arXiv subjects

Jinhyung Park

Publications and source records attributed to Jinhyung Park.

At least 37 records · Page 2Linked to original sources

arXiv2024

Effective gonality theorem on weight-one syzygies of algebraic curves

In 1986, Green-Lazarsfeld raised the gonality conjecture asserting that the gonality $\operatorname{gon}(C)$ of a smooth projective curve $C$ of genus $g\geq 2$ can be read off from weight-one syzygies of a sufficiently positive line bundle $L$ on $C$, and also proposed possible least degree of such a line bundle. In 2015, Ein-Lazarsfeld proved the conjecture when $\operatorname{deg} L$ is sufficiently large, but the effective part of the conjecture remained widely open and was reformulated explicitly by Farkas-Kemeny. In this paper, we establish an effective vanishing theorem for weight-one syzygies, which implies that the gonality conjecture holds if $\operatorname{deg} L \geq 2g+\operatorname{gon}(C)$ or $\operatorname{deg} L = 2g+\operatorname{gon}(C)-1$ and $C$ is not a plane curve. As Castryck observed that the gonality conjecture may not hold for a plane curve when $\operatorname{deg} L = 2g+\operatorname{gon}(C)-1$, our theorem is the best possible and thus gives a complete answer to the gonality conjecture.

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arXiv2024

Some remarks on smooth projective varieties of small degree and codimension

The purpose of this note is twofold. First, we give a quick proof of Ballico-Chiantini's theorem stating that a Fano or Calabi-Yau variety of dimension at least 4 in codimension two is a complete intersection. Second, we improve Barth-Van de Ven's result asserting that if the degree of a smooth projective variety of dimension $n$ is less than approximately $0.63 \cdot n^{1/2}$, then it is a complete intersection. We show that the degree bound can be improved to approximately $0.79 \cdot n^{2/3}$.

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arXiv2024

Generalizable Neural Human Renderer

While recent advancements in animatable human rendering have achieved remarkable results, they require test-time optimization for each subject which can be a significant limitation for real-world applications. To address this, we tackle the challenging task of learning a Generalizable Neural Human Renderer (GNH), a novel method for rendering animatable humans from monocular video without any test-time optimization. Our core method focuses on transferring appearance information from the input video to the output image plane by utilizing explicit body priors and multi-view geometry. To render the subject in the intended pose, we utilize a straightforward CNN-based image renderer, foregoing the more common ray-sampling or rasterizing-based rendering modules. Our GNH achieves remarkable generalizable, photorealistic rendering with unseen subjects with a three-stage process. We quantitatively and qualitatively demonstrate that GNH significantly surpasses current state-of-the-art methods, notably achieving a 31.3% improvement in LPIPS.

cs.CV↗

arXiv2024

Multi-Person 3D Pose Estimation from Multi-View Uncalibrated Depth Cameras

We tackle the task of multi-view, multi-person 3D human pose estimation from a limited number of uncalibrated depth cameras. Recently, many approaches have been proposed for 3D human pose estimation from multi-view RGB cameras. However, these works (1) assume the number of RGB camera views is large enough for 3D reconstruction, (2) the cameras are calibrated, and (3) rely on ground truth 3D poses for training their regression model. In this work, we propose to leverage sparse, uncalibrated depth cameras providing RGBD video streams for 3D human pose estimation. We present a simple pipeline for Multi-View Depth Human Pose Estimation (MVD-HPE) for jointly predicting the camera poses and 3D human poses without training a deep 3D human pose regression model. This framework utilizes 3D Re-ID appearance features from RGBD images to formulate more accurate correspondences (for deriving camera positions) compared to using RGB-only features. We further propose (1) depth-guided camera-pose estimation by leveraging 3D rigid transformations as guidance and (2) depth-constrained 3D human pose estimation by utilizing depth-projected 3D points as an alternative objective for optimization. In order to evaluate our proposed pipeline, we collect three video sets of RGBD videos recorded from multiple sparse-view depth cameras and ground truth 3D poses are manually annotated. Experiments show that our proposed method outperforms the current 3D human pose regression-free pipelines in terms of both camera pose estimation and 3D human pose estimation.

cs.CV↗

arXiv2023

K-polystability of the first secant varieties of rational normal curves

The first secant variety $Σ$ of a rational normal curve of degree $d \geq 3$ is known to be a $\mathbf{Q}$-Fano threefold. In this paper, we prove that $Σ$ is K-polystable, and hence, $Σ$ admits a weak Kähler-Einstein metric. We also show that there exists a $(-K_Σ)$-polar cylinder in $Σ$.

math.AG↗

arXiv2023

A note on an effective bound for the gonality conjecture

The gonality conjecture, proved by Ein--Lazarsfeld, asserts that the gonality of a nonsingular projective curve of genus $g$ can be detected from its syzygies in the embedding given by a line bundle of sufficiently large degree. An effective result obtained by Rathmann says that any line bundle of degree at least 4g-3 would work in the gonality theorem. In this note, we improve the degree bound to 4g-4 with two exceptional cases.

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arXiv2023

Syzygies of secant varieties of smooth projective curves and gonality sequences

The purpose of this paper is to prove that one can read off the gonality sequence of a smooth projective curve from syzygies of secant varieties of the curve embedded by a line bundle of sufficiently large degree. More precisely, together with Ein-Niu-Park's theorem, our main result shows that the gonality sequence of a smooth projective curve completely determines the shape of the minimal free resolutions of secant varieties of the curve of sufficiently large degree. This is a natural generalization of the gonality conjecture on syzygies of smooth projective curves established by Ein-Lazarsfeld and Rathmann to the secant varieties.

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arXiv2023

Asymptotic nonvanishing of syzygies of algebraic varieties

We establish precise nonvanishing results for asymptotic syzygies of smooth projective varieties. This refines Ein-Lazarsfeld's asymptotic nonvanishing theorem. Combining with the author's previous asymptotic vanishing result, we completely determine the asymptotic shapes of the minimal free resolutions of the graded section modules of a line bundle on a smooth projective variety as the positivity of the embedding line bundle grows.

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arXiv2023

F-RDW: Redirected Walking with Forecasting Future Position

In order to serve better VR experiences to users, existing predictive methods of Redirected Walking (RDW) exploit future information to reduce the number of reset occurrences. However, such methods often impose a precondition during deployment, either in the virtual environment's layout or the user's walking direction, which constrains its universal applications. To tackle this challenge, we propose a novel mechanism F-RDW that is twofold: (1) forecasts the future information of a user in the virtual space without any assumptions, and (2) fuse this information while maneuvering existing RDW methods. The backbone of the first step is an LSTM-based model that ingests the user's spatial and eye-tracking data to predict the user's future position in the virtual space, and the following step feeds those predicted values into existing RDW methods (such as MPCRed, S2C, TAPF, and ARC) while respecting their internal mechanism in applicable ways.The results of our simulation test and user study demonstrate the significance of future information when using RDW in small physical spaces or complex environments. We prove that the proposed mechanism significantly reduces the number of resets and increases the traveled distance between resets, hence augmenting the redirection performance of all RDW methods explored in this work.

cs.HC↗

arXiv2022

Syzygies of tangent developable surfaces and K3 carpets via secant varieties

We give simple geometric proofs of Aprodu-Farkas-Papadima-Raicu-Weyman's theorem on syzygies of tangent developable surfaces of rational normal curves and Raicu-Sam's result on syzygies of K3 carpets. As a consequence, we obtain a quick proof of Green's conjecture for general curves of genus $g$ over an algebraically closed field $\mathbf{k}$ with $\operatorname{char}(\mathbf{k}) = 0$ or $\operatorname{char}(\mathbf{k}) \geq \lfloor (g-1)/2 \rfloor$. We also show the arithmetic normality of tangent developable surfaces of arbitrary smooth projective curves of large degree.

math.AG↗

arXiv2022

Time Will Tell: New Outlooks and A Baseline for Temporal Multi-View 3D Object Detection

While recent camera-only 3D detection methods leverage multiple timesteps, the limited history they use significantly hampers the extent to which temporal fusion can improve object perception. Observing that existing works' fusion of multi-frame images are instances of temporal stereo matching, we find that performance is hindered by the interplay between 1) the low granularity of matching resolution and 2) the sub-optimal multi-view setup produced by limited history usage. Our theoretical and empirical analysis demonstrates that the optimal temporal difference between views varies significantly for different pixels and depths, making it necessary to fuse many timesteps over long-term history. Building on our investigation, we propose to generate a cost volume from a long history of image observations, compensating for the coarse but efficient matching resolution with a more optimal multi-view matching setup. Further, we augment the per-frame monocular depth predictions used for long-term, coarse matching with short-term, fine-grained matching and find that long and short term temporal fusion are highly complementary. While maintaining high efficiency, our framework sets new state-of-the-art on nuScenes, achieving first place on the test set and outperforming previous best art by 5.2% mAP and 3.7% NDS on the validation set. Code will be released $\href{https://github.com/Divadi/SOLOFusion}{here.}$

cs.CV↗

arXiv2022

Asymptotic vanishing of syzygies of algebraic varieties

The purpose of this paper is to prove Ein--Lazarsfeld's conjecture on asymptotic vanishing of syzygies of algebraic varieties. This result, together with Ein--Lazarsfeld's asymptotic nonvanishing theorem, describes the overall picture of asymptotic behaviors of the minimal free resolutions of the graded section rings of line bundles on a projective variety as the positivity of the line bundles grows. Previously, Raicu reduced the problem to the case of products of three projective spaces, and we resolve this case here.

math.AG↗

arXiv2022

Singularities of pluri-fundamental divisors on Gorenstein Fano varieties of coindex $4$

Let $X$ be a Gorenstein canonical Fano variety of coindex $4$ and dimension $n$ with $H$ fundamental divisor. Assume $h^0(X, H) \geq n -2$. We prove that a general element of the linear system $|mH|$ has at worst canonical singularities for any integer $m \geq 1$. When $X$ has terminal singularities and $n \geq 5$, we show that a general element of $|mH|$ has at worst terminal singularities for any integer $m \geq 1$. When $n=4$, we give an example of Gorenstein terminal Fano fourfold $X$ such that a general element of $|H|$ does not have terminal singularities.

math.AG↗

arXiv2022

DetMatch: Two Teachers are Better Than One for Joint 2D and 3D Semi-Supervised Object Detection

While numerous 3D detection works leverage the complementary relationship between RGB images and point clouds, developments in the broader framework of semi-supervised object recognition remain uninfluenced by multi-modal fusion. Current methods develop independent pipelines for 2D and 3D semi-supervised learning despite the availability of paired image and point cloud frames. Observing that the distinct characteristics of each sensor cause them to be biased towards detecting different objects, we propose DetMatch, a flexible framework for joint semi-supervised learning on 2D and 3D modalities. By identifying objects detected in both sensors, our pipeline generates a cleaner, more robust set of pseudo-labels that both demonstrates stronger performance and stymies single-modality error propagation. Further, we leverage the richer semantics of RGB images to rectify incorrect 3D class predictions and improve localization of 3D boxes. Evaluating on the challenging KITTI and Waymo datasets, we improve upon strong semi-supervised learning methods and observe higher quality pseudo-labels. Code will be released at https://github.com/Divadi/DetMatch

cs.CV↗

arXiv2022

A Castelnuovo-Mumford regularity bound for threefolds with rational singularities

The purpose of this paper is to establish a Castelnuovo-Mumford regularity bound for threefolds with mild singularities. Let $X$ be a non-degenerate normal projective threefold in $\mathbb{P}^r$ of degree $d$ and codimension $e$. We prove that if $X$ has rational singularities, then $\text{reg}(X) \leq d-e+2$. Our bound is very close to a sharp bound conjectured by Eisenbud-Goto. When $e=2$ and $X$ has Cohen-Macaulay Du Bois singularities, we obtain the conjectured bound $\text{reg}(X) \leq d-1$, and we also classify the extremal cases. To achieve these results, we bound the regularity of fibers of a generic projection of $X$ by using Loewy length, and also bound the dimension of the varieties swept out by secant lines through the singular locus of $X$.

math.AG↗

arXiv2022

Flexible Networks for Learning Physical Dynamics of Deformable Objects

Learning the physical dynamics of deformable objects with particle-based representation has been the objective of many computational models in machine learning. While several state-of-the-art models have achieved this objective in simulated environments, most existing models impose a precondition, such that the input is a sequence of ordered point sets. That is, the order of the points in each point set must be the same across the entire input sequence. This precondition restrains the model from generalizing to real-world data, which is considered to be a sequence of unordered point sets. In this paper, we propose a model named time-wise PointNet (TP-Net) that solves this problem by directly consuming a sequence of unordered point sets to infer the future state of a deformable object with particle-based representation. Our model consists of a shared feature extractor that extracts global features from each input point set in parallel and a prediction network that aggregates and reasons on these features for future prediction. The key concept of our approach is that we use global features rather than local features to achieve invariance to input permutations and ensure the stability and scalability of our model. Experiments demonstrate that our model achieves state-of-the-art performance with real-time prediction speed in both synthetic dataset and real-world dataset. In addition, we provide quantitative and qualitative analysis on why our approach is more effective and efficient than existing approaches.

cs.CV↗

arXiv2021

On subadditivity of Okounkov bodies for algebraic fiber spaces

The purpose of this paper is to establish a subadditivity theorem of Okounkov bodies for algebraic fiber spaces. As applications, we obtain a product formula of the restricted canonical volumes for algebraic fiber spaces and a sufficient condition for an algebraic fiber space to be birationally isotrivial in terms of Okounkov bodies when a general fiber is of general type. Furthermore, we also prove the subadditivity of the numerical Iitaka dimensions for algebraic fiber spaces, and this confirms some numerical variants of the Iitaka conjecture. We hope that our results would provide a new approach toward the Iitaka conjecture.

math.AG↗

arXiv2021

Okounkov bodies associated to abundant divisors and Iitaka fibrations

The aim of this paper is to study the Okounkov bodies associated to abundant divisors. As a main result, we prove that the valuative Okounkov bodies of an abundant divisor encode all the numerical properties. We apply this result to recover the asymptotic base loci of an abundant divisor from the valuative Okounkov bodies. We also give a criterion of when the valuative and limiting Okounkov bodies of an abundant divisor coincide by comparing their Euclidean volumes. To obtain these results, we prove some variants of Fujita's approximations for Okounkov bodies using Iitaka fibrations.

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