Figure 9
From: Cost function for low-dimensional manifold topology assessment
Illustrative demonstration of the effect of changing the cost function’s hyper-parameters, r and b. (a–c) Costs corresponding to toy functions from Fig. 3 when setting r and b to non-unity values. (d) The original 3D swiss roll dataset with the cost reported for the original 3D parameterization using \(r = 1\) and \(b = 1\). (e) 2D swiss roll dataset projections with the resulting costs using \(r = 1\) and \(b = 1\). (f) Comparison of the \(\hat{\mathcal {D}}(\sigma )\) curves between generated 2D swiss roll dataset projections. The \(\hat{\mathcal {D}}(\sigma )\) curve for the original 3D parameters is shown with the red dashed line for reference. (g) Costs for the 3D swiss roll data parameters and its various 2D projections with b set fixed to unity, and with changing r. (h) Costs for the 3D swiss roll data parameters and its various 2D projections with r set fixed to unity, and with changing b. With the circular outlines in (g,h), we mark the lowest costs for any of the 2D projections which consistently happens for the SE manifold, irrespective of the values selected for r and b.
