Boris Hanin


About Me

I am an Assistant Professor at Princeton ORFE working on deep learning, probability, and spectral asymptotics. Prior to Princeton, I was an Assistant Professor in Mathematics at Texas A&M, an NSF Postdoc at MIT Math, and a PhD student in Math at Northwestern, where I was supervised by Steve Zelditch.

Funding: I am grateful to be supported by an NSF CAREER grant DMS-2143754 and NSF grants DMS-1855684, DMS-2133806. I am also a consultant for an ONR MURI on Foundations of Deep Learning. See my CV for more information.

Email: bhanin ‘at’



I am fortunate to supervise Pierfrancesco Beneventano, Samy Jelassi, and Kaiqi Jiang.

Papers ArXiv

Deep Learning and Random Matrix Theory

  1. Bayesian Interpolation with Deep Linear Networks, with A. Zlokapa ArXiv
  2. Maximal Initial Learning Rates in Deep ReLU Networks, with G. Iyer and D. Rolnick ArXiv
  3. Deep Architecture Connectivity Matters for Its Convergence: A Fine-Grained Analysis with W. Chen, W. Huang, X. Gong, Z. Wang, NeurIPS 2022 ArXiv
  4. Random Fully Connected Neural Networks as Perturbatively Solvable Hierarchies (2022) ArXiv
  5. Ridgeless Interpolation with Shallow ReLU Networks in 1D is Nearest Neighbor Curvature Extrapolation and Provably Generalizes on Lipschitz Functions (2021) ArXiv
  6. Random Neural Networks in the Infinite Width Limit as Gaussian Processes, Annals of Applied Probability (2023) ArXiv
  7. Non-asymptotic Results for Singular Values of Gaussian Matrix Products, with G. Paouris. GAFA (2021) ArXiv
  8. Deep ReLU Networks Preserve Expected Length, with R. Jeong and D. Rolnick, ICLR 2022 ArXiv
  9. Neural Network Approximation, with R. DeVore and G. Petrova, Acta Numerica (2020) ArXiv
  10. How Data Augmentation affects Optimization for Linear Regression, with Y. Sun NeurIPS 2021 ArXiv
  11. Products of Many Large Random Matrices and Gradients in Deep Neural Networks, with M. Nica. Communications in Mathematical Physics (2020) ArXiv
  12. Finite Depth and Width Corrections to the Neural Tangent Kernel, with M. Nica, Splotlight at ICLR 2020 ArXiv
  13. Deep ReLU Networks Have Surprisingly Few Activation Patterns, with D. Rolnick, NeurIPS 2019 ArXiv
  14. Nonlinear Approximation and (Deep) ReLU Networks, with I. Daubechies, R. DeVore, S. Foucart, and G. Petrova. Constructive Approximation (Special Issue on Deep Networks in Approximation Theory) (2019) ArXiv
  15. Complexity of Linear Regions in Deep Networks, with D. Rolnick, ICML 2019 ArXiv
  16. How to Start Training: The Effect of Initialization and Architecture, with D. Rolnick. NIPS 2018 ArXiv
  17. Which Neural Net Architectures Give Rise to Vanishing and Exploding Gradients? NIPS 2018 ArXiv
  18. Approximating Continuous Functions by ReLU Nets of Minimal Width, with M. Sellke (2017) ArXiv
  19. Universal Function Approximation by Deep Neural Nets with Bounded Width and ReLU Activations. Mathematics 2019, 7(10), 992 (Special Issue on Computational Mathematics, Algorithms, and Data Processing) ArXiv

Spectral Theory

  1. Scaling Asymptotics of Spectral Wigner Functions, with S. Zelditch. Journal of Physics A (Special Edition on Claritons and the Asymptotics of Ideas: the Physics of Michael Berry) (2022) ArXiv
  2. Interface Asymptotics of Wigner-Weyl Distributions for the Harmonic Oscillator, with S. Zelditch. Journal d’Analyse (2022) ArXiv
  3. Interface Asymptotics of Eigenspace Wigner distributions for the Harmonic Oscillator, with S. Zelditch. Communications in PDE (2020) ArXiv
  4. Level Spacings and Nodal Sets at Infinity for Radial Perturbations of the Harmonic Oscillator, with T. Beck. International Math Research Notices, 2021. ArXiv
  5. Local Universality for Zeros and Critical Points of Monochromatic Random Waves, with Y. Canzani. Communication in Mathematical Physics, 2020. ArXiv
  6. Nodal Sets of Functions with Finite Vanishing Order, with T. Beck and S. Becker-Khan. Calculus of Variations and PDE (2018) ArXiv
  7. Scaling of Harmonic Oscillator Eigenfunctions and Their Nodal Sets Around the Caustic, with S. Zelditch and P. Zhou. Communications in Mathematical Physics. Vol. 350, no. 3, pp. 1147–1183, 2017. ArXiv
  8. C^∞ Scaling Asymptotics for the Spectral Function of the Laplacian, with Y. Canzani. The Journal of Geometric Analysis (2018) ArXiv
  9. Scaling Limit for the Kernel of the Spectral Projector and Remainder Estimates in the Pointwise Weyl Law, with Y. Canzani. Analysis and PDE, Vol. 8 (2015), No. 7, pp. 1707-1731. ArXiv
  10. High Frequency Eigenfunction Immersions and Supremum Norms of Random Waves, with Y. Canzani. Electronic Research Announcements. MS 22, no. 0, January 2015, pp. 76 - 86. ArXiv
  11. Nodal Sets of Random Eigenfunctions for the Isotropic Harmonic Oscillator, with S. Zelditch and P. Zhou. International Mathematics Research Notices, Vol. 2015, No. 13, pp. 4813 - 4839. ArXiv

Zeros and Critical Points of Random Polynomials

  1. The Lemniscate Tree of a Random Polynomial, with M. Epstein and E. Lundberg. Annales Institute Henri Poincare (B), 2018. ArXiv
  2. Pairing of Zeros and Critical Points for Random Polynomials. Annales de l’Institut Henri Poincare (B) Probabilites et Statistiques. Volume 53, Number 3 (2017), 1498-1511. ArXiv
  3. Pairing of Zeros and Critical Points for Random Meromorphic Functions on Riemann Surfaces</b>. Mathematics Research Letters, Vol. 22 (2015), No. 1, pp. 111-140. ArXiv
  4. Correlations and Pairing Between Zeros and Critical Points of Gaussian Random Polynomials. International Math Research Notices (2015), Vol. (2), pp. 381-421. ArXiv


  1. Contributed research to Principles of Deep Learning Theory, written by D. Roberts and S. Yaida, Cambridge University Press (2021) ArXiv
  2. An Intriguing Property of the Center of Mass for Points on Quadradtic Curves and Surfaces, with L. Hanin and R. Fisher. Mathematics Maganize, v. 80, No. 5, pp. 353-362, 2007.