A single thread runs through my work: finding the structure that makes hard
representation problems tractable — from solving structured polynomial
systems to exploring how the retina codes the world.
Journal of the London Mathematical Society · 2007
Solvability of symmetric word equations in positive definite letters
with Scott Armstrong
Let S(X, B) be a symmetric (‘palindromic’) word in two letters X and B. A theorem due to Hillar and Johnson states that for each pair of positive definite matrices B and P, there is a positive definite solution X to the word equation S(X, B)=P. They also conjectured that these solutions are finite and unique. In this paper, we resolve a modified version of this conjecture by showing that the Brouwer degree of such an equation is equal to 1 (in the case of real matrices). It follows that, generically, the number of solutions is odd (and thus finite) in the real case. Our approach allows us to address the more subtle question of uniqueness by exhibiting equations with multiple real solutions, as well as providing a second proof of the result of Hillar and Johnson in the real case.
arXiv
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Proceedings of the American Mathematical Society · 2008
An elementary and constructive solution to Hilbert's 17th Problem for matrices
with Jiawang Nie
We give a short and elementary proof of a theorem of Procesi, Schacher and (independently) Gondard, Ribenboim that generalizes a famous result of Artin.
arXiv
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Bulletin of the London Mathematical Society · 2012
Equations solvable by radicals in a uniquely divisible group
with Lionel Levine amd Darren Rhea
We study equations in groups G with unique mth roots for each positive integer m. We obtain the first known infinite families of word equations not solvable by radicals, and conjecture a complete classification.
arXiv
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Advances in Mathematics · 2012
Finite Gröbner bases in infinite dimensional polynomial rings and applications
with Seth Sullivant
We prove the Independent Set Conjecture in algebraic statistics.
arXiv
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Journal of the Association for Computing Machinery · 2013
Most tensor problems are NP-hard
with Lek-Heng Lim
The natural generalizations of
matrix problems to tensors are, almost without exception, computationally intractable.
arXiv
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Advanced Studies in Pure Mathematics · 2018
Equivariant Gröbner bases
with Robert Krone and Anton Leykin
Algorithmic computation in polynomial rings is a classical topic in mathematics. However, little attention has been given to the case of rings with an infinite number of variables until recently when theoretical efforts have made possible the development of effective routines. Ability to compute relies on finite generation up to symmetry for ideals invariant under a large group or monoid action, such as the permutations of the natural numbers. We summarize the current state of theory and applications for equivariant Gröbner bases, develop several algorithms to compute them, showcase our software implementation, and close with several open problems and computational challenges.
arXiv
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Github
arXiv · 2018
Maximum entropy distributions on graphs
with Andre Wibisono
Inspired by applications to theories of coding and communication in networks of nervous tissue, we
study maximum entropy distributions on weighted graphs with a given expected degree sequence.
arXiv
Journal of Mathematical Neuroscience · 2018
Robust exponential memory in Hopfield networks
with Ngoc Tran
Shows how a recurrent network can store an exponential number
of memories robustly, which also solves the hidden clique problem in computer science.
arXiv
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Nature Scientific Reports · 2018
Active state organization of spontaneous behavioral patterns
with Giorgio Onnis, Darren Rhea, and Laurence Tecott
Mouse genetics are 99% predicted by behavior alone.
PDF
IEEE Transactions Signal Processing · 2019
On the uniqueness and stability of dictionaries for sparse representation of noisy signals
with Charles Garfinkle
We provide very general conditions guaranteeing when dictionaries yielding the sparsest
encodings are unique and stable with respect to measurement or modeling error.
PDF
ICLR · 2023
Bispectral neural networks
with Sophia Sanborn, Christian Shewmake, and Bruno Olshausen
We present a neural network architecture, Bispectral Neural Networks (BNNs),
for learning representations that are invariant to the actions of compact commutative groups.
arXiv
Github
COLT · 2024
Harmonics of learning: Universal Fourier features emerge in invariant networks
with Giovanni Luca Marchetti, Danica Kragic, and Sophia Sanborn
We formally prove that, under certain conditions, if a neural network is invariant to a finite group then its weights recover the Fourier transform on that group.
arXiv
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Github
Entropy · 2025
Detecting signatures of criticality using divergence rate
with Tenzin Chan and De Wen Soh
Guided by the classical theory of rate–distortion (RD) from information theory, we propose a measure for detecting and characterizing critical phenomena from data.
PDF
Preprint · 2026
Implicit bias and invariance: How Hopfield networks efficiently learn graph orbits
with Michael Murray, Tenzin Chan, and Kedar Karhadkar
Many learning problems involve symmetries, and while invariance can be built into neural architectures, it can also emerge implicitly when training on group-structured data. We study this phenomenon in classical Hopfield-Amari networks and illustrate how they can infer the isomorphism class of a graph from a small, random sample. Our results reveal that: (i) graph isomorphism classes can be represented within a three-dimensional invariant subspace, (ii) using gradient descent to minimize energy flow (MEF) has an implicit bias toward norm-efficient solutions, which underpins a polynomial sample complexity bound for learning isomorphism classes, and (iii) across multiple learning rules, parameters converge toward the invariant subspace as sample sizes grow.
arXiv