Absolute continuity of two-dimensional polynomial random vectors

What the math gives to ML

The paper gives a quantitative anti-concentration principle for two-dimensional polynomial maps of independent random inputs: if the highest-degree components of the two outputs are not proportional, the joint law cannot place excessive probability on sets of very small area. Its transferable asset is the connection between algebraic non-degeneracy of top-order coefficient tensors and a lower bound on the expected determinant of the two-output Jacobian Gram matrix. This suggests a regularizer for low-dimensional continuous output heads that discourages local functional dependence and output-collapse onto curves. The idea is most relevant for VAE decoders, diffusion output heads, and learned two-dimensional coordinate maps.