This paper presents new theoretical insights into how diffusion generative models adapt to low-dimensional structure in data distributions. We study two widely used samplers — the denoising diffusion probabilistic model (DDPM) and the denoising diffusion implicit model (DDIM) — and analyze their convergence behavior under the assumption of accurate score estimates. Our main result shows that both DDPM and DDIM require at most $O(k/\varepsilon)$ iterations (up to logarithmic factors) to generate samples that are $\varepsilon$-close to the target distribution in total variation distance, where $k$ captures an intrinsic low-dimensional structure of the distribution. Importantly, our theory holds without assuming smoothness or log-concavity. These results provide the first rigorous guarantees for the low-dimensional adaptation capability of DDIM-type samplers, and significantly improve upon prior TV-based convergence bounds for DDPM. Our analysis also highlights the role of discretization coefficients in exploiting low-dimensional structure, and establishes lower bounds that justify the optimality of commonly used parameter choices originally proposed by Ho et al. (2020); Song et al. (2020).

We study Q-learning with Polyak-Ruppert averaging (a.k.a., averaged Q-learning) in a discounted markov decision process in synchronous and tabular settings. Under a Lipschitz condition, we establish a functional central limit theorem for the averaged iteration $\bar{\mathbf{Q}}_T$ and show that its standardized partial-sum process converges weakly to a rescaled Brownian motion. The FCLT implies a fully online inference method for reinforcement learning. Furthermore, we show that $\bar{\mathbf{Q}}_T$ is the regular asymptotically linear (RAL) estimator for the optimal Q-value function $\mathbf{Q}^*$ that has the most efficient influence function. We present a nonasymptotic analysis for the $\ell_{\infty}$ error, $\mathbb{E}\|\bar{\mathbf{Q}}_T-\mathbf{Q}^*\|_{\infty}$, showing that it matches the instance-dependent lower bound for polynomial step sizes. Similar results are provided for entropy-regularized Q-Learning without the Lipschitz condition.

The crux of text-to-image synthesis stems from the difficulty of preserving the cross-modality semantic consistency between the input text and the synthesized image. Typical methods, which seek to model the text-to-image mapping directly, could only capture keywords in the text that indicates common objects or actions but fail to learn their spatial distribution patterns. An effective way to circumvent this limitation is to generate an image layout as guidance, which is attempted by a few methods. Nevertheless, these methods fail to generate practically effective layouts due to the diversity of input text and object location. In this paper we push for effective modeling in both text-to-layout generation and layout-to-image synthesis. Specifically, we formulate the text-to-layout generation as a sequence-to-sequence modeling task, and build our model upon Transformer to learn the spatial relationships between objects by modeling the sequential dependencies between them. In the stage of layout-to-image synthesis, we focus on learning the textual-visual semantic alignment per object in the layout to precisely incorporate the input text into the layout-to-image synthesizing process. To evaluate the quality of generated layout, we design a new metric specifically, dubbed Layout Quality Score, which considers both the absolute distribution errors of bounding boxes in the layout and the mutual spatial relationships between them. Extensive experiments on three datasets demonstrate the superior performance of our method over state-of-the-art methods on both predicting the layout and synthesizing the image from the given text.

We analyze the novel Local SGD in federated Learning, a multi-round estimation procedure that uses intermittent communication to improve communication efficiency. Under a $2{+}\delta$ moment condition on stochastic gradients, we first establish a {\it functional central limit theorem} that shows the averaged iterates of Local SGD converge weakly to a rescaled Brownian motion. We next provide two iterative inference methods: the {\it plug-in} and the {\it random scaling}. Random scaling constructs an asymptotically pivotal statistic for inference by using the information along the whole Local SGD path. Both the methods are communication efficient and applicable to online data. Our results show that Local SGD simultaneously achieves both statistical efficiency and communication efficiency.

In this paper, we consider linearly constrained stochastic approximation problems with federated learning (FL) as a special case. We propose a stochastic approximation algorithm named by LPSA with probabilistic projections to ensure feasibility so that projections are performed with probability $p_n$ at the $n$-th iteration. Considering a specific family of the probability $p_n$ and step size $\eta_n$, we analyze our algorithm from an asymptotic and continuous perspective. Using a novel jump diffusion approximation, we show that the trajectories consisting of properly rescaled last iterates weakly converge to the solution of specific SDEs. By analyzing the SDEs, we identify the asymptotic behaviors of LPSA for different choices of $(p_n, \eta_n)$. We find the algorithm presents an intriguing asymptotic bias-variance trade-off according to the relative magnitude of $p_n$ w.r.t. $\eta_n$. It provides insights on how to choose appropriate $\{(p_n, \eta_n)\}_{n \geq 1}$ to minimize the projection complexity.

Reconstructing 3D virtual face from a single image has a wide range of applications in virtual reality. Existing approaches synthesize plausible reconstructed virtual faces, however, eye gaze information is usually ignored, which is critical in human-computer interaction. In this paper, we propose to reconstruct 3D virtual face with eye gaze information from a single image. The main challenges lie in two aspects, one is the low reconstruction quality in the eye region, the other one is the lack of an efficient method to obtain precise eye gaze information. To address these problems, we decompose this task into two key steps, i.e., 3D face reconstruction with precise eye region and eye contact guided facial-rotation for eye gaze information. The first step is designed for precise eye region reconstruction through joint optimization on 3D face/eye shapes and textures. The second step consists of two parts: eye contact discriminator and automatic eye contact search algorithm via gradient-based optimization to perform both eye contact and gaze estimation simultaneously. Extensive experiments on different tasks demonstrate the significant gain of the proposed approach, achieving an MSE of (30%), an SSIM of (17.85%), and a PSNR of (8.4%). It also produces lower angular errors (63.01%) in the gaze estimation task compared with human annotations.

In this paper we show that generative adversarial networks (GANs) without restriction on the discriminative function space commonly suffer from the problem that the gradient produced by the discriminator is uninformative to guide the generator. By contrast, Wasserstein GAN (WGAN), where the discriminative function is restricted to 1-Lipschitz, does not suffer from such a gradient uninformativeness problem. We further show in the paper that the model with a compact dual form of Wasserstein distance, where the Lipschitz condition is relaxed, may also theoretically suffer from this issue. This implies the importance of Lipschitz condition and motivates us to study the general formulation of GANs with Lipschitz constraint, which leads to a new family of GANs that we call Lipschitz GANs (LGANs). We show that LGANs guarantee the existence and uniqueness of the optimal discriminative function as well as the existence of a unique Nash equilibrium. We prove that LGANs are generally capable of eliminating the gradient uninformativeness problem. According to our empirical analysis, LGANs are more stable and generate consistently higher quality samples compared with WGAN.

In this paper, under the situation of multiple interference regions, an optimal antenna placement problem for a distributed Multi-Input Multi-Output (MIMO) radar is studied. Considering multiple interference regions, we solve the antenna placement problem by utilizing antenna placement method based on Multi-Objective Particle Swarm Optimization (MOPSO). However, it is not clear when to stop the iteration for which no knowledge about the optimum result is available. Hence, computational resource may be wasted over iterations. Nevertheless, time and computational resource is limited in real application. Therefore, to obtain the optimal placement result with limited time and computational resource, an iteration convergence criterion based on interval distance is proposed. The iteration convergence criterion can be used to stop the optimization process efficiently when the optimal antenna placement algorithm reaches the desired convergence level. Finally, numerical results are provided to verify the validity of the proposed algorithm.