Hopfield network

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 neuron and the QHAM are shown to be resilient to noise and require low qubit overhead and gate complexity. We benchmark the QHAM by testing its effective memory capacity and demonstrate its capabilities in the NISQ-era of quantum hardware. This demonstration of the first functional QHAM to be implemented in NISQ-era quantum hardware is a significant step in machine learning at the leading edge of quantum computing. Since the advent of quantum computing, one of the primary applications which has piqued the interest of both academia and industry is quantum machine learning The characteristics of quantum computing are of particular interest for Hopfield network Several studies theorize quantum associative memories with improved capacity \(\frac\) and spin-1 qubits to demonstrate a theoretical associative memory, and a more recent design In the NISQ era of quantum computing In this work, w...

7. Hopfield Network model of associative memory Book chapters See Python classes Hopfield networks can be analyzed mathematically. In this Python exercise we focus on visualization and simulation to develop our intuition about Hopfield dynamics. We provide a couple of functions to easily create patterns, store them in the network and visualize the network dynamics. Check the modules hopfield_network.network, hopfield_network.pattern_tools and hopfield_network.plot_tools to learn the building blocks we provide. 7.1. Getting started: Run the following code. Read the inline comments and check the documentation. The patterns and the flipped pixels are randomly chosen. Therefore the result changes every time you execute this code. Run it several times and change some parameters like nr_patterns and nr_of_flips. % matplotlib inline from neurodynex3.hopfield_network import network , pattern_tools , plot_tools pattern_size = 5 # create an instance of the class HopfieldNetwork hopfield_net = network . HopfieldNetwork ( nr_neurons = pattern_size ** 2 ) # instantiate a pattern factory factory = pattern_tools . PatternFactory ( pattern_size , pattern_size ) # create a checkerboard pattern and add it to the pattern list checkerboard = factory . create_checkerboard () pattern_list = [ checkerboard ] # add random patterns to the list pattern_list . extend ( factory . create_random_pattern_list ( nr_patterns = 3 , on_probability = 0.5 )) plot_tools . plot_pattern_list ( pattern_list ) # h...

Introduced in the 1970s, Hopfield networks were popularised by John Hopfield in 1982. Hopfield networks, for the most part of machine learning history, have been sidelined due to their own shortcomings and introduction of superior architectures such as the Co-creator of ‘Hopfield networks Is All You Need’, the authors introduce a couple of elements that make Hopfield networks interchangeable with the state-of-the-art transformer models. What’s New About Hopfield Networks Source: Hubert Ramsauer et al. The above figure depicts the relation between binary modern Hopfield networks — the new Hopfield network has continuous states, a new update rule, and the transformer. The standard binary Hopfield network has an The 1| Introduction of a new energy function using the log-sum-exp function 2| The state ξ is updated by the following new update rule: 3| The new energy function offers the following, • Global convergence to a local minimum • Exponential storage capacity • Convergence after one update step In this work, the authors have also provided a new PyTorch layer called “Hopfield” which allows equipping deep learning architectures with modern Hopfield networks as new powerful concepts comprising pooling, memory, and attention. Why Use Them At All “The modern Hopfield network gives the same results as the SOTA Transformer.” The modern Hopfield networks were put to use by Hochreiter and his colleagues to find patterns in the immune repertoire of an individual. Their network call...

a, When the number of stored memories significantly exceeds the number of feature neurons, the traditional Hopfield network acquires spin glass local minima that are uncorrelated with the memory vectors (one such minimum is shown in panel a). b, In the dense associative memory model this spin glass transition happens at a much larger number of memories. Thus, even in situations when the number of memories is significantly larger than the number of feature neurons, each memory has a large basin of attraction around it, and there are no spin glass local minima. The memory patterns (vectors ξ μ) are indexed by µ (going from 1 to the number of memory patterns K mem) and each pattern is an N f-dimensional vector. For continuous variables the feature vector x needs to additionally pass through a bounded activation function, such as sigmoid or layer normalization, to ensure that the energy E is bounded from below. Mathematically, the state of the Hopfield network is described by an N f-dimensional vector of features x, which can be either binary or continuous. The temporal evolution of this state vector is governed by an energy function, which has local minima located at a set of K mem memory vectors ξ μ representing the patterns that the network stores in its weights (each pattern is an N f-dimensional vector). When presented with an initial prompt that resembles one of the memory vectors, the energy descent dynamics finds the most similar memory vector based on the similarity b...

Download a PDF of the paper titled Universal Hopfield Networks: A General Framework for Single-Shot Associative Memory Models, by Beren Millidge and 4 other authors Abstract: 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 and separation functions. We extend the mathematical framework of Krotov et al (2020) to express general associative memory models using neural network dynamics with only second-order interactions between neurons, and derive a general energy function that is a Lyapunov function of the dynamics. Finally, using our framework, we empirically investigate the capacity of using different similarity functions for these associative memory models, beyond the dot product similarity measure, and demonstrate empirically that Euclidean or Manhattan distance similarity metrics perform substantially better in practice on many tasks, enabling a more robust retrieval and higher memory capacity than existing models.