The 8-dimensional data are projected onto a 3-dimensional subspace The 3-dimensional subspace is spanned by their 1st, 3rd, and 8th coordinates. If the data would lie on a linear subspace in ℝ8, then the projected data must show a linear pattern. However, the actual projection of our data does not show a linear pattern but rather two cones next to each other. A nonlinear approach like Laplacian Eigenmaps could be useful to recover nonlinear structure of the data manifold.
Ehler et al. BMC Proceedings 2011 5(Suppl 2):S3 doi:10.1186/1753-6561-5-S2-S3