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.

翻译：深度视觉生成模型（如基于流的模型和自回归模型）的表达能力与计算复杂性因其在生成任务中的广泛应用而受到广泛关注。然而，从电路复杂度的角度对其表达能力进行理论刻画的研究仍显不足，特别是对于[Ren等人，2024]提出的融合基于流与自回归机制的最新架构FlowAR。这一空白限制了我们对其固有计算极限和实际效率的理解。在本研究中，我们通过分析FlowAR架构的电路复杂度来填补这一空白。我们证明，当FlowAR模型生成的最大特征图维度为$n \times n \times c$时，该模型可由一族阈值电路$\mathsf{TC}^0$模拟，这些电路具有恒定深度$O(1)$和多项式宽度$\mathrm{poly}(n)$。这是首个严格揭示FlowAR模型表达能力局限性的研究。此外，我们确定了FlowAR模型计算可实现近乎二次时间效率的条件。为验证理论发现，我们提出了基于低秩近似的高效模型变体构建方法，这些构建符合所推导的标准。我们的工作为未来与其他生成范式的比较奠定了基础，并指导开发更高效、更具表达能力的实现方案。