Boris Hanin

About Me

I am an Associate Professor at Princeton ORFE (and Associated Faculty at Princeton PACM) 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.

I also work part time at Foundry, an incredible AI/computing startup that seeks to orchestrate the world’s compute, where I lead the Foundry Institute.

Funding: I am grateful to be supported by a 2024 Sloan Fellowship in Mathematics, NSF CAREER grant DMS-2143754, and NSF grant DMS-2133806.

Please see my CV for more information.

Email: bhanin ‘at’ princeton.edu

Research Group

• I am fortunate to supervise several excellent PhD students: Boris Shigida (joint with Matias Cattaneo), Alex Negron, Jake Freeman, Tianze Jiang. I am also fortunate that Lorenzo Noci will be visiting from ETH Zurich in Spring 2025.

• I am excited to be working with several phenomenal postdocs: Gage DeZoort, Mufan Li, Noam Razin, and Mike Winer

First Placement of Former Members

• Samy Jelassi (Postdoc at Harvard CMSA)
• Pierfrancesco Beneventano (Postdoc at MIT CBMM)
• Kaiqi Jiang (Research Scientist, Huawei)

Professional Service

I am an Associate Editor of

• Pure and Applied Analysis
• Mathematics of Operations Research
• Advances in Theoretical and Mathematical Physics

All these journals are always looking for high quality submissions on theoretical machine learning.

Short Courses

1. Statistical Physics for Neural Networks. University of Oxford. March 2025.
2. Mathematics Aspects of Deep Learning Theory. University of Luxembourg. June 2024. Lecture 1 video, Lecture 2 video, Lecture 3 video. Lecture Notes pdf
3. Neural Networks and Gaussian Processes. Tor Vergata (Rome). Jan 2023. notes; Lecture 1 video, Lecture 2 video, Lecture 3 video
4. Neural Networks at Large and Infinite Width (joint with Yasaman Bahri). Les Houches Summer School on Statistical Physics of Machine Learning (France). July 2022. Lecture 1 video, Lecture 2 video, Lecture Notes arXiv J. Stat. Mech.

Papers ArXiv

Deep Learning

Preprints

1. Don’t be lazy: CompleteP enables compute-efficient deep transformers, with N. Dey, B. Zhang, L. Noci, M. Li, B. Bordelon, S. Bergsma, C. Pehlevan, J. Hestness. (2025) ArXiv
2. Deep Nets as Hamiltonians, with M. Winer. (2025) ArXiv
3. Global Universality of Singular Values in Products of Many Large Random Matrices, with T. Jiang. (2025) ArXiv
4. Optimizing Model Selection for Compound AI Systems, with L. Chen, J. Q. Davis, P. Bailis, M. Zaharia, J. Zou, I. Stoica (2025) ArXiv
5. BARE: Combining Base and Instruction-Tuned Language Models for Better Synthetic Data Generation, with A. Zhu, P. Asawa, J. Q. Davis, L. Chen, I. Stoica, J. Gonzalez, M. Zaharia (2025) ArXiv
6. Networks of Networks: Complexity Class Principles Applied to Compound AI Systems Design, with J. Q. Davis, L. Chen, P. Bailis, I. Stoica, M. Zaharia (2024) ArXiv
7. Bayesian Inference with Deep Weakly Nonlinear Networks, with A. Zlokapa (2024) ArXiv
8. Depth Dependence of μP Learning Rates in ReLU MLPs, with S. Jelassi, Z. Ji, S. Reddi, S. Bhojanapalli, and S. Kumar (2023) ArXiv
9. Ridgeless Interpolation with Shallow ReLU Networks in 1D is Nearest Neighbor Curvature Extrapolation and Provably Generalizes on Lipschitz Functions (2021) ArXiv
10. Approximating Continuous Functions by ReLU Nets of Minimal Width, with M. Sellke (2017) ArXiv

Journal Articles

1. Quantitative CLTs in Deep Neural Networks, with S. Favaro, D. Marinucci, I. Nourdin, and G. Pecatti. In Press: Probability Theory and Related Fields, Volume 191, pages 933–977, (2025) ArXiv
2. Random Fully Connected Neural Networks as Perturbatively Solvable Hierarchies. Journal of Machine Learning Research, 25(267):1−58, 2024 ArXiv
3. Principles for Initialization and Architecture Selection in Graph Neural Networks with ReLU Activations, with G. DeZoort. SIAM Journal on Mathematics of Data Science, Vol. 7, No. 1 (2025) ArXiv
4. Bayesian Interpolation with Deep Linear Networks, with A. Zlokapa, Proceedings of the National Acacdemy of Science, Volume 120, No. 23 (2023) ArXiv
5. Random Neural Networks in the Infinite Width Limit as Gaussian Processes, Annals of Applied Probability, 2023, Vol. 33, No. 6A, 4798 – 4819 ArXiv
6. Non-asymptotic Results for Singular Values of Gaussian Matrix Products, with G. Paouris. GAFA (2021) ArXiv
7. Products of Many Large Random Matrices and Gradients in Deep Neural Networks, with M. Nica. Communications in Mathematical Physics (2020) ArXiv
8. Neural Network Approximation, with R. DeVore and G. Petrova, Acta Numerica (2020) ArXiv
9. 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
10. 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

Conference Articles

1. Unintentional Unalignment: Likelihood Displacement in Direct Preference Optimization, with N. Razin, S. Malladi, A. Bhaskar, D. Chen, S. Arora. ICLR 2025 ArXiv
2. Are More LLM Calls All You Need? Towards Scaling Laws of Compound Inference Systems, with L. Chen, J. Q. Davis, P. Bailis, M. Zaharia, I. Stoica, and J. Zou (2024). NeurIPS 2024 ArXiv
3. Depthwise Hyperparameter Transfer in Residual Networks: Dynamics and Scaling Limit, with B. Bordelon, L. Noci, M. Li, and C. Pehlevan. ICLR 2024. ArXiv
4. Principled Architecture-Aware Scaling of Hyperparameters, with W. Chen, J. Wu, and Z. Wang. ICLR 2024. Arxiv
5. Maximal Initial Learning Rates in Deep ReLU Networks, with G. Iyer and D. Rolnick, ICML 2023 ArXiv
6. Deep Architecture Connectivity Matters for Its Convergence: A Fine-Grained Analysis with W. Chen, W. Huang, X. Gong, Z. Wang, NeurIPS 2022 ArXiv
7. Finite Depth and Width Corrections to the Neural Tangent Kernel, with M. Nica, Splotlight at ICLR 2020 ArXiv
8. Deep ReLU Networks Preserve Expected Length, with R. Jeong and D. Rolnick, ICLR 2022 ArXiv
9. How Data Augmentation affects Optimization for Linear Regression, with Y. Sun NeurIPS 2021 ArXiv
10. Deep ReLU Networks Have Surprisingly Few Activation Patterns, with D. Rolnick, NeurIPS 2019 ArXiv
11. Complexity of Linear Regions in Deep Networks, with D. Rolnick, ICML 2019 ArXiv
12. Which Neural Net Architectures Give Rise to Vanishing and Exploding Gradients? NIPS 2018 ArXiv
13. How to Start Training: The Effect of Initialization and Architecture, with D. Rolnick. NIPS 2018 ArXiv

Spectral Theory

Journal Articles

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

Journal Articles

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

Other

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.