arXiv subjects

Taehyeong Kim

Publications and source records attributed to Taehyeong Kim.

At least 37 records · Page 2Linked to original sources

arXiv2025

Infinitely badly approximable affine forms

A pair $(A,\mathbf{b})$ of a real $m\times n$ matrix $A$ and $\mathbf{b}\in\mathbb{R}^m$ is said to be $\textit{infinitely badly approximable}$ if \[ \liminf_{\mathbf{q}\in\mathbb{Z}^n, \|\mathbf{q}\|\to\infty} \|\mathbf{q}\|^{\frac{n}{m}}\|A\mathbf{q}-\mathbf{b}\|_{\mathbb{Z}} =\infty, \] where $\|\cdot\|_\mathbb{Z}$ denotes the distance from the nearest integer vector. In this article, we introduce a novel concept of singularity for $(A,\mathbf{b})$ and characterize the infinitely badly approximable property by this singular property. As an application, we compute the Hausdorff dimension of the infinitely badly approximable set. We also discuss dynamical interpretations on the space of grids in $\mathbb{R}^{m+n}$.

math.NT↗

arXiv2025

Optimized Weight Initialization on the Stiefel Manifold for Deep ReLU Neural Networks

Stable and efficient training of ReLU networks with large depth is highly sensitive to weight initialization. Improper initialization can cause permanent neuron inactivation dying ReLU and exacerbate gradient instability as network depth increases. Methods such as He, Xavier, and orthogonal initialization preserve variance or promote approximate isometry. However, they do not necessarily regulate the pre-activation mean or control activation sparsity, and their effectiveness often diminishes in very deep architectures. This work introduces an orthogonal initialization specifically optimized for ReLU by solving an optimization problem on the Stiefel manifold, thereby preserving scale and calibrating the pre-activation statistics from the outset. A family of closed-form solutions and an efficient sampling scheme are derived. Theoretical analysis at initialization shows that prevention of the dying ReLU problem, slower decay of activation variance, and mitigation of gradient vanishing, which together stabilize signal and gradient flow in deep architectures. Empirically, across MNIST, Fashion-MNIST, multiple tabular datasets, few-shot settings, and ReLU-family activations, our method outperforms previous initializations and enables stable training in deep networks.

cs.LG↗

arXiv2025

Knowledge Synthesis of Photosynthesis Research Using a Large Language Model

The development of biological data analysis tools and large language models (LLMs) has opened up new possibilities for utilizing AI in plant science research, with the potential to contribute significantly to knowledge integration and research gap identification. Nonetheless, current LLMs struggle to handle complex biological data and theoretical models in photosynthesis research and often fail to provide accurate scientific contexts. Therefore, this study proposed a photosynthesis research assistant (PRAG) based on OpenAI's GPT-4o with retrieval-augmented generation (RAG) techniques and prompt optimization. Vector databases and an automated feedback loop were used in the prompt optimization process to enhance the accuracy and relevance of the responses to photosynthesis-related queries. PRAG showed an average improvement of 8.7% across five metrics related to scientific writing, with a 25.4% increase in source transparency. Additionally, its scientific depth and domain coverage were comparable to those of photosynthesis research papers. A knowledge graph was used to structure PRAG's responses with papers within and outside the database, which allowed PRAG to match key entities with 63% and 39.5% of the database and test papers, respectively. PRAG can be applied for photosynthesis research and broader plant science domains, paving the way for more in-depth data analysis and predictive capabilities.

cs.CL↗

arXiv2024

Hausdorff dimension of singular vectors in function fields

We compute the Hausdorff dimension of the set of singular vectors in function fields and bound the Hausdorff dimension of the set of $\varepsilon$-Dirichlet improvable vectors in this setting. This is a function field analogue of the results of Cheung and Chevallier [Duke Math. J. 165 (2016), 2273--2329].

math.NT↗

arXiv2024

On the rate of convergence of continued fraction statistics of random rationals

We show that the statistics of the continued fraction expansion of a randomly chosen rational in the unit interval, with a fixed large denominator $q$, approaches the Gauss-Kuzmin statistics with polynomial rate in $q$. This improves on previous results giving the convergence without rate. As an application of this effective rate of convergence, we show that the statistics of a randomly chosen rational in the unit interval, with a fixed large denominator $q$ and prime numerator, also approaches the Gauss-Kuzmin statistics. Our results are obtained as applications of improved non-escape of mass and equidistribution statements for the geodesic flow on the space $SL_2(\mathbb{R})/SL_2(\mathbb{Z})$.

math.DS↗

arXiv2024

Click-Gaussian: Interactive Segmentation to Any 3D Gaussians

Interactive segmentation of 3D Gaussians opens a great opportunity for real-time manipulation of 3D scenes thanks to the real-time rendering capability of 3D Gaussian Splatting. However, the current methods suffer from time-consuming post-processing to deal with noisy segmentation output. Also, they struggle to provide detailed segmentation, which is important for fine-grained manipulation of 3D scenes. In this study, we propose Click-Gaussian, which learns distinguishable feature fields of two-level granularity, facilitating segmentation without time-consuming post-processing. We delve into challenges stemming from inconsistently learned feature fields resulting from 2D segmentation obtained independently from a 3D scene. 3D segmentation accuracy deteriorates when 2D segmentation results across the views, primary cues for 3D segmentation, are in conflict. To overcome these issues, we propose Global Feature-guided Learning (GFL). GFL constructs the clusters of global feature candidates from noisy 2D segments across the views, which smooths out noises when training the features of 3D Gaussians. Our method runs in 10 ms per click, 15 to 130 times as fast as the previous methods, while also significantly improving segmentation accuracy. Our project page is available at https://seokhunchoi.github.io/Click-Gaussian

cs.CV↗

arXiv2024

On a lower bound of Hausdorff dimension of weighted singular vectors

Let $w=(w_1,\dots,w_d)$ be a $d$-tuple of positive real numbers such that $\sum_{i}w_i =1$ and $w_1\geq \cdots \geq w_d$. A $d$-dimensional vector $x=(x_1,\dots,x_d)\in\mathbb{R}^d$ is said to be $w$-singular if for every $ε>0$ there exists $T_0>1$ such that for all $T>T_0$ the system of inequalities \[ \max_{1\leq i\leq d}|qx_i - p_i|^{\frac{1}{w_i}} < \fracε{T} \quad\text{and}\quad 0<q<T \] have an integer solution $(\mathbf{p},q)=(p_1,\dots,p_d,q)\in \mathbb{Z}^d \times \mathbb{Z}$. We prove that the Hausdorff dimension of the set of $w$-singular vectors in $\mathbb{R}^d$ is bounded below by $d-\frac{1}{1+w_1}$. Our result partially extends the previous result of Liao et al. [Hausdorff dimension of weighted singular vectors in $\mathbb{R}^2$, J. Eur. Math. Soc. 22 (2020), 833-875].

math.NT↗

arXiv2024

Geometric mean for T-positive definite tensors and associated Riemannian geometry

In this paper, we generalize the geometric mean of two positive definite matrices to that of third-order tensors using the notion of T-product. Specifically, we define the geometric mean of two T-positive definite tensors and verify several properties that "mean" should satisfy including the idempotence and the commutative property, and so on. Moreover, it is shown that the geometric mean is a unique T-positive definite solution of an algebraic Riccati tensor equation and can be expressed as solutions of algebraic Riccati matrix equations. In addition, we investigate the Riemannian manifold associated with the geometric mean for T-positive definite tensors, considering it as a totally geodesic embedded submanifold of the Riemannian manifold associated with the case of matrices. It is particularly shown that the geometric mean of two T-positive definite tensors is the midpoint of a unique geodesic joining the tensors, and the manifold is a Cartan-Hadamard-Riemannian manifold.

math.NA↗

arXiv2024

On a Kurzweil type theorem via ubiquity

Kurzweil's theorem ('55) is concerned with zero-one laws for well approximable targets in inhomogeneous Diophantine approximation under the badly approximable assumption. In this article, we prove the divergent part of a Kurzweil type theorem via a suitable construction of ubiquitous systems when the badly approximable assumption is relaxed. Moreover, we also discuss some counterparts of Kurzweil's theorem.

math.NT↗

arXiv2023

Blending-NeRF: Text-Driven Localized Editing in Neural Radiance Fields

Text-driven localized editing of 3D objects is particularly difficult as locally mixing the original 3D object with the intended new object and style effects without distorting the object's form is not a straightforward process. To address this issue, we propose a novel NeRF-based model, Blending-NeRF, which consists of two NeRF networks: pretrained NeRF and editable NeRF. Additionally, we introduce new blending operations that allow Blending-NeRF to properly edit target regions which are localized by text. By using a pretrained vision-language aligned model, CLIP, we guide Blending-NeRF to add new objects with varying colors and densities, modify textures, and remove parts of the original object. Our extensive experiments demonstrate that Blending-NeRF produces naturally and locally edited 3D objects from various text prompts. Our project page is available at https://seokhunchoi.github.io/Blending-NeRF/

cs.CV↗

arXiv2023

A new formula for the determinant of $4 \times 4$ matrices

In this paper, we present a new formula for the determinant of a $4 \times 4$ matrix. We approach via the sparse optimization problem and derive the formula through the Least Absolute Shrinkage and Selection Operator (LASSO). Our formula has the potential to advance understanding of the algebraic structure of determinants, such as an upper bound of the tensor rank, various notions to measure complexity, and effective computational tools in exterior algebras. We also address several numerical experiments which compare our formula with built-in functions in a computer-algebra system.

math.AC↗

arXiv2023

Finding tensor decompositions with sparse optimization

In this paper, we suggest a new method for a given tensor to find CP decompositions using a less number of rank $1$ tensors. The main ingredient is the Least Absolute Shrinkage and Selection Operator (LASSO) by considering the decomposition problem as a sparse optimization problem. As applications, we design experiments to find some CP decompositions of the matrix multiplication and determinant tensors. In particular, we find a new formula for the $4 \times 4$ determinant tensor as a sum of $12$ rank $1$ tensors.

math.AC↗

arXiv2023

On Hausdorff dimension in inhomogeneous Diophantine approximation over global function fields

In this paper, we study inhomogeneous Diophantine approximation over the completion $K_v$ of a global function field $K$ (over a finite field) for a discrete valuation $v$, with affine algebra $R_v$. We obtain an effective upper bound for the Hausdorff dimension of the set \[ \mathbf{Bad}_A(ε)=\left\{\boldsymbolθ\in K_v^{\,m} : \liminf_{(\mathbf{p},\mathbf{q})\in R_v^{\,m} \times R_v^{\,n}, \|\mathbf{q}\|\to \infty} \|\mathbf{q}\|^n \|A\mathbf{q}-\boldsymbolθ-\mathbf{p}\|^m \geq ε\right\}, \] of $ε$-badly approximable targets $\boldsymbolθ\in K_v^{\,m}$ for a fixed matrix $A\in\mathscr{M}_{m,n}(K_v)$, using an effective version of entropy rigidity in homogeneous dynamics for an appropriate diagonal action on the space of $R_v$-grids. We further characterize matrices $A$ for which $\mathbf{Bad}_A(ε)$ has full Hausdorff dimension for some $ε>0$ by a Diophantine condition of singularity on average. Our methods also work for the approximation using weighted ultrametric distances.

math.NT↗

arXiv2023

A new formula of the determinant tensor with symmetries

In this paper, we present a new formula of the determinant tensor $det_n$ for $n \times n$ matrices. In \cite{kim2023newdet4}, Kim, Ju, and Kim found a new formula of $4 \times 4$ determinant tensor $det_4$ which is available when the base field is not of characteristic $2$. Considering some symmetries in that formula, we found a new formula so that \begin{equation*} \operatorname{Crank}(det_n) \leq \operatorname{rank}(det_n) \leq \frac{n!}{2^{\lfloor(n-2)/2 \rfloor}} \end{equation*} when the base field is not of characteristic $2$.

math.AC↗

arXiv2022

Dimension estimates for badly approximable affine forms

For given $ε>0$ and $b\in\mathbb{R}^m$, we say that a real $m\times n$ matrix $A$ is $ε$-badly approximable for the target $b$ if $$\liminf_{q\in\mathbb{Z}^n, \|q\|\to\infty} \|q\|^n \langle Aq-b \rangle^m \geq ε,$$ where $\langle \cdot \rangle$ denotes the distance from the nearest integral point. In this article, we obtain upper bounds for the Hausdorff dimensions of the set of $ε$-badly approximable matrices for fixed target $b$ and the set of $ε$-badly approximable targets for fixed matrix $A$. Moreover, we give an equivalent Diophantine condition of $A$ for which the set of $ε$-badly approximable targets for fixed $A$ has full Hausdorff dimension for some $ε>0$. The upper bounds are established by effectivizing entropy rigidity in homogeneous dynamics, which is of independent interest. For the $A$-fixed case, our method also works for the weighted setting where the supremum norms are replaced by certain weighted quasinorms.

math.DS↗

arXiv2022

Cross-Modal Alignment Learning of Vision-Language Conceptual Systems

Human infants learn the names of objects and develop their own conceptual systems without explicit supervision. In this study, we propose methods for learning aligned vision-language conceptual systems inspired by infants' word learning mechanisms. The proposed model learns the associations of visual objects and words online and gradually constructs cross-modal relational graph networks. Additionally, we also propose an aligned cross-modal representation learning method that learns semantic representations of visual objects and words in a self-supervised manner based on the cross-modal relational graph networks. It allows entities of different modalities with conceptually the same meaning to have similar semantic representation vectors. We quantitatively and qualitatively evaluate our method, including object-to-word mapping and zero-shot learning tasks, showing that the proposed model significantly outperforms the baselines and that each conceptual system is topologically aligned.

cs.CV↗

arXiv2022

Hausdorff measure of sets of Dirichlet non-improvable affine forms

For a decreasing real valued function $ψ$, a pair $(A,\mathbf{b})$ of a real $m\times n$ matrix $A$ and $\mathbf{b}\in\mathbb{R}^m$ is said to be $ψ$-Dirichlet improvable if the system $$\|A\mathbf{q}+\mathbf{b}-\mathbf{p}\|^m < ψ(T)\quad\text{and}\quad\|\mathbf{q}\|^n < T$$ has a solution $\mathbf{p}\in\mathbb{Z}^m$, $\mathbf{q}\in\mathbb{Z}^n$ for all sufficiently large $T$, where $\|\cdot\|$ denotes the supremum norm. Kleinbock and Wadleigh (2019) established an integrability criterion for the Lebesgue measure of the $ψ$-Dirichlet non-improvable set. In this paper, we prove a similar criterion for the Hausdorff measure of the $ψ$-Dirichlet non-improvable set. Also, we extend this result to the singly metric case that $\mathbf{b}$ is fixed. As an application, we compute the Hausdorff dimension of the set of pairs $(A,\mathbf{b})$ with uniform Diophantine exponents $\widehat{w}(A,\mathbf{b})\leq w$.

math.DS↗

arXiv2021

Message Passing Adaptive Resonance Theory for Online Active Semi-supervised Learning

Active learning is widely used to reduce labeling effort and training time by repeatedly querying only the most beneficial samples from unlabeled data. In real-world problems where data cannot be stored indefinitely due to limited storage or privacy issues, the query selection and the model update should be performed as soon as a new data sample is observed. Various online active learning methods have been studied to deal with these challenges; however, there are difficulties in selecting representative query samples and updating the model efficiently without forgetting. In this study, we propose Message Passing Adaptive Resonance Theory (MPART) that learns the distribution and topology of input data online. Through message passing on the topological graph, MPART actively queries informative and representative samples, and continuously improves the classification performance using both labeled and unlabeled data. We evaluate our model in stream-based selective sampling scenarios with comparable query selection strategies, showing that MPART significantly outperforms competitive models.

cs.LG↗