Selected work

Representative Papers

msPCA: An R Package for Sparse PCA with Multiple Components

Jul 06, 2026

This study addresses the challenge that traditional sparse principal component analysis (PCA) struggles to simultaneously achieve high explained variance, sparsity, and non-redundancy among multiple principal components in high-dimensional data. To overcome this limitation, the authors propose msPCA, a novel method based on an alternating maximization algorithm that enforces two forms of non-redundancy constraints—either orthogonality of loadings or zero correlation among principal components—while preserving high variance explanation and sparsity. The accompanying open-source R package efficiently scales to datasets with thousands of features, demonstrating superior performance over existing approaches by striking an effective balance between controllable sparsity and computational efficiency.

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Trajectory Geometry of Transformer Representations Across Layers

Jun 08, 2026

This work investigates the dynamic evolution of Transformer representations across layers, moving beyond static analyses of encoded content. It introduces a probe-free geometric framework that treats forward propagation as a discrete trajectory on a high-dimensional representation manifold and directly computes five geometric metrics—trajectory length, curvature, semantic convergence index, inter-layer cosine similarity, and representational stability—in the original embedding space. The study systematically uncovers the geometric signatures of semantic convergence, reasoning complexity, and ambiguity resolution: semantically related prompts exhibit pronounced convergence in middle-to-late layers, reasoning tasks yield trajectories with higher curvature, and ambiguous tokens induce trajectory bifurcations. Furthermore, it identifies a universal three-stage representational evolution pattern across architectures, substantially enhancing the interpretability of model mechanisms.

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A formal proof of the Ramanujan--Nagell theorem in Lean 4

Apr 10, 2026

This work presents a complete formalization of the Ramanujan–Nagell theorem in Lean 4, leveraging the Mathlib library to establish that the Diophantine equation $x^2 + 7 = 2^n$ admits exactly five integer solutions: $(n, x) = (3, \pm1), (4, \pm3), (5, \pm5), (7, \pm11), (15, \pm181)$. The proof is carried out within an interactive theorem prover by developing foundational structures from algebraic number theory, including the ring of integers, class group, and unit group of the quadratic field $\mathbb{Q}(\sqrt{-7})$. This effort not only provides the first machine-verified proof of the theorem but also significantly extends the reusable infrastructure for algebraic number theory in formalized mathematics.

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Distribution-Free Equilibrium in Search Contests

Mar 21, 2026

This study investigates strategic behavior in sequential search contests where participants incur a cost per draw, draws are memoryless and unlimited in number, and the highest score wins. Using game-theoretic and sequential search models, the paper establishes the existence of a unique symmetric equilibrium whose acceptance threshold depends solely on the number of contestants, the search cost, and the prize—remarkably independent of the underlying score distribution. The key contribution lies in demonstrating this distribution-free property of equilibrium strategies and showing that total expected expenditure exactly equals the prize value. The analysis extends these insights to settings with multiple prizes and hierarchical team competitions. Furthermore, equilibrium efficiency is shown to be governed by the hazard rate of the score distribution, and under finite horizons, the selective effect can dominate the discouragement effect when search costs are sufficiently low.

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Compact Lifted Relaxations for Low-Rank Optimization

Mar 05, 2026

This work addresses the challenge of lacking a general computable convex relaxation for quadratic matrix optimization problems with rank constraints by proposing a lifted semidefinite relaxation framework that does not rely on spectral structure assumptions. By analyzing block redundancies in moment matrices, the authors derive a compact equivalent formulation involving only two small-scale semidefinite constraints. They further design projection-based cutting planes that exploit rank inheritance under linear mappings to strengthen the low-rank constraint. This approach significantly enhances scalability and enables efficient solutions for problems such as matrix completion and reduced-rank regression, handling instances with dimensions up to \(n + m\).

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Recent publications

Latest Papers

msPCA: An R Package for Sparse PCA with Multiple Components

Jul 06, 2026

This study addresses the challenge that traditional sparse principal component analysis (PCA) struggles to simultaneously achieve high explained variance, sparsity, and non-redundancy among multiple principal components in high-dimensional data. To overcome this limitation, the authors propose msPCA, a novel method based on an alternating maximization algorithm that enforces two forms of non-redundancy constraints—either orthogonality of loadings or zero correlation among principal components—while preserving high variance explanation and sparsity. The accompanying open-source R package efficiently scales to datasets with thousands of features, demonstrating superior performance over existing approaches by striking an effective balance between controllable sparsity and computational efficiency.

0 citationsRead paper

Trajectory Geometry of Transformer Representations Across Layers

Jun 08, 2026

This work investigates the dynamic evolution of Transformer representations across layers, moving beyond static analyses of encoded content. It introduces a probe-free geometric framework that treats forward propagation as a discrete trajectory on a high-dimensional representation manifold and directly computes five geometric metrics—trajectory length, curvature, semantic convergence index, inter-layer cosine similarity, and representational stability—in the original embedding space. The study systematically uncovers the geometric signatures of semantic convergence, reasoning complexity, and ambiguity resolution: semantically related prompts exhibit pronounced convergence in middle-to-late layers, reasoning tasks yield trajectories with higher curvature, and ambiguous tokens induce trajectory bifurcations. Furthermore, it identifies a universal three-stage representational evolution pattern across architectures, substantially enhancing the interpretability of model mechanisms.

0 citationsRead paper

A formal proof of the Ramanujan--Nagell theorem in Lean 4

Apr 10, 2026

This work presents a complete formalization of the Ramanujan–Nagell theorem in Lean 4, leveraging the Mathlib library to establish that the Diophantine equation $x^2 + 7 = 2^n$ admits exactly five integer solutions: $(n, x) = (3, \pm1), (4, \pm3), (5, \pm5), (7, \pm11), (15, \pm181)$. The proof is carried out within an interactive theorem prover by developing foundational structures from algebraic number theory, including the ring of integers, class group, and unit group of the quadratic field $\mathbb{Q}(\sqrt{-7})$. This effort not only provides the first machine-verified proof of the theorem but also significantly extends the reusable infrastructure for algebraic number theory in formalized mathematics.

0 citationsRead paper

Distribution-Free Equilibrium in Search Contests

Mar 21, 2026

This study investigates strategic behavior in sequential search contests where participants incur a cost per draw, draws are memoryless and unlimited in number, and the highest score wins. Using game-theoretic and sequential search models, the paper establishes the existence of a unique symmetric equilibrium whose acceptance threshold depends solely on the number of contestants, the search cost, and the prize—remarkably independent of the underlying score distribution. The key contribution lies in demonstrating this distribution-free property of equilibrium strategies and showing that total expected expenditure exactly equals the prize value. The analysis extends these insights to settings with multiple prizes and hierarchical team competitions. Furthermore, equilibrium efficiency is shown to be governed by the hazard rate of the score distribution, and under finite horizons, the selective effect can dominate the discouragement effect when search costs are sufficiently low.

0 citationsRead paper

Compact Lifted Relaxations for Low-Rank Optimization

Mar 05, 2026

This work addresses the challenge of lacking a general computable convex relaxation for quadratic matrix optimization problems with rank constraints by proposing a lifted semidefinite relaxation framework that does not rely on spectral structure assumptions. By analyzing block redundancies in moment matrices, the authors derive a compact equivalent formulation involving only two small-scale semidefinite constraints. They further design projection-based cutting planes that exploit rank inheritance under linear mappings to strengthen the low-rank constraint. This approach significantly enhances scalability and enables efficient solutions for problems such as matrix completion and reduced-rank regression, handling instances with dimensions up to \(n + m\).

0 citationsRead paper