Fast Differentiable SVD on GPU via Polar Decomposition

Uliana Parkina, Askar Tsyganov, Sergei Kudriashov, Sergey Samsonov, Maxim Rakhuba

We present a fully GPU-oriented SVD pipeline based on polar decomposition, motivated by iterative methods that rely solely on matrix multiplications, such as the Newton-Schulz iteration. We show that this approach enables up to a $2\times$ speedup compared to standard implementations. Furthermore, we derive a numerically stable backward pass for the polar decomposition and leverage it to obtain a fully differentiable SVD. Our methods are released as open-source implementations in both PyTorch and JAX: https://github.com/fallnlove/cans_svd.