On the Expressive Power of Modern Hopfield Networks
This work addresses a foundational gap in deep learning theory for researchers and practitioners, providing insights to optimize MHN integration, though it is incremental in extending known complexity theory to MHNs.
The paper tackled the problem of understanding the computational expressiveness of modern Hopfield networks (MHNs) by establishing rigorous theoretical bounds, showing that MHNs are DLOGTIME-uniform TC^0 and cannot solve NC^1-hard problems like undirected graph connectivity and tree isomorphism unless TC^0 = NC^1.
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.