The Swiss roll and S-curve manifolds of Gashler, Ventura and Martinez (2008), with their intrinsic (two-dimensional) coordinates, to test dimensionality reduction methods.
Value
A tibble with n rows and the columns x, y, z (the
coordinates in three dimensions) and u, v (the intrinsic
coordinates). The points are ordered by u.
Details
For the Swiss roll, \(t_i = 8i/n + 2\) for \(0 \le i < n\), and $$x_i = t_i \sin(t_i), \quad y_i \sim U(-6, 6), \quad z_i = t_i \cos(t_i).$$ Its intrinsic coordinates are the arc length along the roll, \(u_i = (\sinh^{-1}(t_i) + t_i \sqrt{t_i^2 + 1}) / 2\), and \(v_i = y_i\).
For the S-curve, \(t_i = (2.2i - 0.1)\pi / n\), and $$x_i = t_i, \quad y_i = \sin(t_i), \quad z_i \sim U(0, 2).$$ Its intrinsic coordinates are the arc length along the curve, \(u_i = \int_0^{t_i} \sqrt{\cos^2(w) + 1} \, dw\), and \(v_i = z_i\). (The paper gives \(v_i = y_i\), a misprint since \(y_i\) is a function of \(u_i\).)
References
Gashler, M., Ventura, D. and Martinez, T. (2008). Iterative non-linear dimensionality reduction with manifold sculpting. In Advances in Neural Information Processing Systems 20, 513–520.
Examples
set.seed(1)
sim_swiss_roll(5)
#> # A tibble: 5 × 5
#> x y z u v
#> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 1.82 -2.81 -0.832 2.96 -2.81
#> 2 -1.59 -1.53 -3.23 7.72 -1.53
#> 3 -4.59 0.874 2.44 14.9 0.874
#> 4 3.36 4.90 5.91 24.7 4.90
#> 5 7.18 -3.58 -4.36 36.9 -3.58
sim_s_curve(5)
#> # A tibble: 5 × 5
#> x y z u v
#> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 -0.0628 -0.0628 1.80 -0.0888 1.80
#> 2 1.32 0.969 1.89 1.66 1.89
#> 3 2.70 0.426 1.32 3.21 1.32
#> 4 4.08 -0.809 1.26 5.07 1.26
#> 5 5.47 -0.729 0.124 6.54 0.124