Regression | |||||||
Case | Case 1: a regularised step function | Case 2: a high dimensional function | |||||
Purpose | Find $p$ such that the constant $\mathbb{E}_\omega[\frac{|\hat f(\omega)|^2}{(2\pi)^{d}p^2(\omega)}]$ does not become large |
Study the ability of Alg. 1 to find a high dimensional dimensional function | |||||
Target $f(x)$ | $\text{Si}\left(\frac{x}{a}\right)e^{-\frac{x^2}{2}}$ $a = 10^{-3}$ |
$\text{Si}\left(\frac{x_1}{a}\right)e^{-\frac{|x|^2}{2}}$ $a = 10^{-1}$ |
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$d$ | $1$ | $5$ | |||||
$K$ | $2^i, \, i = 1, 2, ..., 11$ | $2^i, \, i = 1, 2, ..., 10$ | |||||
Experiment | Exp. 1 | Exp. 2 | Exp. 3 | Exp. 4 | Exp. 5 | Exp. 1 | Exp. 4 |
Method | Alg. 1 | Alg. 2 | RFF$^{1}$ | SGM | Alg. 1 tabular |
Alg. 1 | SGM |
$N$ | $10^4$ | $10^4$ | $10^4$ | $10^4$ | $10^4$ | $10^4$ | $10^4$ |
$\tilde{N}$ | $10^4$ | $10^4$ | $10^4$ | $10^4$ | $10^4$ | $10^4$ | $10^4$ |
$γ$ | $3d-2$ | $3d-2$ | $3d-2$ | $3d-2$ | |||
$\lambda$ | $0.1$ | $0.1$ | $0.1$ | $0$ | $0.1$ | $0.1$ | 0 |
$M$ | $10^3$ | $5000$ | N/A | $10^7$ | $10^4$ | $2.5\times 10^3$ | $10^7$ |
$\bar{M}$ | $10$ | $10$ | $10$ | $10$ | $10$ | $10$ | $10$ |
$\delta$ | $2.4^2/d$ | $0.1$ | $2.4^2/d$ | $\frac{2.4^2}{10d}$ | |||
$\Delta t$ | $\mathtt{1.5e-4}$ | $\mathtt{3.0e-4}$ | |||||
$t_0$ | $M/10$ | ||||||
$ω_{\text{max}}$ | $\infty$ | ||||||
$m$ | $10$ | $50$ | $100$ | $25$ | |||
Regression | Classification | ||||||
Case | Case 3: anisotropic Gaussian function |
Case 4: The MNIST data set |
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Purpose | Computational complexity comparison |
Study the ability of Alg. 1 to work with non synthetic data | |||||
Target $f(x)$ |
$e^{-(32 x_1)^2/2}e^{-(32^{-1} x_2)^2/2}$ | ||||||
$d$ | $2$ | $784$ | |||||
$K$ | $256$ | $2^i, \, i = 1, 2, ..., 13$ | |||||
Experiment | Exp. 1 | Exp. 2 | Exp. 4 | Exp. 1 | Exp. 3 | ||
Method | Alg. 1 | Alg. 2 | SGM | Alg. 1 | RFF$^{2}$ | ||
$N$ | $10^4$ | $10^4$ | $3\times 10^7$ | $6\times 10^4$ | $6 \times 10^4$ | ||
$\tilde{N}$ | $10^4$ | $10^4$ | $10^4$ | $10^4$ | $10^4$ | ||
$γ$ | $3d-2$ | $3d-2$ | $3d-2$ | ||||
$\lambda$ | $0.1$ | $0.1$ | $0$ | $0.1$ | $0.1$ | ||
$M$ | $10^4$ | $10^4$ | $3\times 10^7$ | $10^2$ | |||
$\bar{M}$ | $1$ | $1$ | $1$ | $1$ | $1$ | ||
$\delta$ | $0.5$ | $0.1$ | $0.1$ | ||||
$\Delta t$ | $\mathtt{1.5e-3}$ | ||||||
$t_0$ | $M/10$ | ||||||
$ω_{\text{max}}$ | $\infty$ | ||||||
$m$ | $100$ | $100$ | $M+1$ | ||||