On the Computational Capability of Graph Neural Networks: A Circuit Complexity Bound Perspective

Graph Neural Networks (GNNs) have become the standard approach for learning and reasoning over relational data, leveraging the message-passing mechanism that iteratively propagates node embeddings through graph structures. While GNNs have achieved significant empirical success, their theoretical limitations remain an active area of research. Existing studies primarily focus on characterizing GNN expressiveness through Weisfeiler-Lehman (WL) graph isomorphism tests. In this paper, we take a fundamentally different approach by exploring the computational limitations of GNNs through the lens of circuit complexity. Specifically, we analyze the circuit complexity of common GNN architectures and prove that under constraints of constant-depth layers, linear or sublinear embedding sizes, and polynomial precision, GNNs cannot solve key problems such as graph connectivity and graph isomorphism unless $\mathsf{TC}^0 = \mathsf{NC}^1$. These results reveal the intrinsic expressivity limitations of GNNs behind their empirical success and introduce a novel framework for analyzing GNN expressiveness that can be extended to a broader range of GNN models and graph decision problems.

On Computational Limits and Provably Efficient Criteria of Visual Autoregressive Models: A Fine-Grained Complexity Analysis

Recently, Visual Autoregressive ($\mathsf{VAR}$) Models introduced a groundbreaking advancement in the field of image generation, offering a scalable approach through a coarse-to-fine "next-scale prediction" paradigm. However, the state-of-the-art algorithm of $\mathsf{VAR}$ models in [Tian, Jiang, Yuan, Peng and Wang, NeurIPS 2024] takes $O(n^4)$ time, which is computationally inefficient. In this work, we analyze the computational limits and efficiency criteria of $\mathsf{VAR}$ Models through a fine-grained complexity lens. Our key contribution is identifying the conditions under which $\mathsf{VAR}$ computations can achieve sub-quadratic time complexity. Specifically, we establish a critical threshold for the norm of input matrices used in $\mathsf{VAR}$ attention mechanisms. Above this threshold, assuming the Strong Exponential Time Hypothesis ($\mathsf{SETH}$) from fine-grained complexity theory, a sub-quartic time algorithm for $\mathsf{VAR}$ models is impossible. To substantiate our theoretical findings, we present efficient constructions leveraging low-rank approximations that align with the derived criteria. This work initiates the study of the computational efficiency of the $\mathsf{VAR}$ model from a theoretical perspective. Our technique will shed light on advancing scalable and efficient image generation in $\mathsf{VAR}$ frameworks.

Circuit Complexity Bounds for Visual Autoregressive Model

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.

Theoretical Constraints on the Expressive Power of $\mathsf{RoPE}$-based Tensor Attention Transformers

Tensor Attention extends traditional attention mechanisms by capturing high-order correlations across multiple modalities, addressing the limitations of classical matrix-based attention. Meanwhile, Rotary Position Embedding ($\mathsf{RoPE}$) has shown superior performance in encoding positional information in long-context scenarios, significantly enhancing transformer models' expressiveness. Despite these empirical successes, the theoretical limitations of these technologies remain underexplored. In this study, we analyze the circuit complexity of Tensor Attention and $\mathsf{RoPE}$-based Tensor Attention, showing that with polynomial precision, constant-depth layers, and linear or sublinear hidden dimension, they cannot solve fixed membership problems or $(A_{F,r})^*$ closure problems, under the assumption that $\mathsf{TC}^0 \neq \mathsf{NC}^1$. These findings highlight a gap between the empirical performance and theoretical constraints of Tensor Attention and $\mathsf{RoPE}$-based Tensor Attention Transformers, offering insights that could guide the development of more theoretically grounded approaches to Transformer model design and scaling.

Fast Gradient Computation for RoPE Attention in Almost Linear Time

The Rotary Position Embedding (RoPE) mechanism has become a powerful enhancement to the Transformer architecture, which enables models to capture token relationships when encoding positional information. However, the RoPE mechanisms make the computations of attention mechanisms more complicated, which makes efficient algorithms challenging. Earlier research introduced almost linear time, i.e., $n^{1+o(1)}$ where $n$ is the number of input tokens, algorithms for the forward computation under specific parameter settings. However, achieving a subquadratic time algorithm for other parameter regimes remains impossible unless the widely accepted Strong Exponential Time Hypothesis (SETH) is disproven. In this work, we develop the first almost linear time algorithm for backward computations in the RoPE-based attention under bounded entries. Our approach builds on recent advancements in fast RoPE attention computations, utilizing a novel combination of the polynomial method and the Fast Fourier Transform. Furthermore, we show that with lower bounds derived from the SETH, the bounded entry condition is necessary for subquadratic performance.

The Computational Limits of State-Space Models and Mamba via the Lens of Circuit Complexity

In this paper, we analyze the computational limitations of Mamba and State-space Models (SSMs) by using the circuit complexity framework. Despite Mamba's stateful design and recent attention as a strong candidate to outperform Transformers, we have demonstrated that both Mamba and SSMs with $\mathrm{poly}(n)$-precision and constant-depth layers reside within the $\mathsf{DLOGTIME}$-uniform $\mathsf{TC}^0$ complexity class. This result indicates Mamba has the same computational capabilities as Transformer theoretically, and it cannot solve problems like arithmetic formula problems, boolean formula value problems, and permutation composition problems if $\mathsf{TC}^0 \neq \mathsf{NC}^1$. Therefore, it challenges the assumption Mamba is more computationally expressive than Transformers. Our contributions include rigorous proofs showing that Selective SSM and Mamba architectures can be simulated by $\mathsf{DLOGTIME}$-uniform $\mathsf{TC}^0$ circuits, and they cannot solve problems outside $\mathsf{TC}^0$.

Curse of Attention: A Kernel-Based Perspective for Why Transformers Fail to Generalize on Time Series Forecasting and Beyond

The application of transformer-based models on time series forecasting (TSF) tasks has long been popular to study. However, many of these works fail to beat the simple linear residual model, and the theoretical understanding of this issue is still limited. In this work, we propose the first theoretical explanation of the inefficiency of transformers on TSF tasks. We attribute the mechanism behind it to {\bf Asymmetric Learning} in training attention networks. When the sign of the previous step is inconsistent with the sign of the current step in the next-step-prediction time series, attention fails to learn the residual features. This makes it difficult to generalize on out-of-distribution (OOD) data, especially on the sign-inconsistent next-step-prediction data, with the same representation pattern, whereas a linear residual network could easily accomplish it. We hope our theoretical insights provide important necessary conditions for designing the expressive and efficient transformer-based architecture for practitioners.

On the Expressive Power of Modern Hopfield Networks

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.

Circuit Complexity Bounds for RoPE-based Transformer Architecture

Characterizing the express power of the Transformer architecture is critical to understanding its capacity limits and scaling law. Recent works provide the circuit complexity bounds to Transformer-like architecture. On the other hand, Rotary Position Embedding ($\mathsf{RoPE}$) has emerged as a crucial technique in modern large language models, offering superior performance in capturing positional information compared to traditional position embeddings, which shows great potential in application prospects, particularly for the long context scenario. Empirical evidence also suggests that $\mathsf{RoPE}$-based Transformer architectures demonstrate greater generalization capabilities compared to conventional Transformer models. In this work, we establish a tighter circuit complexity bound for Transformers with $\mathsf{RoPE}$ attention. Our key contribution is that we show that unless $\mathsf{TC}^0 = \mathsf{NC}^1$, a $\mathsf{RoPE}$-based Transformer with $\mathrm{poly}(n)$-precision, $O(1)$ layers, hidden dimension $d \leq O(n)$ cannot solve the arithmetic problem or the Boolean formula value problem. This result significantly demonstrates the fundamental limitation of the expressivity of the $\mathsf{RoPE}$-based Transformer architecture, although it achieves giant empirical success. Our theoretical framework not only establishes tighter complexity bounds but also may instruct further work on the $\mathsf{RoPE}$-based Transformer.

Advancing the Understanding of Fixed Point Iterations in Deep Neural Networks: A Detailed Analytical Study

Recent empirical studies have identified fixed point iteration phenomena in deep neural networks, where the hidden state tends to stabilize after several layers, showing minimal change in subsequent layers. This observation has spurred the development of practical methodologies, such as accelerating inference by bypassing certain layers once the hidden state stabilizes, selectively fine-tuning layers to modify the iteration process, and implementing loops of specific layers to maintain fixed point iterations. Despite these advancements, the understanding of fixed point iterations remains superficial, particularly in high-dimensional spaces, due to the inadequacy of current analytical tools. In this study, we conduct a detailed analysis of fixed point iterations in a vector-valued function modeled by neural networks. We establish a sufficient condition for the existence of multiple fixed points of looped neural networks based on varying input regions. Additionally, we expand our examination to include a robust version of fixed point iterations. To demonstrate the effectiveness and insights provided by our approach, we provide case studies that looped neural networks may exist $2^d$ number of robust fixed points under exponentiation or polynomial activation functions, where $d$ is the feature dimension. Furthermore, our preliminary empirical results support our theoretical findings. Our methodology enriches the toolkit available for analyzing fixed point iterations of deep neural networks and may enhance our comprehension of neural network mechanisms.