  We study approximation of probability measures supported on $n$-dimensional
manifolds embedded in $\mathbb{R}^m$ by injective flows -- neural networks
composed of invertible flows and injective layers. We show that in general,
injective flows between $\mathbb{R}^n$ and $\mathbb{R}^m$ universally
approximate measures supported on images of extendable embeddings, which are a
subset of standard embeddings: when the embedding dimension m is small,
topological obstructions may preclude certain manifolds as admissible targets.
When the embedding dimension is sufficiently large, $m \ge 3n+1$, we use an
argument from algebraic topology known as the clean trick to prove that the
topological obstructions vanish and injective flows universally approximate any
differentiable embedding. Along the way we show that the studied injective
flows admit efficient projections on the range, and that their optimality can
be established "in reverse," resolving a conjecture made in Brehmer and Cranmer
2020.
