Figure 3
From: Cost function for low-dimensional manifold topology assessment
Cost function behavior on toy functions. (a) The response of the cost function to increasing feature size. The dependent variable \(\phi\) is generated from the bivariate Gaussian normal distribution with varying standard deviation, s. (b) The \(\hat{\mathcal {D}}(\sigma )\) curves corresponding to functions with increasing feature size. The curves are plotted in gray-scale denoting an increasing value of s from the darkest curve (\(s = 0.05\)) to the lightest (\(s = 0.6\)). (c) The response of the cost function to multiple feature sizes. The dependent variable \(\phi\) is generated from a superposition of sine functions with varying frequencies. (d) The \(\hat{\mathcal {D}}(\sigma )\) curves corresponding to functions with increasing number of feature sizes. The curves are plotted in gray-scale, such that the darkest curve corresponds to five different feature sizes and the lightest curve corresponds to a single feature size. (e) The response of the cost function to increasing non-uniqueness. The dependent variable \(\phi\) is generated as a linear function of x with an additional set of overlapping observations for which \(\phi = 0\). (f) The \(\hat{\mathcal {D}}(\sigma )\) curves corresponding to functions with increasing non-uniqueness. The curves are plotted in gray-scale denoting an increasing value of the non-uniqueness depth, d, from the lightest curve (\(d=0\)) to the darkest (\(d=90\)). In all examples, the dependent variables, \(\phi\), have been normalized to a [0, 1] range.
