This sound like projecting data into the linear space spanned by {x_i, x_i*x_j} where x_i are the features variables, and then applying standard regularization methods to remove noise and low value coefficients.

Anisotropy and the cone ideas may explain why PCA underperforms, but it does not uniquely justify this particular quadratic decoder. The geometric story is not doing explanatory work beyond “data is nonlinear,” and the real substance is simply that second-order reconstruction empirically helps.