Spatiotemporal convolutional network for time-series prediction and causal inference

The paper claims that, by delay-embedding, there exist smooth encoder/decoder maps F and F^{-1} satisfying the STI equations (its Eq. (2)) for windows [X_{t-w},…,X_t] and delay vectors Y_t, and even that the decoder recovers the original high-dimensional window, invoking Takens/Sauer generically . This justification omits key conditions (notably invertibility of f to reconstruct past states in the window and genericity of the measurement y), and conflates approximate reconstruction (hatted X) with exact recovery; hence the theoretical argument is incomplete. The model’s solution correctly leverages the delay map to define an encoder on the attractor and uses bump-function extensions to satisfy the identities on the finitely many observed samples, but it also implicitly assumes invertibility (via f^{-i} in the window map) and overstates smoothness of the inverse on fractal attractors. Thus, both sides require additional assumptions or clarifications to be fully correct.