2210.12497
Deep Linear Networks for Matrix Completion – An Infinite Depth Limit
Nadav Cohen, Govind Menon, Zsolt Veraszto
correcthigh confidence
- Category
- Not specified
- Journal tier
- Specialist/Solid
- Processed
- Sep 28, 2025, 12:56 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper proves the SVD-volume formulas for the DLN metric by diagonalizing A_N in SVD coordinates and multiplying reciprocal eigenvalues, then converting to SVD variables via the SVD Jacobian; the candidate derives the same formulas by pulling back the metric to SVD coordinates, computing block determinants for (δΣ, Ω_U, Ω_V) directions, and calibrating with N=1. Both yield Theorem 1.1 (finite N and N=∞) exactly. Minor stylistic and normalization differences aside, the results coincide.
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} specialist/solid
\textbf{Justification:}
The core statements (metric diagonalization, volume forms for finite and infinite depth) are correct and proven with clean spectral computations. The results are relevant for understanding implicit regularization via intrinsic volume. Minor clarifications would improve the exposition (square-root vs. reciprocal eigenvalue products, explicit normalization choices, and handling of coincident singular values).