# A Spectral Phase Admissibility Certificate for Complex Linear Maps

arXiv: https://arxiv.org/abs/2609.02911 · HTML: https://mlsanity.com/paper/2609.02911/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 25 references that are in the map.
- First posted on arXiv 13 Jul 2026.

## Builds on (most cited first, 19 of 19)

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- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Exact solutions to the nonlinear dynamics of learning in deep linear neural networks](https://mlsanity.com/paper/1312.6120/) (2013)
- [Hopfield Networks is All You Need](https://mlsanity.com/paper/2008.02217/) (2020)
- [Unitary Evolution Recurrent Neural Networks](https://mlsanity.com/paper/1511.06464/) (2015)
- [Parseval Networks: Improving Robustness to Adversarial Examples](https://mlsanity.com/paper/1704.08847/) (2017)
- [Cheap Orthogonal Constraints in Neural Networks: A Simple Parametrization of the Orthogonal and Unitary Group](https://mlsanity.com/paper/1901.08428/) (2019)
- [Resurrecting the sigmoid in deep learning through dynamical isometry: theory and practice](https://mlsanity.com/paper/1711.04735/) (2017)
- [AntisymmetricRNN: A Dynamical System View on Recurrent Neural Networks](https://mlsanity.com/paper/1902.09689/) (2019)
- [Full-Capacity Unitary Recurrent Neural Networks](https://mlsanity.com/paper/1611.00035/) (2016)
- [Lipschitz Recurrent Neural Networks](https://mlsanity.com/paper/2006.12070/) (2020)
- [A Survey of Complex-Valued Neural Networks](https://mlsanity.com/paper/2101.12249/) (2021)
- [Learning Nonsymmetric Determinantal Point Processes](https://mlsanity.com/paper/1905.12962/) (2019)

## Abstract (arXiv)

The paper imports the Kontsevich Segal Witten criterion from quantum gravity into machine learning to evaluate complex linear maps Standard techniques analyze magnitude or positive definiteness whereas this method exclusively limits the collective phase of a spectrum The researchers create three distinct differentiable certificates comprising a determinant sector a subset product envelope and the full criterion The subset envelope prevents all exterior power eigenvalues from touching the negative real axis This constraint precisely matches the accept or reject choices of an exponential minor enumeration while reducing processing expenses drastically The team provides a differentiable enforcement application via a Schur parameterization The document also identifies crucial boundaries regarding where this system works The constraint cannot balance deep linear propagation since restricting the phase budget damages eigenvector conditioning Furthermore the technique remains completely blind to magnitude based targets like normalizing flow likelihoods Thus researchers must restrict this tool specifically to models that process the argument of a spectral product