arXiv subjects

Zhijie Chen

Publications and source records attributed to Zhijie Chen.

At least 37 records · Page 2Linked to original sources

arXiv2025

Show-o: One Single Transformer to Unify Multimodal Understanding and Generation

We present a unified transformer, i.e., Show-o, that unifies multimodal understanding and generation. Unlike fully autoregressive models, Show-o unifies autoregressive and (discrete) diffusion modeling to adaptively handle inputs and outputs of various and mixed modalities. The unified model flexibly supports a wide range of vision-language tasks including visual question-answering, text-to-image generation, text-guided inpainting/extrapolation, and mixed-modality generation. Across various benchmarks, it demonstrates comparable or superior performance to existing individual models with an equivalent or larger number of parameters tailored for understanding or generation. This significantly highlights its potential as a next-generation foundation model. Code and models are released at https://github.com/showlab/Show-o.

cs.CV↗

arXiv2025

Sovability of curvature equations with multiple singular sources on torus via Painleve VI equations

We study the curvature equation with multiple singular sources on a torus \[Δu+e^{u}=8π\sum_{k=0}^{3}n_{k}δ_{\frac{ω_{k}}{2}}% +4π\left( δ_{p}+δ_{-p}\right) \quad \text{ on }\;E_τ:=\mathbb{C}/(\mathbb Z+\mathbb{Z}τ),\] where $n_k\in\mathbb N$ and $δ_a$ denotes the Dirac measure at $a$. This is known as a critical case for which the apriori estimate does not hold, and the existence of solutions has been a long-standing problem. In this paper, by establishing a deep connection with Painlevé VI equations, we show that the existence of even solutions (i.e. $u(z)=u(-z)$) depends on the location of the singular point $p$, and we give a sharp criterion of $p$ in terms of Painlevé VI equations.

math.AP↗

arXiv2025

On the distribution of critical points of the Eisenstein series $E_6$ and monodromy interpretation

In previous works joint with Lin, we proved that the Eisenstein series $E_4$ (resp. $E_2$) has at most one critical point in every fundamental domain $γ(F_0)$ of $Γ_{0}(2)$, where $γ(F_0)$ are translates of the basic fundamental domain $F_0$ via the Möbius transformation of $γ\inΓ_{0}(2)$. But the method can not work for the Eisenstein series $E_6$. In this paper, we develop a new approach to show that $E_6'(τ)$ has exactly either $1$ or $2$ zeros in every fundamental domain $γ(F_0)$ of $Γ_{0}(2)$. A criterion for $γ(F_0)$ containing exactly $2$ zeros is also given. Furthermore, by mapping all zeros of $E_6'(τ)$ into $F_0$ via the Möbius transformations of $Γ_{0}(2)$ action, the images give rise to a dense subset of the union of three disjoint smooth curves in $F_0$. A monodromy interpretation of these curves from a complex linear ODE is also given. As a consequence, we give a complete description of the distribution of the zeros of $E_6'(τ)$ in fundamental domains of $SL(2,\mathbb{Z})$.

math.NT↗

arXiv2025

Critical points of the Moser-Trudinger functional on conical singular surfaces, I: compactness

Let $(Σ, g_1)$ be a compact Riemann surface with conical singularites of angles in $(0, 2π)$, and $f: Σ\to\mathbb R$ be a positive smooth function. In this paper, by establishing a sharp quantization result, we prove the compactness of the set of positive critical points for the Moser-Trudinger functional \[F_1(u)=\int_Σ(e^{u^2}-1)f dv_{g_1}\] constrained to $u\in\mathcal E_β:=\{u\in H^1(Σ,g_1) : \|u\|_{H^1(Σ,g_1)}^2=β\}$ for any $β>0$. This result is a generalization of the compactness result for the Moser-Trudinger functional on regular compact surfaces, proved by De Marchis-Malchiodi-Martinazzi-Thizy (Inventiones Mathematicae, 2022, 230: 1165-1248). The presence of conical singularities brings many additional difficulties and we need to develop different ideas and techniques. The compactness lays the foundation for proving the existence of critical points of the Moser-Trudinger functional on conical singular surfaces in a sequel work.

math.AP↗

arXiv2025

Sketched Adaptive Federated Deep Learning: A Sharp Convergence Analysis

Combining gradient compression methods (e.g., CountSketch, quantization) and adaptive optimizers (e.g., Adam, AMSGrad) is a desirable goal in federated learning (FL), with potential benefits on both fewer communication rounds and less per-round communication. In spite of the preliminary empirical success of sketched adaptive methods, existing convergence analyses show the communication cost to have a linear dependence on the ambient dimension, i.e., number of parameters, which is prohibitively high for modern deep learning models. In this work, we introduce specific sketched adaptive federated learning (SAFL) algorithms and, as our main contribution, provide theoretical convergence analyses in different FL settings with guarantees on communication cost depending only logarithmically (instead of linearly) on the ambient dimension. Unlike existing analyses, we show that the entry-wise sketching noise existent in the preconditioners and the first moments of SAFL can be implicitly addressed by leveraging the recently-popularized anisotropic curvatures in deep learning losses, e.g., fast decaying loss Hessian eigen-values. In the i.i.d. client setting of FL, we show that SAFL achieves asymptotic $O(1/\sqrt{T})$ convergence, and converges faster in the initial epochs. In the non-i.i.d. client setting, where non-adaptive methods lack convergence guarantees, we show that SACFL (SAFL with clipping) algorithms can provably converge in spite of the additional heavy-tailed noise. Our theoretical claims are supported by empirical studies on vision and language tasks, and in both fine-tuning and training-from-scratch regimes. Surprisingly, as a by-product of our analysis, the proposed SAFL methods are competitive with the state-of-the-art communication-efficient federated learning algorithms based on error feedback.

cs.LG↗

arXiv2025

Generic non-degeneracy of critical points of multiple Green functions on torus and applications to curvature equations

Let $E_τ:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}τ)$ with $\operatorname{Im}τ>0$ be a flat torus and $G(z;τ)$ be the Green function on $E_τ$ with the singularity at $0$. Consider the multiple Green function $G_{n}$ on $(E_τ)^{n}$: \[ G_{n}(z_{1},\cdots,z_{n};τ):=\sum_{i 0\}$ such that $G_n(\cdot;τ)$ has degenerate critical points for any $τ$ on the union of these curves. In this paper, we prove that there is a measure zero subset $\mathcal{O}_n\subset \mathbb H$ (containing these curves) such that for any $τ\in \mathbb H\setminus\mathcal{O}_n$, all critical points of $G_n(\cdot;τ)$ are non-degenerate. Applications to counting the exact number of solutions of the curvature equation $Δu+e^{u}=ρδ_{0}$ on $E_τ$ will be given.

math.AP↗

arXiv2025

OpenVid-1M: A Large-Scale High-Quality Dataset for Text-to-video Generation

Text-to-video (T2V) generation has recently garnered significant attention thanks to the large multi-modality model Sora. However, T2V generation still faces two important challenges: 1) Lacking a precise open sourced high-quality dataset. The previous popular video datasets, e.g. WebVid-10M and Panda-70M, are either with low quality or too large for most research institutions. Therefore, it is challenging but crucial to collect a precise high-quality text-video pairs for T2V generation. 2) Ignoring to fully utilize textual information. Recent T2V methods have focused on vision transformers, using a simple cross attention module for video generation, which falls short of thoroughly extracting semantic information from text prompt. To address these issues, we introduce OpenVid-1M, a precise high-quality dataset with expressive captions. This open-scenario dataset contains over 1 million text-video pairs, facilitating research on T2V generation. Furthermore, we curate 433K 1080p videos from OpenVid-1M to create OpenVidHD-0.4M, advancing high-definition video generation. Additionally, we propose a novel Multi-modal Video Diffusion Transformer (MVDiT) capable of mining both structure information from visual tokens and semantic information from text tokens. Extensive experiments and ablation studies verify the superiority of OpenVid-1M over previous datasets and the effectiveness of our MVDiT.

cs.CV↗

arXiv2025

Optimization and Generalization Guarantees for Weight Normalization

Weight normalization (WeightNorm) is widely used in practice for the training of deep neural networks and modern deep learning libraries have built-in implementations of it. In this paper, we provide the first theoretical characterizations of both optimization and generalization of deep WeightNorm models with smooth activation functions. For optimization, from the form of the Hessian of the loss, we note that a small Hessian of the predictor leads to a tractable analysis. Thus, we bound the spectral norm of the Hessian of WeightNorm networks and show its dependence on the network width and weight normalization terms--the latter being unique to networks without WeightNorm. Then, we use this bound to establish training convergence guarantees under suitable assumptions for gradient decent. For generalization, we use WeightNorm to get a uniform convergence based generalization bound, which is independent from the width and depends sublinearly on the depth. Finally, we present experimental results which illustrate how the normalization terms and other quantities of theoretical interest relate to the training of WeightNorm networks.

cs.LG↗

arXiv2024

Optimal Control Operator Perspective and a Neural Adaptive Spectral Method

Optimal control problems (OCPs) involve finding a control function for a dynamical system such that a cost functional is optimized. It is central to physical systems in both academia and industry. In this paper, we propose a novel instance-solution control operator perspective, which solves OCPs in a one-shot manner without direct dependence on the explicit expression of dynamics or iterative optimization processes. The control operator is implemented by a new neural operator architecture named Neural Adaptive Spectral Method (NASM), a generalization of classical spectral methods. We theoretically validate the perspective and architecture by presenting the approximation error bounds of NASM for the control operator. Experiments on synthetic environments and a real-world dataset verify the effectiveness and efficiency of our approach, including substantial speedup in running time, and high-quality in- and out-of-distribution generalization.

eess.SY↗

arXiv2024

Concentration phenomena of positive solutions to weakly coupled Schrödinger systems with large exponents in dimension two

We study the weakly coupled nonlinear Schrödinger system \begin{equation*} \begin{cases} -Δu_1 = μ_1 u_1^{p} +βu_1^{\frac{p-1}{2}} u_2^{\frac{p+1}{2}}\text{ in } Ω,\\ -Δu_2 = μ_2 u_2^{p} +βu_2^{\frac{p-1}{2}}u_1^{\frac{p+1}{2}} \text{ in } Ω,\\ u_1,u_2>0\quad\text{in }\;Ω;\quad u_1=u_2=0 \quad\text { on } \;\partialΩ, \end{cases} \end{equation*} where $p>1, μ_1, μ_2, β>0$ and $Ω$ is a smooth bounded domain in $\mathbb{R}^2$. Under the natural condition that holds automatically for all positive solutions in star-shaped domains \begin{align*} p\int_Ω|\nabla u_{1,p}|^2+|\nabla u_{2,p}|^2 dx \leq C, \end{align*} we give a complete description of the concentration phenomena of positive solutions $(u_{1,p},u_{2,p})$ as $p\rightarrow+\infty$, including the $L^{\infty}$-norm quantization $\|u_{k,p}\|_{L^\infty(Ω)}\to \sqrt{e}$ for $k=1,2$, the energy quantization $p\int_Ω|\nabla u_{1,p}|^2+|\nabla u_{2,p}|^2dx\to 8nπe $ with $n\in\mathbb{N}_{\geq 2}$, and so on. In particular, we show that the ``local mass'' contributed by each concentration point must be one of $\{(8π,8π), (8π,0),(0,8π)\}$.

math.AP↗

arXiv2024

Co-Axial Metrics on the Sphere and Algebraic Numbers

In this paper, we consider the following curvature equation $$Δu+{\rm e}^u=4π\biggl((θ_0-1)δ_0+(θ_1-1)δ_1 +\sum_{j=1}^{n+m}\bigl(θ_j'-1\bigr)δ_{t_j}\biggr)\qquad \text{in}\ \mathbb R^2,$$ $$u(x)=-2(1+θ_\infty)\ln|x|+O(1)\qquad \text{as} \ |x|\to\infty,$$ where $θ_0$, $θ_1$, $θ_\infty$, and $θ_{j}'$ are positive non-integers for $1\le j\le n$, while $θ_{j}'\in\mathbb{N}_{\geq 2}$ are integers for $n+1\le j\le n+m$. Geometrically, a solution $u$ gives rise to a conical metric ${\rm d}s^2=\frac12 {\rm e}^u|{\rm d}x|^2$ of curvature $1$ on the sphere, with conical singularities at $0$, $1$, $\infty$, and $t_j$, $1\le j\le n+m$, with angles $2πθ_0$, $2πθ_1$, $2πθ_\infty$, and $2πθ_{j}'$ at $0$, $1$, $\infty$, and $t_j$, respectively. The metric ${\rm d}s^2$ or the solution $u$ is called co-axial, which was introduced by Mondello and Panov, if there is a developing map $h(x)$ of $u$ such that the projective monodromy group is contained in the unit circle. The sufficient and necessary conditions in terms of angles for the existence of such metrics were obtained by Mondello-Panov (2016) and Eremenko (2020). In this paper, we fix the angles and study the locations of the singularities $t_1,\dots,t_{n+m}$. Let $A\subset\mathbb{C}^{n+m}$ be the set of those $(t_1,\dots,t_{n+m})$'s such that a co-axial metric exists, among other things we prove that (i) If $m=1$, i.e., there is only one integer $θ_{n+1}'$ among $θ_j'$, then $A$ is a finite set. Moreover, for the case $n=0$, we obtain a sharp bound of the cardinality of the set $A$. We apply a result due to Eremenko, Gabrielov, and Tarasov (2016) and the monodromy of hypergeometric equations to obtain such a bound. (ii) If $m\ge 2$, then $A$ is an algebraic set of dimension $\leq m-1$.

math.CA↗

arXiv2024

Monodromy of generalized Lame equations with Darboux-Treibich-Verdier potentials: A universal law

The Darboux-Treibich-Verdier (DTV) potential $\sum_{k=0}^{3}n_{k}(n_{k}+1)\wp(z+\tfrac{ ω_{k}}{2};τ)$ is well-known as doubly-periodic solutions of the stationary KdV hierarchy (Treibich-Verdier, Duke Math. J. {\bf 68} (1992), 217-236). In this paper, we study the generalized Lamé equation with the DTV potential \begin{equation*} y^{\prime \prime }(z)=\bigg[ \sum_{k=0}^{3}n_{k}(n_{k}+1)\wp(z+\tfrac{ ω_{k}}{2};τ)+B\bigg] y(z),\quad n_{k}\in \mathbb{N} \end{equation*} from the monodromy aspect. We prove that the map from $(τ, B)$ to the monodromy data $(r,s)$ satisfies a surprising universal law $dτ\wedge dB\equiv8π^2 dr\wedge ds.$ Our proof applies Panlevé VI equation and modular forms. We also give applications to the algebraic multiplicity of (anti)periodic eigenvalues for the associated Hill operator.

math.CA↗

arXiv2023

Effects of detection-beam focal offset on displacement detection in optical tweezers

A high-resolution displacement detection can be achieved by analyzing the scattered light of the trapping beams from the particle in optical tweezers. In some applications where trapping and displacement detection need to be separated, a detection beam can be introduced for independent displacement detection. However, the detection beam focus possibly deviates from the centre of the particle, which will affect the performance of the displacement detection. In this paper, we detect the radial displacement of the particle by utilizing the forward scattered light of the detection beam from the particle. The effects of the lateral and axial offsets between the detection beam focus and the particle centre on the displacement detection are analyzed by the simulation and experiment. The results show that the lateral offsets will decrease the detection sensitivity and linear range and aggravate the crosstalk between the x-direction signal and y-direction signal of QPD. The axial offsets also affect the detection sensitivity, an optimal axial offset can improve the sensitivity of the displacement detection substantially. In addition, the influence of system parameters, such as particle radius a, numerical aperture of the condenser NAc and numerical aperture of the objective NAo on the optimal axial offset are discussed. A combination of conventional optical tweezers instrument and a detection beam provides a more flexible working point, allowing for the active modulation of the sensitivity and linear range of the displacement detection. This work would be of great interest for improving the accuracy of the displacement and force detection performed by the optical tweezers.

physics.optics↗

arXiv2023

Energy quantization of the two dimensional Lane-Emden equation with vanishing potentials

We study the concentration phenomenon of the Lane-Emden equation with vanishing potentials \[\begin{cases} -Δu_n=W_n(x)u_n^{p_n},\quad u_n>0,\quad\text{in}~Ω, u_n=0,\quad\text{on}~\partialΩ, \int_Ωp_n W_n(x)u_n^{p_n}dx\le C, \end{cases}\] where $Ω$ is a smooth bounded domain in $\mathbb{R}^2$, $W_n(x)\geq 0$ are bounded functions with zeros in $Ω$, and $p_n\to\infty$ as $n\to\infty$. A typical example is $W_n(x)=|x|^{2α}$ with $0\inΩ$, i.e. the equation turns to be the well-known Hénon equation. The asymptotic behavior for $α=0$ has been well studied in the literature. While for $α>0$, the problem becomes much more complicated since a singular Liouville equation appears as a limit problem. In this paper, we study the case $α>0$ and prove a quantization property (suppose $0$ is a concentration point) \[p_n|x|^{2α}u_n(x)^{p_n-1+t}\to 8πe^{\frac{t}{2}}\sum_{i=1}^kδ_{a_i}+8π(1+α)e^{\frac{t}{2}}c^tδ_0, \quad t=0,1,2,\] for some $k\ge0$, $a_i\inΩ\setminus\{0\}$ and some $c\ge1$. Moreover, for $α\not\in\mathbb{N}$, we show that the blow up must be simple, i.e. $c=1$. As applications, we also obtain the complete asymptotic behavior of ground state solutions for the Hénon equation.

math.AP↗

arXiv2023

Machine-Learning Recognition of Dzyaloshinskii-Moriya Interaction from Magnetometry

The Dzyaloshinskii-Moriya interaction (DMI), which is the antisymmetric part of the exchange interaction between neighboring local spins, winds the spin manifold and can stabilize non-trivial topological spin textures. Since topology is a robust information carrier, characterization techniques that can extract the DMI magnitude are important for the discovery and optimization of spintronic materials. Existing experimental techniques for quantitative determination of DMI, such as high-resolution magnetic imaging of spin textures and measurement of magnon or transport properties, are time consuming and require specialized instrumentation. Here we show that a convolutional neural network can extract the DMI magnitude from minor hysteresis loops, or magnetic "fingerprints" of a material. These hysteresis loops are readily available by conventional magnetometry measurements. This provides a convenient tool to investigate topological spin textures for next-generation information processing.

cond-mat.mtrl-sci↗

arXiv2023

Ionically-Driven Synthesis and Exchange Bias in Mn$_{4}$N/MnN$_{x}$ Heterostructures

Ferrimagnets have received renewed attention as a promising platform for spintronic applications. Of particular interest is the Mn4N from the $ε$-phase of the manganese nitride as an emergent rare-earth-free spintronic material due to its perpendicular magnetic anisotropy, small saturation magnetization, high thermal stability, and large domain wall velocity. We have achieved high-quality (001)-ordered Mn$_{4}$N thin film by sputtering Mn onto $η$-phase Mn$_{3}$N$_{2}$ seed layers on Si substrates. As the deposited Mn thickness varies, nitrogen ion migration across the Mn$_{3}$N$_{2}$/Mn layers leads to a continuous evolution of the layers to Mn$_{3}$N$_{2}$/Mn$_{2}$N/Mn$_{4}$N, Mn$_{2}$N/Mn$_{4}$N, and eventually Mn$_{4}$N alone. The ferrimagnetic Mn$_{4}$N indeed exhibits perpendicular magnetic anisotropy, and forms via a nucleation-and-growth mechanism. The nitrogen ion migration is also manifested in a significant exchange bias, up to 0.3 T at 5 K, due to the interactions between ferrimagnetic Mn$_{4}$N and antiferromagnetic Mn$_{3}$N$_{2}$ and Mn$_{2}$N. These results demonstrate a promising all-nitride magneto-ionic platform with remarkable tunability for device applications.

cond-mat.mtrl-sci↗

arXiv2023

Monodromy of a generalized Lame equation of third order

We study the monodromy of the following third order linear differential equation \[y'''(z)-(α\wp(z;τ)+B)y'(z)+β\wp'(z;τ)y(z)=0, \] where $B\in\mathbb{C}$ is a parameter, $\wp(z;τ)$ is the Weierstrass $\wp$-function with periods $1$ and $τ$, and $α,β$ are constants such that the local exponents at the singularity $0$ are three distinct integers, which can always be written as $-n-l, 1-l, n+2l+2$ after a dual transformation, where $n,l\in\mathbb{N}$. This ODE can be seen as the third order version of the well-known Lamé equation $y''(z)-(m(m+1)\wp(z;τ)+B)y(z)=0$. We say that the monodromy is unitary if the monodromy group is conjugate to a subgroup of the unitary group. We show that \begin{itemize} \item[(i)] if $n, l$ are both odd, then the monodromy can not be unitary; \item[(ii)] if $n$ is odd and $l$ is even, then there exist finite values of $B$ such that the monodromy is the Klein four-group and hence unitary; \item[(iii)] if $n$ is even, then whether there exists $B$ such that the monodromy is unitary depends on the choice of the period $τ$. \end{itemize} The methods of studying the second order Lamé equation can not work here, and we need to develop different approaches to treat these different cases separately. These results have interesting applications to the integrable $SU(3)$ Toda system in another work (Chen-Lin, J. Differ. Geom. to appear).

math.CA↗

arXiv2023

Sharp estimates and non-degeneracy of low energy nodal solutions for the Lane-Emden equation in dimension two

We study the Lane-Emden problem \[\begin{cases} -Δu_p =|u_p|^{p-1}u_p&\text{in}\quad Ω, u_p=0 &\text{on}\quad\partialΩ, \end{cases}\] where $Ω\subset\mathbb R^2$ is a smooth bounded domain and $p>1$ is sufficiently large. We obtain sharp estimates and non-degeneracy of low energy nodal solutions $u_p$ (i.e. nodal solutions satisfying $\lim_{p\to+\infty}p\int_Ω|u_p|^{p+1}dx=16πe$). As applications, we prove that the comparable condition $p(\|u_p^+\|_{\infty}-\|u_p^-\|_{\infty})=O(1)$ holds automatically for least energy nodal solutions, which confirms a conjecture raised by (Grossi-Grumiau-Pacella, Ann.I.H. Poincare-AN, 30 (2013), 121-140) and (Grossi-Grumiau-Pacella, J.Math.Pures Appl. 101 (2014), 735-754).

math.AP↗