Differentiating densities on smooth manifolds

Adam A. Śliwiak, Qiqi Wang

correctmedium confidence

Category: math.DS
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper derives gi = ∂s_i log ρ = − tr(Q^T ∂_{ξ_i}∇ξx R^{-1}) / ||∂_{ξ_i}x|| and its Einstein form directly from ρ|det R|=1, the matrix identity ∂ log det = tr(A^{-1}∂A), and the differentiated QR relation; the candidate solution reproduces exactly this chain of arguments, adds a clean remark to bypass the absolute value via C = (∇ξx)^T∇ξx, and states the same discrete-time recursion as the paper.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The derivation is correct and well motivated, connecting geometric measure on manifolds with a computable density-gradient formula and a recursion suitable for trajectories. The results are validated numerically. Minor clarifications about QR sign conventions, smoothness assumptions, and points where the formula may degenerate would improve rigor and readability.