arXiv subjects

Dongsung Huh

Publications and source records attributed to Dongsung Huh.

10 recordsLinked to original sources

arXiv2026

Open Problem: Separating Geometric and Algorithmic Compression via Cayley-Table Completion

Modern statistical learning theory and deep learning characterize generalization primarily in terms of continuous capacity control (e.g., norm-based regularization, margin maximization, low-rank bias). While highly successful in continuous domains, deep learning consistently fails to extrapolate exact algorithmic or discrete algebraic rules, reflecting a missing inductive bias toward algorithmic complexity minimization. We propose the Cayley-table completion as the canonical testbed for this missing bias, serving as the discrete algebraic counterpart to matrix completion. Just as matrix factorization combined with weight decay yields an implicit geometric bias toward low linear rank, recent results demonstrate that operator-valued tensor factorizations paired with a flatness prior yield an implicit algorithmic bias toward exact discrete associativity. We pose the open problem of establishing formal exact recovery bounds for Cayley-table completion, and challenge the community to generalize continuous flatness priors to autonomously discover broader discrete algorithmic axioms without combinatorial search.

cs.LG↗

arXiv2026

Exact Symmetry as Algebra: A Machine-Verified Tensor Calculus that Enforces Physical Selection Rules

Symmetry is central to the physical sciences, yet machine learning usually captures it only approximately, leaving a residual per-step equivariance error $\varepsilon$ that compounds with depth $M$ as $M\varepsilon$, whereas exact equivariance holds at unbounded depth; we demonstrate this divergence at fourteen orders of magnitude. We show that a symmetry can be made exact by construction, as the multiplication rule of a tensor algebra. In the resulting $\starG$ algebra, defined by any finite group $G$, the group-Fourier transform block-diagonalizes every tensor into irreducible-representation blocks, making equivariance intrinsic; requiring equivariance conversely \emph{forces} this suitably normalized transform, so the algebra is determined by $G$ rather than chosen. The standard matrix toolbox, including a Frobenius-optimal low-rank factorization, transfers blockwise, machine-checked in Lean~4 under an explicit axiom budget, and extends unchanged to band-limited compact groups and, under periodic boundary conditions, to all 230 crystallographic space groups and the compact little-group fibers of Euclidean and Poincar\'e symmetry. This exactness is an applied capability: on inorganic-crystal elastic tensors the algebra enforces point-group selection rules exactly on the output of \emph{any} predictor, driving a trained graph network's forbidden-channel leakage from $10^{-2}$ to machine zero, eliminating mechanically unstable predictions, and recovering viable materials that an unconstrained screen discards; on molecular data, with no quantum-mechanical input, it exposes octahedral selection-rule signatures consistent with the Wigner--Eckart theorem. Matched networks lead on pooled molecular accuracy, which we report plainly: the contribution is a complementary algebraic calculus, structural and diagnostic, exact at any depth.

cs.LG↗

arXiv2025

A Differentiable Measure of Algebraic Complexity: Provably Exact Discovery of Group Structures

Discovering discrete algebraic rules from data is a fundamental challenge in machine learning. We formalize this problem through Cayley-table completion -- an algebraic counterpart to classical matrix completion -- where the degree of associativity violation replaces linear rank as the intrinsic measure of complexity. We provide a rigorous landscape analysis of HyperCube, an operator-valued tensor factorization, on the fully observed target table $\delta$, proving that its global infimum $H_{\inf}(\delta) := \inf_{\Theta \in F_\delta} H(\Theta)$ implicitly defines an exact differentiable measure for this complexity. We show that HyperCube's native objective $H(\Theta)$ decomposes into two components: geometric alignment (collinearity) and an inverse $\ell_2$ penalty. We establish that these continuous variational pressures induce core discrete properties: collinearity enforces associativity (Collinearity--Associativity Equivalence), and the inverse $\ell_2$ penalty reduces to an exact inverse rank penalty within the collinear manifold, driving the parameters toward full-rank unitarity. Consequently, we derive an absolute lower bound $H(\Theta) \ge H_{\inf}(\delta) \ge 3 \, |\delta|$, where $|\delta|$ is the target table size. We prove this absolute floor is attained if and only if the target is isotopic to a group, and characterize the global minimizer as the regular representation of the underlying group (up to unitary gauge), resolving the central open conjecture of Huh (2025). This work serves as an existence proof that certain discrete algebraic structures can be exactly characterized by differentiable measures, enabling gradient-based discovery without the need for combinatorial search. All theoretical results are mechanically verified in Lean 4 and confirmed via small-scale experiments.

cs.LG↗

arXiv2025

Interpretable epistemic uncertainty decomposition in sequential generative models via polynomial chaos surrogates

Sequential generative models conditioned on uncertain rewards are central to AI-driven scientific discovery, yet the epistemic uncertainty they inherit from imperfect reward estimates remains unquantified. We propagate this uncertainty through generative flow networks (GFlowNets) by fitting polynomial chaos expansions (PCEs) to small ensembles of trained models. The PCE coefficients yield analytical Sobol sensitivity indices, providing the first interpretable decomposition of which reward components drive which generative decisions, a capability unavailable from deep ensembles, Bayesian neural networks, or Monte Carlo dropout. Convergence guarantees are established theoretically and four of five are formally verified in the Lean 4 proof assistant. Across three real-world tasks the framework reveals actionable structure invisible to ensembles alone. On the Doyle-Dreher Buchwald-Hartwig dataset catalyst selection is robust ($D_{\mathrm{catalyst}}\approx 71$) while additive selection is fragile ($D_{\mathrm{additive}}\approx 179$, $2.5\times$ higher). In fragment-based molecular design the linker position is the most sensitive ($D_{\mathrm{linker}}\approx 28$) while decoration positions are the most robust ($D\approx 14$-$18$), reversing the conventional scaffold-robust / decoration-fragile assumption. On the Sachs protein signalling network, MAPK-cascade edges and PKA/PKC hub edges separate into distinct sensitivity regimes, providing a targeted map for perturbation experiments. Calibration coverage at the 95% level reaches 0.97-1.00 across the dominant steps, and the surrogate evaluates 10{,}000 policy samples in milliseconds - $10^{3}$-$10^{4}\times$ faster than exhaustive retraining.

cs.LG↗

arXiv2024

Discovering Abstract Symbolic Relations by Learning Unitary Group Representations

We investigate a principled approach for symbolic operation completion (SOC), a minimal task for studying symbolic reasoning. While conceptually similar to matrix completion, SOC poses a unique challenge in modeling abstract relationships between discrete symbols. We demonstrate that SOC can be efficiently solved by a minimal model - a bilinear map - with a novel factorized architecture. Inspired by group representation theory, this architecture leverages matrix embeddings of symbols, modeling each symbol as an operator that dynamically influences others. Our model achieves perfect test accuracy on SOC with comparable or superior sample efficiency to Transformer baselines across most datasets, while boasting significantly faster learning speeds (100-1000$\times$). Crucially, the model exhibits an implicit bias towards learning general group structures, precisely discovering the unitary representations of underlying groups. This remarkable property not only confers interpretability but also significant implications for automatic symmetry discovery in geometric deep learning. Overall, our work establishes group theory as a powerful guiding principle for discovering abstract algebraic structures in deep learning, and showcases matrix representations as a compelling alternative to traditional vector embeddings for modeling symbolic relationships.

cs.LG↗

arXiv2023

ISAAC Newton: Input-based Approximate Curvature for Newton's Method

We present ISAAC (Input-baSed ApproximAte Curvature), a novel method that conditions the gradient using selected second-order information and has an asymptotically vanishing computational overhead, assuming a batch size smaller than the number of neurons. We show that it is possible to compute a good conditioner based on only the input to a respective layer without a substantial computational overhead. The proposed method allows effective training even in small-batch stochastic regimes, which makes it competitive to first-order as well as second-order methods.

cs.LG↗

arXiv2023

The Missing Invariance Principle Found -- the Reciprocal Twin of Invariant Risk Minimization

Machine learning models often generalize poorly to out-of-distribution (OOD) data as a result of relying on features that are spuriously correlated with the label during training. Recently, the technique of Invariant Risk Minimization (IRM) was proposed to learn predictors that only use invariant features by conserving the feature-conditioned label expectation $\mathbb{E}_e[y|f(x)]$ across environments. However, more recent studies have demonstrated that IRM-v1, a practical version of IRM, can fail in various settings. Here, we identify a fundamental flaw of IRM formulation that causes the failure. We then introduce a complementary notion of invariance, MRI, based on conserving the label-conditioned feature expectation $\mathbb{E}_e[f(x)|y]$, which is free of this flaw. Further, we introduce a simplified, practical version of the MRI formulation called MRI-v1. We prove that for general linear problems, MRI-v1 guarantees invariant predictors given sufficient number of environments. We also empirically demonstrate that MRI-v1 strongly out-performs IRM-v1 and consistently achieves near-optimal OOD generalization in image-based nonlinear problems.

cs.LG↗

arXiv2017

Generalized Phase Representation of Integrate-and-Fire Models

The quadratic integrate-and-fire (QIF) model captures the normal form bifurcation dynamics of Type-I neurons found in cortex. Remarkably, this model is known to have a dual equivalent representation in terms of phase, called the $θ$-model, which has advantages for numerical simulations and analysis over the QIF model. Here, I investigate the nature of the dual representation and derive the general phase model expression for all integrate-and-fire models. Moreover, I show the condition for which the phase models exhibit discontinuous onset firing rate, the hallmark of Type-II spiking neurons.

math.DS↗

arXiv2017

Gradient Descent for Spiking Neural Networks

Much of studies on neural computation are based on network models of static neurons that produce analog output, despite the fact that information processing in the brain is predominantly carried out by dynamic neurons that produce discrete pulses called spikes. Research in spike-based computation has been impeded by the lack of efficient supervised learning algorithm for spiking networks. Here, we present a gradient descent method for optimizing spiking network models by introducing a differentiable formulation of spiking networks and deriving the exact gradient calculation. For demonstration, we trained recurrent spiking networks on two dynamic tasks: one that requires optimizing fast (~millisecond) spike-based interactions for efficient encoding of information, and a delayed memory XOR task over extended duration (~second). The results show that our method indeed optimizes the spiking network dynamics on the time scale of individual spikes as well as behavioral time scales. In conclusion, our result offers a general purpose supervised learning algorithm for spiking neural networks, thus advancing further investigations on spike-based computation.

q-bio.NC↗

arXiv2015

The Vector Space of Convex Curves: How to Mix Shapes

We present a novel, log-radius profile representation for convex curves and define a new operation for combining the shape features of curves. Unlike the standard, angle profile-based methods, this operation accurately combines the shape features in a visually intuitive manner. This method have implications in shape analysis as well as in investigating how the brain perceives and generates curved shapes and motions.

cs.GR↗