Xiaoyu Li

Yekun Ke, Xiaoyu Li, Yingyu Liang et al.

Understanding the expressive ability of a specific model is essential for grasping its capacity limitations. Recently, several studies have established circuit complexity bounds for Transformer architecture. Besides, the Visual AutoRegressive (VAR) model has risen to be a prominent method in the field of image generation, outperforming previous techniques, such as Diffusion Transformers, in generating high-quality images. We investigate the circuit complexity of the VAR model and establish a bound in this study. Our primary result demonstrates that the VAR model is equivalent to a simulation by a uniform $\mathsf{TC}^0$ threshold circuit with hidden dimension $d \leq O(n)$ and $\mathrm{poly}(n)$ precision. This is the first study to rigorously highlight the limitations in the expressive power of VAR models despite their impressive performance. We believe our findings will offer valuable insights into the inherent constraints of these models and guide the development of more efficient and expressive architectures in the future.

Xiaoyu Li, Yuanpeng Li, Yingyu Liang et al.

Modern Hopfield networks (MHNs) have emerged as powerful tools in deep learning, capable of replacing components such as pooling layers, LSTMs, and attention mechanisms. Recent advancements have enhanced their storage capacity, retrieval speed, and error rates. However, the fundamental limits of their computational expressiveness remain unexplored. Understanding the expressive power of MHNs is crucial for optimizing their integration into deep learning architectures. In this work, we establish rigorous theoretical bounds on the computational capabilities of MHNs using circuit complexity theory. Our key contribution is that we show that MHNs are $\mathsf{DLOGTIME}$-uniform $\mathsf{TC}^0$. Hence, unless $\mathsf{TC}^0 = \mathsf{NC}^1$, a $\mathrm{poly}(n)$-precision modern Hopfield networks with a constant number of layers and $O(n)$ hidden dimension cannot solve $\mathsf{NC}^1$-hard problems such as the undirected graph connectivity problem and the tree isomorphism problem. We also extended our results to Kernelized Hopfield Networks. These results demonstrate the limitation in the expressive power of the modern Hopfield networks. Moreover, Our theoretical analysis provides insights to guide the development of new Hopfield-based architectures.

Yifang Chen, Xiaoyu Li, Yingyu Liang et al.

We investigate the fundamental limits of transformer-based foundation models, extending our analysis to include Visual Autoregressive (VAR) transformers. VAR represents a big step toward generating images using a novel, scalable, coarse-to-fine ``next-scale prediction'' framework. These models set a new quality bar, outperforming all previous methods, including Diffusion Transformers, while having state-of-the-art performance for image synthesis tasks. Our primary contributions establish that, for single-head VAR transformers with a single self-attention layer and single interpolation layer, the VAR Transformer is universal. From the statistical perspective, we prove that such simple VAR transformers are universal approximators for any image-to-image Lipschitz functions. Furthermore, we demonstrate that flow-based autoregressive transformers inherit similar approximation capabilities. Our results provide important design principles for effective and computationally efficient VAR Transformer strategies that can be used to extend their utility to more sophisticated VAR models in image generation and other related areas.

Chengyue Gong, Yekun Ke, Xiaoyu Li et al.

The expressive power and computational complexity of deep visual generative models, such as flow-based and autoregressive (AR) models, have gained considerable interest for their wide-ranging applications in generative tasks. However, the theoretical characterization of their expressiveness through the lens of circuit complexity remains underexplored, particularly for the state-of-the-art architecture like FlowAR proposed by [Ren et al., 2024], which integrates flow-based and autoregressive mechanisms. This gap limits our understanding of their inherent computational limits and practical efficiency. In this study, we address this gap by analyzing the circuit complexity of the FlowAR architecture. We demonstrate that when the largest feature map produced by the FlowAR model has dimensions $n \times n \times c$, the FlowAR model is simulable by a family of threshold circuits $\mathsf{TC}^0$, which have constant depth $O(1)$ and polynomial width $\mathrm{poly}(n)$. This is the first study to rigorously highlight the limitations in the expressive power of FlowAR models. Furthermore, we identify the conditions under which the FlowAR model computations can achieve almost quadratic time. To validate our theoretical findings, we present efficient model variant constructions based on low-rank approximations that align with the derived criteria. Our work provides a foundation for future comparisons with other generative paradigms and guides the development of more efficient and expressive implementations.

Yifang Chen, Xuyang Guo, Xiaoyu Li et al.

Recently, Large Language Models (LLMs) have achieved remarkable success. A key factor behind this success is the scaling law observed by OpenAI. Specifically, for models with Transformer architecture, the test loss exhibits a power-law relationship with model size, dataset size, and the amount of computation used in training, demonstrating trends that span more than seven orders of magnitude. This scaling law challenges traditional machine learning wisdom, notably the Oscar Scissors principle, which suggests that an overparametrized algorithm will overfit the training datasets, resulting in poor test performance. Recent research has also identified the scaling law in simpler machine learning contexts, such as linear regression. However, fully explaining the scaling law in large practical models remains an elusive goal. In this work, we advance our understanding by demonstrating that the scaling law phenomenon extends to multiple regression and kernel regression settings, which are significantly more expressive and powerful than linear methods. Our analysis provides deeper insights into the scaling law, potentially enhancing our understanding of LLMs.

Xiaoyu Li, Zhao Song, Jiahao Zhang

The explosive growth of AI research has driven paper submissions at flagship AI conferences to unprecedented levels, necessitating many venues in 2025 (e.g., CVPR, ICCV, KDD, AAAI, IJCAI, WSDM) to enforce strict per-author submission limits and to desk-reject any excess papers by simple ID order. While this policy helps reduce reviewer workload, it may unintentionally discard valuable papers and penalize authors' efforts. In this paper, we ask an essential research question on whether it is possible to follow submission limits while minimizing needless rejections. We first formalize the current desk-rejection policies as an optimization problem, and then develop a practical algorithm based on linear programming relaxation and a rounding scheme. Under extensive evaluation on 11 years of real-world ICLR (International Conference on Learning Representations) data, our method preserves up to $19.23\%$ more papers without violating any author limits. Moreover, our algorithm is highly efficient in practice, with all results on ICLR data computed within at most 53.64 seconds. Our work provides a simple and practical desk-rejection strategy that significantly reduces unnecessary rejections, demonstrating strong potential to improve current CS conference submission policies.

Bo Chen, Xiaoyu Li, Yekun Ke et al.

A fundamental challenge in Visual Autoregressive models is the substantial memory overhead required during inference to store previously generated representations. Despite various attempts to mitigate this issue through compression techniques, prior works have not explicitly formalized the problem of KV-cache compression in this context. In this work, we take the first step in formally defining the KV-cache compression problem for Visual Autoregressive transformers. We then establish a fundamental negative result, proving that any mechanism for sequential visual token generation under attention-based architectures must use at least $Ω(n^2 d)$ memory, when $d = Ω(\log n)$, where $n$ is the number of tokens generated and $d$ is the embedding dimensionality. This result demonstrates that achieving truly sub-quadratic memory usage is impossible without additional structural constraints. Our proof is constructed via a reduction from a computational lower bound problem, leveraging randomized embedding techniques inspired by dimensionality reduction principles. Finally, we discuss how sparsity priors on visual representations can influence memory efficiency, presenting both impossibility results and potential directions for mitigating memory overhead.