbert/Banach spaces. Metric entropy characterizes the information complexity of function spaces and has implications for their approximability and learnability. Classical results show that embedding a function space into a reproducing kernel Hilbert space (RKHS) implies a bound on its metric entropy growth. Surprisingly, we prove a converse: a bound on the metric entropy growth of a function space allows its embedding to a

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