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.

We present the first application of modern Hopfield networks to the problem of portfolio optimization. We performed an extensive study based on combinatorial purged cross-validation over several datasets and compared our results to both traditional and deep-learning-based methods for portfolio selection. Compared to state-of-the-art deep-learning methods such as Long-Short Term Memory networks and Transformers, we find that the proposed approach performs on par or better, while providing faster training times and better stability. Our results show that Modern Hopfield Networks represent a promising approach to portfolio optimization, allowing for an efficient, scalable, and robust solution for asset allocation, risk management, and dynamic rebalancing.

Current event-/frame-event based trackers undergo evaluation on short-term tracking datasets, however, the tracking of real-world scenarios involves long-term tracking, and the performance of existing tracking algorithms in these scenarios remains unclear. In this paper, we first propose a new long-term and large-scale frame-event single object tracking dataset, termed FELT. It contains 742 videos and 1,594,474 RGB frames and event stream pairs and has become the largest frame-event tracking dataset to date. We re-train and evaluate 15 baseline trackers on our dataset for future works to compare. More importantly, we find that the RGB frames and event streams are naturally incomplete due to the influence of challenging factors and spatially sparse event flow. In response to this, we propose a novel associative memory Transformer network as a unified backbone by introducing modern Hopfield layers into multi-head self-attention blocks to fuse both RGB and event data. Extensive experiments on RGB-Event (FELT), RGB-Thermal (RGBT234, LasHeR), and RGB-Depth (DepthTrack) datasets fully validated the effectiveness of our model. The dataset and source code can be found at \url{https://github.com/Event-AHU/FELT_SOT_Benchmark}.

An essential step in the discovery of new drugs and materials is the synthesis of a molecule that exists so far only as an idea to test its biological and physical properties. While computer-aided design of virtual molecules has made large progress, computer-assisted synthesis planning (CASP) to realize physical molecules is still in its infancy and lacks a performance level that would enable large-scale molecule discovery. CASP supports the search for multi-step synthesis routes, which is very challenging due to high branching factors in each synthesis step and the hidden rules that govern the reactions. The central and repeatedly applied step in CASP is reaction prediction, for which machine learning methods yield the best performance. We propose a novel reaction prediction approach that uses a deep learning architecture with modern Hopfield networks (MHNs) that is optimized by contrastive learning. An MHN is an associative memory that can store and retrieve chemical reactions in each layer of a deep learning architecture. We show that our MHN contrastive learning approach enables few- and zero-shot learning for reaction prediction which, in contrast to previous methods, can deal with rare, single, or even no training example(s) for a reaction. On a well established benchmark, our MHN approach pushes the state-of-the-art performance up by a large margin as it improves the predictive top-100 accuracy from $0.858\pm0.004$ to $0.959\pm0.004$. This advance might pave the way to large-scale molecule discovery.

The transformer and BERT models pushed the performance on NLP tasks to new levels via their attention mechanism. We show that this attention mechanism is the update rule of a modern Hopfield network with continuous states. This new Hopfield network can store exponentially (with the dimension) many patterns,converges with one update, and has exponentially small retrieval errors. The number of stored patterns must be traded off against convergence speed and retrieval error. Transformers learn an attention mechanism by constructing an embedding of patterns and queries into an associative space. Transformer and BERT models operate in their first layers preferably in the global averaging regime, while they operate in higher layers in metastable states.

We show that the transformer attention mechanism is the update rule of a modern Hopfield network with continuous states. This new Hopfield network can store exponentially (with the dimension) many patterns, converges with one update, and has exponentially small retrieval errors. The number of stored patterns is traded off against convergence speed and retrieval error. The new Hopfield network has three types of energy minima (fixed points of the update): (1) global fixed point averaging over all patterns, (2) metastable states averaging over a subset of patterns, and (3) fixed points which store a single pattern. Transformer and BERT models operate in their first layers preferably in the global averaging regime, while they operate in higher layers in metastable states. The gradient in transformers is maximal for metastable states, is uniformly distributed for global averaging, and vanishes for a fixed point near a stored pattern. Using the Hopfield network interpretation, we analyzed learning of transformer and BERT models. Learning starts with attention heads that average and then most of them switch to metastable states. However, the majority of heads in the first layers still averages and can be replaced by averaging, e.g. our proposed Gaussian weighting. In contrast, heads in the last layers steadily learn and seem to use metastable states to collect information created in lower layers. These heads seem to be a promising target for improving transformers. Neural networks with Hopfield networks outperform other methods on immune repertoire classification, where the Hopfield net stores several hundreds of thousands of patterns. We provide a new PyTorch layer called "Hopfield", which allows to equip deep learning architectures with modern Hopfield networks as a new powerful concept comprising pooling, memory, and attention. GitHub: https://github.com/ml-jku/hopfield-layers