Hongrui Chen

Hongrui Chen, Holden Lee, Jianfeng Lu

We give an improved theoretical analysis of score-based generative modeling. Under a score estimate with small $L^2$ error (averaged across timesteps), we provide efficient convergence guarantees for any data distribution with second-order moment, by either employing early stopping or assuming smoothness condition on the score function of the data distribution. Our result does not rely on any log-concavity or functional inequality assumption and has a logarithmic dependence on the smoothness. In particular, we show that under only a finite second moment condition, approximating the following in reverse KL divergence in $ε$-accuracy can be done in $\tilde O\left(\frac{d \log (1/δ)}ε\right)$ steps: 1) the variance-$δ$ Gaussian perturbation of any data distribution; 2) data distributions with $1/δ$-smooth score functions. Our analysis also provides a quantitative comparison between different discrete approximations and may guide the choice of discretization points in practice.

Ruotong Wang, Hongrui Chen, Zihao Zhu et al.

Deep neural networks (DNNs) can be manipulated to exhibit specific behaviors when exposed to specific trigger patterns, without affecting their performance on benign samples, dubbed \textit{backdoor attack}. Currently, implementing backdoor attacks in physical scenarios still faces significant challenges. Physical attacks are labor-intensive and time-consuming, and the triggers are selected in a manual and heuristic way. Moreover, expanding digital attacks to physical scenarios faces many challenges due to their sensitivity to visual distortions and the absence of counterparts in the real world. To address these challenges, we define a novel trigger called the \textbf{V}isible, \textbf{S}emantic, \textbf{S}ample-Specific, and \textbf{C}ompatible (VSSC) trigger, to achieve effective, stealthy and robust simultaneously, which can also be effectively deployed in the physical scenario using corresponding objects. To implement the VSSC trigger, we propose an automated pipeline comprising three modules: a trigger selection module that systematically identifies suitable triggers leveraging large language models, a trigger insertion module that employs generative models to seamlessly integrate triggers into images, and a quality assessment module that ensures the natural and successful insertion of triggers through vision-language models. Extensive experimental results and analysis validate the effectiveness, stealthiness, and robustness of the VSSC trigger. It can not only maintain robustness under visual distortions but also demonstrates strong practicality in the physical scenario. We hope that the proposed VSSC trigger and implementation approach could inspire future studies on designing more practical triggers in backdoor attacks.

Hongrui Chen, Aditya Joglekar, Kate S. Whitefoot et al.

We propose a neural network-based approach to topology optimization that aims to reduce the use of support structures in additive manufacturing. Our approach uses a network architecture that allows the simultaneous determination of an optimized: (1) part segmentation, (2) the topology of each part, and (3) the build direction of each part that collectively minimize the amount of support structure. Through training, the network learns a material density and segment classification in the continuous 3D space. Given a problem domain with prescribed load and displacement boundary conditions, the neural network takes as input 3D coordinates of the voxelized domain as training samples and outputs a continuous density field. Since the neural network for topology optimization learns the density distribution field, analytical solutions to the density gradient can be obtained from the input-output relationship of the neural network. We demonstrate our approach on several compliance minimization problems with volume fraction constraints, where support volume minimization is added as an additional criterion to the objective function. We show that simultaneous optimization of part segmentation along with the topology and print angle optimization further reduces the support structure, compared to a combined print angle and topology optimization without segmentation.

Hongrui Chen, Jihao Long, Lei Wu

In this work, we analyze the learnability of reproducing kernel Hilbert spaces (RKHS) under the $L^\infty$ norm, which is critical for understanding the performance of kernel methods and random feature models in safety- and security-critical applications. Specifically, we relate the $L^\infty$ learnability of a RKHS to the spectrum decay of the associate kernel and both lower bounds and upper bounds of the sample complexity are established. In particular, for dot-product kernels on the sphere, we identify conditions when the $L^\infty$ learning can be achieved with polynomial samples. Let $d$ denote the input dimension and assume the kernel spectrum roughly decays as $λ_k\sim k^{-1-β}$ with $β>0$. We prove that if $β$ is independent of the input dimension $d$, then functions in the RKHS can be learned efficiently under the $L^\infty$ norm, i.e., the sample complexity depends polynomially on $d$. In contrast, if $β=1/\mathrm{poly}(d)$, then the $L^\infty$ learning requires exponentially many samples.

Hongrui Chen, Lexing Ying

Diffusion models have achieved huge empirical success in data generation tasks. Recently, some efforts have been made to adapt the framework of diffusion models to discrete state space, providing a more natural approach for modeling intrinsically discrete data, such as language and graphs. This is achieved by formulating both the forward noising process and the corresponding reversed process as Continuous Time Markov Chains (CTMCs). In this paper, we investigate the theoretical properties of the discrete diffusion model. Specifically, we introduce an algorithm leveraging the uniformization of continuous Markov chains, implementing transitions on random time points. Under reasonable assumptions on the learning of the discrete score function, we derive Total Variation distance and KL divergence guarantees for sampling from any distribution on a hypercube. Our results align with state-of-the-art achievements for diffusion models in $\mathbb{R}^d$ and further underscore the advantages of discrete diffusion models in comparison to the $\mathbb{R}^d$ setting.

Hongrui Chen, Xingchen Liu, Levent Burak Kara

A long-standing challenge is designing multi-scale structures with good connectivity between cells while optimizing each cell to reach close to the theoretical performance limit. We propose a new method for direct multi-scale topology optimization using neural networks. Our approach focuses on inverse homogenization that seamlessly maintains compatibility across neighboring microstructure cells. Our approach consists of a topology neural network that optimizes the microstructure shape and distribution across the design domain as a continuous field. Each microstructure cell is optimized based on a specified elasticity tensor that also accommodates in-plane rotations. The neural network takes as input the local coordinates within a cell to represent the density distribution within a cell, as well as the global coordinates of each cell to design spatially varying microstructure cells. As such, our approach models an n-dimensional multi-scale optimization problem as a 2n-dimensional inverse homogenization problem using neural networks. During the inverse homogenization of each unit cell, we extend the boundary of each cell by scaling the input coordinates such that the boundaries of neighboring cells are combined. Inverse homogenization on the combined cell improves connectivity. We demonstrate our method through the design and optimization of graded multi-scale structures.

Hongrui Chen, Aditya Joglekar, Levent Burak Kara

We propose conditioning field initialization for neural network based topology optimization. In this work, we focus on (1) improving upon existing neural network based topology optimization, (2) demonstrating that by using a prior initial field on the unoptimized domain, the efficiency of neural network based topology optimization can be further improved. Our approach consists of a topology neural network that is trained on a case by case basis to represent the geometry for a single topology optimization problem. It takes in domain coordinates as input to represent the density at each coordinate where the topology is represented by a continuous density field. The displacement is solved through a finite element solver. We employ the strain energy field calculated on the initial design domain as an additional conditioning field input to the neural network throughout the optimization. The addition of the strain energy field input improves the convergence speed compared to standalone neural network based topology optimization.

Hongrui Chen, Jihao Long, Lei Wu

We consider the problem of learning functions within the $\mathcal{F}_{p,π}$ and Barron spaces, which play crucial roles in understanding random feature models (RFMs), two-layer neural networks, as well as kernel methods. Leveraging tools from information-based complexity (IBC), we establish a dual equivalence between approximation and estimation, and then apply it to study the learning of the preceding function spaces. The duality allows us to focus on the more tractable problem between approximation and estimation. To showcase the efficacy of our duality framework, we delve into two important but under-explored problems: 1) Random feature learning beyond kernel regime: We derive sharp bounds for learning $\mathcal{F}_{p,π}$ using RFMs. Notably, the learning is efficient without the curse of dimensionality for $p>1$. This underscores the extended applicability of RFMs beyond the traditional kernel regime, since $\mathcal{F}_{p,π}$ with $p<2$ is strictly larger than the corresponding reproducing kernel Hilbert space (RKHS) where $p=2$. 2) The $L^\infty$ learning of RKHS: We establish sharp, spectrum-dependent characterizations for the convergence of $L^\infty$ learning error in both noiseless and noisy settings. Surprisingly, we show that popular kernel ridge regression can achieve near-optimal performance in $L^\infty$ learning, despite it primarily minimizing square loss. To establish the aforementioned duality, we introduce a type of IBC, termed $I$-complexity, to measure the size of a function class. Notably, $I$-complexity offers a tight characterization of learning in noiseless settings, yields lower bounds comparable to Le Cam's in noisy settings, and is versatile in deriving upper bounds. We believe that our duality framework holds potential for broad application in learning analysis across more scenarios.

Aditya Joglekar, Hongrui Chen, Levent Burak Kara

We propose a direct mesh-free method for performing topology optimization by integrating a density field approximation neural network with a displacement field approximation neural network. We show that this direct integration approach can give comparable results to conventional topology optimization techniques, with an added advantage of enabling seamless integration with post-processing software, and a potential of topology optimization with objectives where meshing and Finite Element Analysis (FEA) may be expensive or not suitable. Our approach (DMF-TONN) takes in as inputs the boundary conditions and domain coordinates and finds the optimum density field for minimizing the loss function of compliance and volume fraction constraint violation. The mesh-free nature is enabled by a physics-informed displacement field approximation neural network to solve the linear elasticity partial differential equation and replace the FEA conventionally used for calculating the compliance. We show that using a suitable Fourier Features neural network architecture and hyperparameters, the density field approximation neural network can learn the weights to represent the optimal density field for the given domain and boundary conditions, by directly backpropagating the loss gradient through the displacement field approximation neural network, and unlike prior work there is no requirement of a sensitivity filter, optimality criterion method, or a separate training of density network in each topology optimization iteration.