We introduce Graph Hopfield Networks, whose energy function couples associative memory retrieval with graph Laplacian smoothing for node classification. Gradient descent on this joint energy yields an iterative update interleaving Hopfield retrieval with Laplacian propagation. Memory retrieval provides regime-dependent benefits: up to 2.0~pp on sparse citation networks and up to 5 pp additional robustness under feature masking; the iterative energy-descent architecture itself is a strong inductive bias, with all variants (including the memory-disabled NoMem ablation) outperforming standard baselines on Amazon co-purchase graphs. Tuning enables graph sharpening for heterophilous benchmarks without architectural changes.
MRC Brain Network Dynamics Unit（MRC脑网络动力学单元）；University of Oxford（牛津大学）
A large number of neural network models of associative memory have been proposed in the literature. These include the classical Hopfield networks (HNs), sparse distributed memories (SDMs), and more recently the modern continuous Hopfield networks (MCHNs), which possesses close links with self-attention in machine learning. In this paper, we propose a general framework for understanding the operation of such memory networks as a sequence of three operations: similarity, separation, and projection. We derive all these memory models as instances of our general framework with differing similarity …
In their recent paper titled "Large Associative Memory Problem in Neurobiology and Machine Learning" [arXiv:2008.06996] the authors gave a biologically plausible microscopic theory from which one can recover many dense associative memory models discussed in the literature. We show that the layers of the recent "MLP-mixer" [arXiv:2105.01601] as well as the essentially equivalent model in [arXiv:2105.02723] are amongst them.
Hopfield-Fenchel-Young Networks: A Unified Framework for Associative Memory Retrieval
Instituto Superior Técnico, Universidade de Lisboa（里斯本大学技术学院）；Instituto de Telecomunicações（电信研究所）；Language Technology Lab, University of Amsterdam（阿姆斯特丹大学语言技术实验室）；Champalimaud Research（恰帕拉姆德研究）；ELLIS Unit Lisbon（里斯本ELLIS单位）
National Institute of Standards and Technology（美国国家标准与技术研究院）
Simulations of complex-valued Hopfield networks based on spin-torque oscillators can recover phase-encoded images. Sequences of memristor-augmented inverters provide tunable delay elements that implement complex weights by phase shifting the oscillatory output of the oscillators. Pseudo-inverse training suffices to store at least 12 images in a set of 192 oscillators, representing 16$\times$12 pixel images. The energy required to recover an image depends on the desired error level. For the oscillators and circuitry considered here, 5 % root mean square deviations from the ideal image require a…
Redundancy Maximization as a Principle of Associative Memory Learning in Hopfield Networks
Complex Systems Theory, Max Planck Institute for Dynamics and Self-Organization（复杂系统理论，马克斯·普朗克动态与自组织研究所）；Faculty of Physics, Institute for the Dynamics of Complex Systems, University of Göttingen（物理系，复杂系统动力学研究所，哥廷根大学）；Göttingen Campus Institute for Dynamics of Biological Networks, University of Göttingen（哥廷根校区生物网络动力学研究所，哥廷根大学）；Champalimaud Centre for the Unknown, Lisbon, Portugal（查帕利马德未知中心，里斯本，葡萄牙）；Max Planck Institute for Dynamics and Self-Organization（马克斯·普朗克动态与自组织研究所）
In this work, we present a Quantum Hopfield Associative Memory (QHAM) and demonstrate its capabilities in simulation and hardware using IBM Quantum Experience. The QHAM is based on a quantum neuron design which can be utilized for many different machine learning applications and can be implemented on real quantum hardware without requiring mid-circuit measurement or reset operations. We analyze the accuracy of the neuron and the full QHAM considering hardware errors via simulation with hardware noise models as well as with implementation on the 15-qubit ibmq_16_melbourne device. The quantum ne…
Collaborative Filtering using Variational Quantum Hopfield Associative Memory
Pasargad Institue of Advanced Innovative Solutions（帕萨格德先进创新解决方案研究所）；Department of Mathematics and Computer Science（数学与计算机科学系）；Iran University of Science and Technology（伊朗科学技术大学）；Qatar Center for Quantum Computing（卡塔尔量子计算中心）；College of Science and Engineering（科学与工程学院）；Hamad Bin Khalifa University（哈马德·本·卡伊夫大学）
Unsupervised learning in a generalized Hopfield associative-memory network is investigated in this work. First, we prove that the (generalized) Hopfield model is equivalent to a semi-restricted Boltzmann machine with a layer of visible neurons and another layer of hidden binary neurons, so it could serve as the building block for a multilayered deep-learning system. We then demonstrate that the Hopfield network can learn to form a faithful internal representation of the observed samples, with the learned memory patterns being prototypes of the input data. Furthermore, we propose a spectral met…
Phase Kernel Lifts Capacity of Dense Associative Memory
Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China（电子科技大学基础与前沿研究院）；School of Physics, University of Electronic Science and Technology of China（电子科技大学物理学院）；Non-classical Information Science Basic Discipline Research Center of Sichuan Province, University of Electronic Science and Technology of China（电子科技大学四川省非经典信息科学基础学科研究中心）
Federal Institute of Education, Science and Technology of Espírito Santo（圣埃斯皮里图联邦教育、科学与技术学院）；University of Campinas（坎皮纳斯大学）
In this paper, we address the stability of a broad class of discrete-time hypercomplex-valued Hopfield-type neural networks. To ensure the neural networks belonging to this class always settle down at a stationary state, we introduce novel hypercomplex number systems referred to as real-part associative hypercomplex number systems. Real-part associative hypercomplex number systems generalize the well-known Cayley-Dickson algebras and real Clifford algebras and include the systems of real numbers, complex numbers, dual numbers, hyperbolic numbers, quaternions, tessarines, and octonions as parti…
MIT-IBM Watson AI Lab（MIT-IBM沃森人工智能实验室）；IBM Research（IBM研究院）；Princeton Neuroscience Institute（普林斯顿神经科学研究所）；Princeton University（普林斯顿大学）
Dense Associative Memories or modern Hopfield networks permit storage and reliable retrieval of an exponentially large (in the dimension of feature space) number of memories. At the same time, their naive implementation is non-biological, since it seemingly requires the existence of many-body synaptic junctions between the neurons. We show that these models are effective descriptions of a more microscopic (written in terms of biological degrees of freedom) theory that has additional (hidden) neurons and only requires two-body interactions between them. For this reason our proposed microscopic …
CY Cergy Paris Université（CY塞尔吉-巴黎大学）；ENSEA（国立高等电子及应用工程师学院）；CNRS（法国国家科学研究中心）；ETIS UMR 8051（信息处理与系统实验室（联合研究单位8051））
In this article, we propose a variational inference formulation of auto-associative memories, allowing us to combine perceptual inference and memory retrieval into the same mathematical framework. In this formulation, the prior probability distribution onto latent representations is made memory dependent, thus pulling the inference process towards previously stored representations. We then study how different neural network approaches to variational inference can be applied in this framework. We compare methods relying on amortized inference such as Variational Auto Encoders and methods relyin…
Thadomal Shahani Engineering College（塔多马尔·沙哈尼工程学院）；Indian Institute of Technology, Bombay（印度理工学院孟买分校）
We propose a novel algorithm for using Hopfield networks to denoise QR codes. Hopfield networks have mostly been used as a noise tolerant memory or to solve difficult combinatorial problems. One of the major drawbacks in their use in noise tolerant associative memory is their low capacity of storage, scaling only linearly with the number of nodes in the network. A larger capacity therefore requires a larger number of nodes, thereby reducing the speed of convergence of the network in addition to increasing hardware costs for acquiring more precise data to be fed to a larger number of nodes. Our…
Hopfield networks and Boltzmann machines (BMs) are fundamental energy-based neural network models. Recent studies on modern Hopfield networks have broaden the class of energy functions and led to a unified perspective on general Hopfield networks including an attention module. In this letter, we consider the BM counterparts of modern Hopfield networks using the associated energy functions, and study their salient properties from a trainability perspective. In particular, the energy function corresponding to the attention module naturally introduces a novel BM, which we refer to as the attentio…
Learning to remember long sequences remains a challenging task for recurrent neural networks. Register memory and attention mechanisms were both proposed to resolve the issue with either high computational cost to retain memory differentiability, or by discounting the RNN representation learning towards encoding shorter local contexts than encouraging long sequence encoding. Associative memory, which studies the compression of multiple patterns in a fixed size memory, were rarely considered in recent years. Although some recent work tries to introduce associative memory in RNN and mimic the en…
As a mathematical model of associative memories, the Hopfield model was now well-established and a lot of studies to reveal the pattern-recalling process have been done from various different approaches. As well-known, a single neuron is itself an uncertain, noisy unit with a finite unnegligible error in the input-output relation. To model the situation artificially, a kind of 'heat bath' that surrounds neurons is introduced. The heat bath, which is a source of noise, is specified by the 'temperature'. Several studies concerning the pattern-recalling processes of the Hopfield model governed by…
Thermodynamics of bidirectional associative memories
Università del Salento（萨伦托大学）；Istituto Nazionale di Fisica Nucleare（国家核物理研究所）；Universidad Complutense de Madrid（马德里康普顿斯大学）；Université Paris-Saclay（巴黎萨克雷大学）；CNRS（法国国家科学研究中心）；INRIA（法国国家信息与自动化研究所）
Multitask algorithms typically use task similarity information as a bias to speed up and improve the performance of learning processes. Tasks are learned jointly, sharing information across them, in order to construct models more accurate than those learned separately over single tasks. In this contribution, we present the first multitask model, to our knowledge, based on Hopfield Networks (HNs), named HoMTask. We show that by appropriately building a unique HN embedding all tasks, a more robust and effective classification model can be learned. HoMTask is a transductive semi-supervised parame…
Associative Memory and Generative Diffusion in the Zero-noise Limit
Computational & Systems Biology, Sloan Kettering Institute, Memorial Sloan Kettering Cancer Center（计算生物学与系统生物学，斯隆凯特灵研究所，纪念斯隆凯特灵癌症中心）；Tri-Institutional Training Program in Computational Biology & Medicine（计算生物学与医学三机构联合培训计划）
Johannes Kepler University Linz（林茨约翰开普勒大学）；Institute of Advanced Research in Artificial Intelligence (IARAI)（高级人工智能研究所（IARAI））；University of Oslo（奥斯陆大学）
We introduce a modern Hopfield network with continuous states and a corresponding update rule. The new Hopfield network can store exponentially (with the dimension of the associative space) many patterns, retrieves the pattern with one update, and has exponentially small retrieval errors. It 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. The new update rule is equivalent to the attention mechanism used in transformers.…
Phases in a class of associative memories via hidden neurons
CyberAgent AI Lab（CyberAgent AI实验室）；RIKEN iTHEMS（RIKEN跨学科理论科学研究所）；Graduate School of Artificial Intelligence and Science, Rikkyo University（立教大学人工智能与科学研究科）
The network embedding task is to represent the node in the network as a low-dimensional vector while incorporating the topological and structural information. Most existing approaches solve this problem by factorizing a proximity matrix, either directly or implicitly. In this work, we introduce a network embedding method from a new perspective, which leverages Modern Hopfield Networks (MHN) for associative learning. Our network learns associations between the content of each node and that node's neighbors. These associations serve as memories in the MHN. The recurrent dynamics of the network m…
Active Adversarial Perturbation-driven Associative Memory Retrieval for RGB-Event Visual Object Tracking
School of Computer Science and Technology, Anhui University（安徽大学计算机科学与技术学院）；School of Electronic and Information Engineering, Anhui University（安徽大学电子信息工程学院）；Peng Cheng Laboratory（鹏城实验室）；Harbin Institute of Technology, Shenzhen（哈尔滨工业大学（深圳））；School of Computer Science, Peking University（北京大学计算机学院）；School of Electronic and Computer Engineering, Shenzhen Graduate School, Peking University（北京大学深圳研究生院电子与计算机工程学院）
Deep Convolutional features extracted from a comprehensive labeled dataset, contain substantial representations which could be effectively used in a new domain. Despite the fact that generic features achieved good results in many visual tasks, fine-tuning is required for pretrained deep CNN models to be more effective and provide state-of-the-art performance. Fine tuning using the backpropagation algorithm in a supervised setting, is a time and resource consuming process. In this paper, we present a new architecture and an approach for unsupervised object recognition that addresses the above m…
John A. Paulson School of Engineering and Applied Sciences（约翰·A·保尔森工程与应用科学学院）；Center for Brain Science（脑科学中心）；Department of Physics（物理系）；MIT-IBM Watson AI Lab, IBM Research（MIT-IBM沃森人工智能实验室，IBM研究院）；Kempner Institute for the Study of Natural and Artificial Intelligence（肯普纳自然与人工智能研究所）
MIT-IBM Watson AI Lab（麻省理工学院-IBM沃森人工智能实验室）；IBM Research（IBM研究院）
Dense Associative Memories or Modern Hopfield Networks have many appealing properties of associative memory. They can do pattern completion, store a large number of memories, and can be described using a recurrent neural network with a degree of biological plausibility and rich feedback between the neurons. At the same time, up until now all the models of this class have had only one hidden layer, and have only been formulated with densely connected network architectures, two aspects that hinder their machine learning applications. This paper tackles this gap and describes a fully recurrent mo…
