Searcharxiv⌕ Search

arXiv subjects

Seok Hyeong Lee

Publications and source records attributed to Seok Hyeong Lee.

10 recordsLinked to original sources
arXiv2026

Decoupling Dynamical Richness from Representation Learning: Towards Practical Measurement

Dynamic feature transformation (the rich regime) does not always align with predictive performance (better representation), yet accuracy is often used as a proxy for richness, limiting analysis of their relationship. We propose a computationally efficient, performance-independent metric of richness grounded in the low-rank bias of rich dynamics, which recovers neural collapse as a special case. The metric is empirically more stable than existing alternatives and captures known lazy-torich transitions (e.g., grokking) without relying on accuracy. We further use it to examine how training factors (e.g., learning rate) relate to richness, confirming recognized assumptions and highlighting new observations (e.g., batch normalization promotes rich dynamics). An eigendecomposition-based visualization is also introduced to support interpretability, together providing a diagnostic tool for studying the relationship between training factors, dynamics, and representations.

stat.ML↗
arXiv2025

Cotype zeta functions enumerating subalgebras of $R$-algebras

We introduce and study subalgebra cotype zeta functions, multivariate zeta functions enumerating fixed-index subalgebras of $R$-algebras of a given cotype. This generalizes and unifies previous works on subalgebra zeta functions and cotype zeta functions of $R$-algebras. We prove the local functional equations for the generic Euler factors of these zeta functions, and give an explicit formula for the subalgebra cotype zeta function of a general $\mathbb{Z}$-Lie algebra $L$ of rank 3. We also give an asymptotic formula for the number of subalgebras $Λ$ of $L$ of index at most $X$ for which $L/Λ$ has rank at most, answering a question of Chinta, Kaplan, and Koplewitz. In particular, we show that unlike $\mathbb{Z}^3$, $\mathbb{Z}$-Lie algebras of rank 3 with additional multiplication structure exhibit different distribution of cocyclic subalgebras.

math.RA↗
arXiv2025

Position: Solve Layerwise Linear Models First to Understand Neural Dynamical Phenomena (Neural Collapse, Emergence, Lazy/Rich Regime, and Grokking)

In physics, complex systems are often simplified into minimal, solvable models that retain only the core principles. In machine learning, layerwise linear models (e.g., linear neural networks) act as simplified representations of neural network dynamics. These models follow the dynamical feedback principle, which describes how layers mutually govern and amplify each other's evolution. This principle extends beyond the simplified models, successfully explaining a wide range of dynamical phenomena in deep neural networks, including neural collapse, emergence, lazy and rich regimes, and grokking. In this position paper, we call for the use of layerwise linear models retaining the core principles of neural dynamical phenomena to accelerate the science of deep learning.

stat.ML↗
arXiv2025

Zeta functions of quadratic lattices of a hyperbolic plane

In this paper, we study the Dirichlet series that enumerates proper equivalence classes of full-rank sublattices of a given quadratic lattice in a hyperbolic plane -- that is, a nondegenerate isotropic quadratic space of dimension $2$. We derive explicit formulas for the associated zeta functions and obtain a combinatorial way to compute them. Their analytic properties lead to the intriguing consequence that a large proportion of proper classes are one-lattice classes.

math.NT↗
arXiv2024

An exactly solvable model for emergence and scaling laws in the multitask sparse parity problem

Deep learning models can exhibit what appears to be a sudden ability to solve a new problem as training time, training data, or model size increases, a phenomenon known as emergence. In this paper, we present a framework where each new ability (a skill) is represented as a basis function. We solve a simple multi-linear model in this skill-basis, finding analytic expressions for the emergence of new skills, as well as for scaling laws of the loss with training time, data size, model size, and optimal compute. We compare our detailed calculations to direct simulations of a two-layer neural network trained on multitask sparse parity, where the tasks in the dataset are distributed according to a power-law. Our simple model captures, using a single fit parameter, the sigmoidal emergence of multiple new skills as training time, data size or model size increases in the neural network.

cs.LG↗
arXiv2024

Zeta functions enumerating subforms of quadratic forms

In this paper, we introduce and study the Dirichlet series enumerating (proper) equivalence classes of full rank subforms/sublattices of a given quadratic form/lattice, focusing on the positive definite binary case. We obtain formulas linking this Dirichlet series with Dirichlet series counting ideal classes of the imaginary quadratic field associated with the quadratic form. Utilizing the result, we provide explicit formulas of the Dirichlet series for several lattices, including square lattice and hexagonal lattice. Moreover, we investigate some analytic properties of this Dirichlet series.

math.NT↗
arXiv2024

Universality lifting from a general base field

Given a totally real number field $F$, we show that there are only finitely many totally real extensions of $K$ of a fixed degree that admit a universal quadratic form defined over $F$. We further obtain several explicit classification results in the case of relative quadratic extensions.

math.NT↗
arXiv2023

Lifting problem for universal quadratic forms over totally real cubic number fields

Lifting problem for universal quadratic forms asks for totally real number fields $K$ that admit a positive definite quadratic form with coefficients in $\mathbb{Z}$ that is universal over the ring of integers of $K$. In this paper, we show that $K=\mathbb{Q}(ζ_7+ζ_7^{-1})$ is the only such totally real cubic field. Moreover, we show that there is no such biquadratic field.

math.NT↗
arXiv2012

Mnev-Sturmfels universality for schemes

We prove a scheme-theoretic version of Mnev-Sturmfels Universality, suitable to be used in the proof of Murphy's Law in Algebraic Geometry. Somewhat more precisely, we show that any singularity type of finite type over Z appears on some incidence scheme of points and lines, subject to some particular further constraints.

math.AG↗