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Shiwen Zhang

Publications and source records attributed to Shiwen Zhang.

At least 37 records · Page 2Linked to original sources
arXiv2024

Spectrum and Lifshitz tails for the Anderson model on the Sierpinski gasket graph

In this work, we study the Anderson model on the Sierpinski gasket graph. We first identify the almost sure spectrum of the Anderson model when the support of the random potential has no gaps. We then prove the existence of the integrated density states of the Anderson model and show that it has Lifshitz tails with Lifshitz exponent determined by the ratio of the volume growth rate and the random walk dimension of the Sierpinski gasket graph.

math-ph↗
arXiv2024

Hyper-parameter tuning for text guided image editing

The test-time finetuning text-guided image editing method, Forgedit, is capable of tackling general and complex image editing problems given only the input image itself and the target text prompt. During finetuning stage, using the same set of finetuning hyper-paramters every time for every given image, Forgedit remembers and understands the input image in 30 seconds. During editing stage, the workflow of Forgedit might seem complicated. However, in fact, the editing process of Forgedit is not more complex than previous SOTA Imagic, yet completely solves the overfitting problem of Imagic. In this paper, we will elaborate the workflow of Forgedit editing stage with examples. We will show how to tune the hyper-parameters in an efficient way to obtain ideal editing results.

cs.CV↗
arXiv2024

Landscape estimates of the integrated density of states for Jacobi operators on graphs

We show the integrated density of states for a variety of Jacobi operators on graphs, such as the Anderson model and random hopping models on graphs with Gaussian heat kernel bounds, can be estimated from above and below in terms of the localization landscape counting function. Specific examples of these graphs include stacked and decorated lattices, graphs corresponding to band matrices, and aperiodic tiling graphs. The upper bound part of the landscape law also applies to the fractal Sierpinski gasket graph. As a consequence of the landscape law, we obtain landscape-based proofs of the Lifshitz tails in several models including random band matrix models, certain bond percolation Hamiltonians on $\mathbb{Z}^d$, and Jacobi operators on certain stacks of graphs. We also present intriguing numerical simulations exploring the behavior of the landscape counting function across various models.

math-ph↗
arXiv2024

Landscape approximation of the ground state eigenvalue for graphs and random hopping models

We consider the localization landscape function $u$ and ground state eigenvalue $λ$ for operators on graphs. We first show that the maximum of the landscape function is comparable to the reciprocal of the ground state eigenvalue if the operator satisfies certain semigroup kernel upper bounds. This implies general upper and lower bounds on the landscape product $λ\|u\|_\infty$ for several models, including the Anderson model and random hopping (bond-disordered) models, on graphs that are roughly isometric to $\mathbb{Z}^d$, as well as on some fractal-like graphs such as the Sierpinski gasket graph. Next, we specialize to a random hopping model on $\mathbb{Z}$, and show that as the size of the chain grows, the landscape product $λ\|u\|_\infty$ approaches $π^2/8$ for Bernoulli off-diagonal disorder, and has the same upper bound of $π^2/8$ for Uniform([0,1]) off-diagonal disorder. We also numerically study the random hopping model when the band width (hopping distance) is greater than one, and provide strong numerical evidence that a similar approximation holds for low-lying energies in the spectrum.

math-ph↗
arXiv2024

Forgedit: Text Guided Image Editing via Learning and Forgetting

Text-guided image editing on real or synthetic images, given only the original image itself and the target text prompt as inputs, is a very general and challenging task. It requires an editing model to estimate by itself which part of the image should be edited, and then perform either rigid or non-rigid editing while preserving the characteristics of original image. In this paper, we design a novel text-guided image editing method, named as Forgedit. First, we propose a vision-language joint optimization framework capable of reconstructing the original image in 30 seconds, much faster than previous SOTA and much less overfitting. Then we propose a novel vector projection mechanism in text embedding space of Diffusion Models, which is capable to control the identity similarity and editing strength seperately. Finally, we discovered a general property of UNet in Diffusion Models, i.e., Unet encoder learns space and structure, Unet decoder learns appearance and identity. With such a property, we design forgetting mechanisms to successfully tackle the fatal and inevitable overfitting issues when fine-tuning Diffusion Models on one image, thus significantly boosting the editing capability of Diffusion Models. Our method, Forgedit, built on Stable Diffusion, achieves new state-of-the-art results on the challenging text-guided image editing benchmark: TEdBench, surpassing the previous SOTA methods such as Imagic with Imagen, in terms of both CLIP score and LPIPS score. Codes are available at https://github.com/witcherofresearch/Forgedit

cs.CV↗
arXiv2024

Cyclic sieving on noncrossing (1,2)-configurations

Verifying a suspicion of Propp and Reiner concerning the cyclic sieving phenomenon (CSP), M. Thiel introduced a Catalan object called noncrossing $(1,2)$-configurations (denoted by $X_n$), which is a class of set partitions of $[n-1]$. More precisely, Thiel proved that, with a natural action of the cyclic group $C_{n-1}$ on $X_n$, the triple $\left(X_n,C_{n-1},\text{Cat}_n(q)\right)$ exhibits the CSP, where $\text{Cat}_n(q):=\frac{1}{[n+1]_q}\begin{bmatrix} 2n\\ n \end{bmatrix}_q$ is MacMahon's $q$-Catalan number. Recently, in a study of the fermionic diagonal coinvariant ring $FDR_n$, J. Kim found a combinatorial basis for $FDR_n$ indexed by $X_n$. In this paper, we continue to study $X_n$ and obtain the following results: (1) We define a statistic $cwt$ on $X_n$ whose generating function is $\text{Cat}_n(q)$, which answers a problem of Thiel. (2) We show that $\text{Cat}_n(q)$ is equivalent to $$\sum_{\substack{k,x,y\\2k+x+y=n-1}}\begin{bmatrix} n-1 2k,x,y \end{bmatrix}_q\text{Cat}_k (q)q^{k+\binom{x}{2}+\binom{y}{2}+\binom{n}{2}}$$ modulo $q^{n-1}-1$, which answers a problem of Kim. As mentioned by Kim, this result leads to a representation theoretic proof of the above cyclic sieving result of Thiel. (3) We consider the dihedral sieving, a generalization of the CSP, which was recently introduced by Rao and Suk. Under a natural action of the dihedral group $I_2(n-1)$ (for even $n$), we prove a dihedral sieving result on $X_n$.

math.CO↗
arXiv2023

On some zero-sum invariants for abelian groups of rank three

Let $G$ be an additive finite abelian group with exponent $\exp(G)$. For $L\subseteq \mathbb N$, let $\mathsf{s}_{L}(G)$ be the smallest integer $\ell$ such that every sequence $S$ over $G$ of length $\ell$ has a zero-sum subsequence $T$ of length $|T|\in L$. In this paper, we consider the invariants $\mathsf{s}_{[1,t]}(G)$ and $\mathsf{s}_{\{k\exp(G)\}}(G)$ (with $k\in \mathbb N$). We obtain precise values as well as upper bounds of the above invariants for some abelian groups of rank three. Some of these results improve previous results of Gao-Thangadurai and Han-Zhang.

math.CO↗
arXiv2022

Approximating the ground state eigenvalue via the effective potential

In this paper, we study 1-d random Schrödinger operators on a finite interval with Dirichlet boundary conditions. We are interested in the approximation of the ground state energy using the minimum of the effective potential. For the 1-d continuous Anderson Bernoulli model, we show that the ratio of the ground state energy and the minimum of the effective potential approaches $\frac{π^2}{8}$ as the domain size approaches infinity. Besides, we will discuss various approximations to the ratio in different situations. There will be numerical experiments supporting our main results for the ground state energy and also supporting approximations for the excited states energies.

math.SP↗
arXiv2022

TFCNet: Temporal Fully Connected Networks for Static Unbiased Temporal Reasoning

Temporal Reasoning is one important functionality for vision intelligence. In computer vision research community, temporal reasoning is usually studied in the form of video classification, for which many state-of-the-art Neural Network structures and dataset benchmarks are proposed in recent years, especially 3D CNNs and Kinetics. However, some recent works found that current video classification benchmarks contain strong biases towards static features, thus cannot accurately reflect the temporal modeling ability. New video classification benchmarks aiming to eliminate static biases are proposed, with experiments on these new benchmarks showing that the current clip-based 3D CNNs are outperformed by RNN structures and recent video transformers. In this paper, we find that 3D CNNs and their efficient depthwise variants, when video-level sampling strategy is used, are actually able to beat RNNs and recent vision transformers by significant margins on static-unbiased temporal reasoning benchmarks. Further, we propose Temporal Fully Connected Block (TFC Block), an efficient and effective component, which approximates fully connected layers along temporal dimension to obtain video-level receptive field, enhancing the spatiotemporal reasoning ability. With TFC blocks inserted into Video-level 3D CNNs (V3D), our proposed TFCNets establish new state-of-the-art results on synthetic temporal reasoning benchmark, CATER, and real world static-unbiased dataset, Diving48, surpassing all previous methods.

cs.CV↗
arXiv2021

Sharp estimates for the integrated density of states in Anderson tight-binding models

Recent work [G. David, M. Filoche, and S. Mayboroda, arXiv:1909.10558[Adv. Math. (to be published)]] has proved the existence of bounds from above and below for the Integrated Density of States (IDOS) of the Schrödinger operator throughout the spectrum, called the landscape law. These bounds involve dimensional constants whose optimal values are yet to be determined. Here, we investigate the accuracy of the landscape law in 1D and 2D tight-binding Anderson models, with binary or uniform random distributions. We show, in particular, that in 1D, the IDOS can be approximated with high accuracy through a single formula involving a remarkably simple multiplicative energy shift. In 2D, the same idea applies but the prefactor has to be changed between the bottom and top parts of the spectrum.

math-ph↗
arXiv2021

On a Novel Effective Equation of the Reduced Hartree-Fock Theory

We show that there is an one-to-one correspondence between solutions to the Poisson-Landscape equations and the reduced Hartree-Fock equations in the semi-classical limit at low temperature. Moreover, we prove that the difference between the two corresponding solutions is small by providing explicit estimates.

math.AP↗
arXiv2021

The landscape law for tight binding Hamiltonians

The present paper extends the landscape theory pioneered in [FM, ADFJM2, DFM] to the tight-binding Schrödinger operator on $\Z^d$. In particular, we establish upper and lower bounds for the integrated density of states in terms of the counting function based upon the localization landscape.

math-ph↗
arXiv2020

Large deviation estimates and Hölder regularity of the Lyapunov exponents for quasi-periodic Schrödinger cocycles

We consider one-dimensional quasi-periodic Schrödinger operators with analytic potentials. In the positive Lyapunov exponent regime, we prove large deviation estimates which lead to optimal Hölder continuity of the Lyapunov exponents and the integrated density of states, in both small Lyapunov exponent and large coupling regimes. Our results cover all the Diophantine frequencies and some Liouville frequencies.

math-ph↗
arXiv2020

The exponential decay of eigenfunctions for tight binding Hamiltonians via landscape and dual landscape functions

We consider the discrete Schrödinger operator $H=-Δ+V$ on a cube $M\subset \mathbb{Z}^d$, with periodic or Dirichlet (simple) boundary conditions. We use a hidden landscape function $u$, defined as the solution of an inhomogeneous boundary problem with uniform right-hand side for $H$, to predict the location of the localized eigenfunctions of $H$. Explicit bounds on the exponential decay of Agmon type for low energy modes are obtained. This extends the recent work of Agmon type of localization in [5] for $\mathbb{R}^d$ to a tight-binding Hamiltonian on $\mathbb{Z}^d$ lattice. Contrary to the continuous case, high energy modes are as localized as the low energy ones in discrete lattices. We show that exponential decay estimates of Agmon type also appear near the top of the spectrum, where the location of the localized eigenfunctions is predicted by a different landscape function. Our results are deterministic and are independent of the size of the cube. We also provide numerical experiments to confirm the conditional results effectively, for some random potentials.

math-ph↗
arXiv2020

V4D:4D Convolutional Neural Networks for Video-level Representation Learning

Most existing 3D CNNs for video representation learning are clip-based methods, and thus do not consider video-level temporal evolution of spatio-temporal features. In this paper, we propose Video-level 4D Convolutional Neural Networks, referred as V4D, to model the evolution of long-range spatio-temporal representation with 4D convolutions, and at the same time, to preserve strong 3D spatio-temporal representation with residual connections. Specifically, we design a new 4D residual block able to capture inter-clip interactions, which could enhance the representation power of the original clip-level 3D CNNs. The 4D residual blocks can be easily integrated into the existing 3D CNNs to perform long-range modeling hierarchically. We further introduce the training and inference methods for the proposed V4D. Extensive experiments are conducted on three video recognition benchmarks, where V4D achieves excellent results, surpassing recent 3D CNNs by a large margin.

cs.CV↗
arXiv2020

Knowledge Integration Networks for Action Recognition

In this work, we propose Knowledge Integration Networks (referred as KINet) for video action recognition. KINet is capable of aggregating meaningful context features which are of great importance to identifying an action, such as human information and scene context. We design a three-branch architecture consisting of a main branch for action recognition, and two auxiliary branches for human parsing and scene recognition which allow the model to encode the knowledge of human and scene for action recognition. We explore two pre-trained models as teacher networks to distill the knowledge of human and scene for training the auxiliary tasks of KINet. Furthermore, we propose a two-level knowledge encoding mechanism which contains a Cross Branch Integration (CBI) module for encoding the auxiliary knowledge into medium-level convolutional features, and an Action Knowledge Graph (AKG) for effectively fusing high-level context information. This results in an end-to-end trainable framework where the three tasks can be trained collaboratively, allowing the model to compute strong context knowledge efficiently. The proposed KINet achieves the state-of-the-art performance on a large-scale action recognition benchmark Kinetics-400, with a top-1 accuracy of 77.8%. We further demonstrate that our KINet has strong capability by transferring the Kinetics-trained model to UCF-101, where it obtains 97.8% top-1 accuracy.

cs.CV↗
arXiv2019

Diffusion in the mean for a periodic Schrödinger equation perturbed by a fluctuating potential

We consider the evolution of a quantum particle hopping on a cubic lattice in any dimension and subject to a potential consisting of a periodic part and a random part that fluctuates stochastically in time. If the random potential evolves according to a stationary Markov process, we obtain diffusive scaling for moments of the position displacement, with a diffusion constant that grows as the inverse square of the disorder strength at weak coupling. More generally, we show that a central limit theorem holds such that the square amplitude of the wave packet converges, after diffusive rescaling, to a solution of a heat equation.

math-ph↗