The Geometry of the Deep Linear Network

The paper’s Theorem 10 gives vol(OW) = c_d^{N−1} sqrt(van(Σ^2)/van(Σ^{2/N})) (see the displayed formula and its product form under Theorem 10) . The candidate solution re-derives exactly this expression by pulling back the Frobenius metric to O(d)^{N−1}, diagonalizing a tridiagonal Toeplitz form in each (i,j)-block via the discrete-sine basis, and evaluating the determinant using a Chebyshev/roots-of-unity identity. This is the same structural argument used in the paper’s Section 8 (block tridiagonal metric and Chebyshev diagonalization) , together with the SVD-based parametrization of the balanced manifold M . A consistency check at N=2 yields vol(OW) = c_d ∏_{i<j} √(σ_i+σ_j), which both approaches satisfy. The mention of a denominator van(Σ^{2N}) in the prompt appears to be a typo not present in the provided PDF (which has Σ^{2/N}); with the correct exponent and the square root spanning the entire fraction (as seen from the product expression), the paper and model agree.