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Associative recall of memory without errors
Abstract
A neural network which is capable of recalling without errors any set of linearly independent patterns is studied. The network is based on a Hamiltonian version of the model of Personnaz et al. The energy of a state of N (\ifmmode\pm\else\textpm\fi{}1) neurons is the square of the Euclidean distance---in phase space---between the state and the linear subspace spanned by the patterns. This energy corresponds to nonlocal updatings of the synapses in the learning mode. Results of the mean-field theory (MFT) of the system as well as computer simulations are presented. The stable and metastable states of the network are studied as a function of ``temperature'' T and \ensuremath{\alpha}=p/N, where p is the number of embedded patterns. The maximum capacity of the network is \ensuremath{\alpha}=1. For all \ensuremath{\alpha} (0\ensuremath{\le}\ensuremath{\alpha}<1) the embedded patterns are not only locally stable but are global minima of the energy. The patterns appear, as metastable states, below a temperature T=${T}_{M}$(\ensuremath{\alpha}). The temperature ${T}_{M}$(\ensuremath{\alpha}) decreases to zero as \ensuremath{\alpha}\ensuremath{\rightarrow}1. The spurious states of the network are studied in detail in the case of random uncorrelated patterns. At finite p, they are identical to the mixture states of Hopfield's model. At finite \ensuremath{\alpha}, a spin-glass phase exists as a metastable state. According to the replica symmetric MFT the spin-glass state becomes degenerate with the patterns at \ensuremath{\alpha}=${\ensuremath{\alpha}}_{g}$=1-2/\ensuremath{\pi} and disappears above it. Possible interpretations of this unusual result are discussed. The average radius of attraction R of the patterns has been determined by computer simulations, for sizes up to N=400. The value of R for 0<\ensuremath{\alpha}<1 depends on the details of the dynamics. Results for both parallel and serial dynamics are presented. In both cases R is unity (the largest distance in phase space by definition) at \ensuremath{\alpha}\ensuremath{\rightarrow}0 and decreases monotonically to zero as \ensuremath{\alpha}\ensuremath{\rightarrow}1. Contrary to the MFT, simulations have not revealed, so far, any singularity in the properties of the spurious states at an intermediate value of \ensuremath{\alpha}.
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“The approaches that have been explored lately also include systems for automatic analysis and synthesis of facial expressions [86], [56], [39], [47], [89], [88], [53], [14], [19]”
From Automatic analysis of facial expressions: the state of the art (2000) · cites this paper as [39]
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Improved Hopfield networks by training with noisy data
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“…p = N in learning algorithms where N is the number of nodes 2 Current learning algorithms tend to produce poorly networks when learning linearly dependent work in this paper focuses on improving overall recall (basin size) and on increasing the number of it is possible to store in a Hopfield…”
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A model of orthogonal auto-associative networks
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“combination $2-C2, with which the memory capacity rises up to around the number of the neurons or its half ( Kanter and Sompolinski 1987; Personnaz et al. 1985).”
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“In particular, we call the reader attention to the following contributions: 1a) Kanter and Sompolinsky enhanced the storage capacity as well as the noise tolerance of the Hopfield network by replacing the outer product rule by the generalized inverse learning [13].”
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Replica-symmetry breaking in neural networks
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Ensemble of Binary Classifiers Combined Using Recurrent Correlation Associative Memories
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Dynamics of Spin Glasses and Related Models of Neural Networks
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An Experimental Assessment of the Performance of Several Associative Memory Models
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MethodHierarchical Organization of Memory
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A Continuous-Time Model of Autoassociative Neural Memories Utilizing the Noise-Subspace Dynamics
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“For the notational convenience, this ANM is denoted by PANM-1.”
InfluentialAssociative memory in damaged neural networks
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“The results presented here would seem to be in agreement with previous simulations of iterative learning networks (Forrest 1988) and the Pseudoinverse-model (Kanter and Sompolinsky 1987).”
ResultThe number of metastable states in the projection rule neural network
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A novel model of autoassociative memory and its self-organization
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Retrieval properties of diluted attractor neural networks
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Finding signatures of low-dimensional geometric landscapes in high-dimensional cell fate transitionsPreprint
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Yet another exponential Hopfield model
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Local fields in optimal perceptrons with correlated patterns
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Learning in Associative Networks Through Pavlovian Dynamics
Lotito, D., Aquaro, M., & Marullo, C. (2024) · Neural Computation · doi:10.1162/neco_a_01730 · Cited by 2
“Various approaches have been used to solve this problem, we cite unlearning algorithms [Kohonen, 1984, Personnaz et al., 1985, Kanter and Sompolinsky, 1987a, Hemmen et al., 1990, Plakhov and Semenov, 1992, Fachechi et al., 2019b], which are based on the idea of increasing the energy of spurious…”
MethodPhysical Considerations in Memory and Information Storage
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Computational models of learning and synaptic plasticity
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“For example, the pseudo-inverse rule (Kanter & Sompolinsky, 1987; Personnaz et al., 1985), where C is the correlation matrix of the dataset, increases the associative memory network’s capacity to P max = N .”
An Introduction to Quaternion-Valued Recurrent Projection Neural Networks
Valle, M. E., & Lobo, R. A. (2019) · Brazilian Conference on Intelligent Systems · doi:10.1109/bracis.2019.00151 · Cited by 2
“[12] as well as Kanter and Sompolinsky [13] proposed the projection rule to determine the synaptic weights of the Hopfield networks.”
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Dekhtyarenko, O. K. (2006) · International Joint Conference on Neural Networks (IJCNN) · doi:10.1109/ijcnn.2005.1556020 · Cited by 2
“In our experiments we use the normalized attraction radius R [12], which is defined as”
MethodA polynomial training algorithm for calculating perceptrons of optimal stability
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“Like the pseudo-inverse solution of the perceptron problem [9,10] this algorithm uses inversion of pattern correlation matrices for searching (optimal) perceptron couplings.”
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Self-citationSolvable neural network model for input-output associations: Optimal recall at the onset of chaos
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SCRAM: statistically converging recurrent associative memory
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“Those models were proposed by Begin and Proulx [4], Diederich and Opper [5], Kanter and Sompolinsky [ 6 ], Storkey and Valebregue [7].”
MethodSome results on convergent unlearning algorithm
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Entropy Based Associative Model
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Pattern classification by an exponental response neural net
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Bollé, D., & Huyghebaert, J. (1995) · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics · doi:10.1103/physreve.52.870 · Cited by 2
Simulated quantum-optical object recognition from high-resolution images
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A pseudo-inverse neural net with storage capacity exceeding N
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A sequential pseudo-inverse learning rule for networks of formal neurons
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Storage capacity of the truncated projection rule
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Performance enhancement of Willshaw type networks through the use of limit cycles
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“Personnaz [ 131 and Karitner [ 6 ] developed a model with non-local connections, and Lee et a1 [9] and Peretto and Niez [ 121 described models with higher order connections.”
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Functional connectivity-based attractor dynamics of the human brain in rest, task, and disease
Englert, R., Kincses, B., Kotikalapudi, R., Gallitto, G., Li, J., Hoffschlag, K., Woo, C., Wager, T. D., Timmann, D., Bingel, U., et al. (2024) · eLife · doi:10.1101/2023.11.03.565516 · Cited by 4
“…FEP-ANNs - through their emergent predictive coding-based learning rule - have been shown to develop approximately orthogonal attractor representations, and thereby approximate the so-called Kanter-Sompolinsky projector neural networks (K-S network, for short) (Kanter & Sompolinsky, 1987).”
A spectral approach to Hebbian-like neural networks
Agliari, E., Fachechi, A., & Luongo, D. (2024) · Applied Mathematics and Computation · doi:10.1016/j.amc.2024.128689 · Cited by 1
“At t → ∞ , the basic storing setting and supervised/unsupervised cases at r = 1 reproduce the spectrum of the projector model [64], consisting in two δ -peaks at λ = 0 and λ = 1 , with mass resp.”
BackgroundMethodSupervised and unsupervised protocols for hetero-associative neural networks
Alessandrelli, A., Adriano Barra, A., Ladiana, A., Lepre, A., & Ricci‐Tersenghi, F. (2025) · Physica A Statistical Mechanics and its Applications · doi:10.1016/j.physa.2025.130871 · Cited by 4
Content-Addressable Memory with a Content-Free Energy Function
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Hopfield Networks as Models of Emergent Function in Biology
Yampolskaya, M., & Mehta, P. (2026) · Annual Review of Biophysics · doi:10.1146/annurev-biophys-101425-023356 · Cited by 4
He, C., Jiang, M., Shan, K., Yang, S., Li, Z., Wang, S., Pedretti, G., Ignowski, J., & Li, C. (2026) · Nature Communications · doi:10.1038/s41467-026-69958-0 · Cited by 2
Convergent unlearning algorithm for the Hopfield neural network
Plakhov, A. Y., Semenov, S. A., & Shuvalova, I. B. (2002) · New Zealand International Two-Stream Conference on Artificial Neural Networks and Expert Systems · doi:10.1109/annes.1995.499432 · Cited by 1
“0 (N-'12) . Since Pii has CY as mean and variance vanishing in the limit N -+ CO [ 5 ], we thus conclude that the stability does not almost differ from that for the projector rule (s = s').”
BackgroundInfluentialHow to Compress Sequential Memory Patterns into Periodic Oscillations: General Reduction Rules
Zhang, K. (2014) · Neural Computation · doi:10.1162/neco_a_00618 · Cited by 1
“The pseudoinverse weight matrix in equation 2.4 works for any probability distribution and is therefore more general (Kohonen & Ruohonen, 1973; Kohonen, 1977; Personnaz, Guyon, & Dreyfus, 1985; Kanter & Sompolinsky, 1987).”
InfluentialCapacity bounds for the CDMA system and a neural network: a moderate deviations approach
Löwe, M., & Vermet, F. (2008) · ESAIM Probability and Statistics · doi:10.1051/ps:2008016 · Cited by 1
Random symmetry breaking and freezing in chaotic networks
Peleg, Y., Kinzel, W., & Kanter, I. (2012) · Physical Review E · doi:10.1103/physreve.86.036212 · Cited by 1
Self-citationGrowing associative memory based on annealed preferential attachment mechanism
Yang, J., Li-xin, H., & Can, W. (2016) · IEEE Advanced Information Management, Communicates, Electronic and Automation Control Conference (IMCEC) · doi:10.1109/imcec.2016.7867352 · Cited by 1
“Equilibrium properties of fully connected networks have been discussed through mean field [2] and spin glass [3] theory.”
MethodGabbiani, F., & Cox, S. J. (2017) · Elsevier eBooks · doi:10.1016/b978-0-12-801895-8.00030-0 · Cited by 1
Metadata
- Authors
- Kanter, I., & Sompolinsky, H.
- Affiliations
- Bar-Ilan University (IL)
- Venue
- Physical Review A
- Publisher
- American Physical Society (APS)
- Published
- 1987-01-01
- Volume
- 35
- Issue
- 1
- Pages
- 380-392
- ISSN
- 0556-2791 (ISSN-L)
- OpenAlex
- W2029182425
- Cited by
- 379
- References
- 13
- Citations by year
- 2026: 9 · 2025: 11 · 2024: 13 · 2023: 6 · 2022: 10 · 2021: 9 · 2020: 6 · 2019: 6 · 2018: 1 · 2017: 4 · 2016: 4 · 2015: 2 · 2014: 7 · 2013: 6 · 2012: 6
- FWCI
- 30.30
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Physical Review A (publishedVersion)
- PubMed (publishedVersion)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Neural Networks and Applications
- Physical Sciences › Physics and Astronomy › Condensed Matter Physics › Theoretical and Computational Physics
- Life Sciences › Neuroscience › Cognitive Neuroscience › Neural dynamics and brain function
- Keywords
- associative memory, mean-field theory, metastable states, spin glass phase, spurious states, Hopfield neural network, capacity
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