Exponential Capacity in Multilayer Hetero-Associative Neural Networks

• URL: http://arxiv.org/abs/2607.29554v1
• Date: Fri, 31 Jul 2026 15:46:21 GMT
• Title: Exponential Capacity in Multilayer Hetero-Associative Neural Networks
• Abstract summary: We study an exponential neural network of $L$ layers of $N$ binary neurons.<n>We find that the exponential capacity and the predicted basins survive correlated, many-to-one patterns.
• Score: 1.5749416770494706
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Exponential Hopfield networks store a number of patterns that grows exponentially with the number of neurons, and in their classical formulation they are auto-associative: they complete a corrupted copy of a memory into the memory itself. Many of the tasks one wants such a network to perform are instead hetero-associative, mapping a cue to a different target. We introduce and analyse an exponential neural network of $L$ layers of $N$ binary neurons, each layer carrying its own dataset, whose energy is an exponential of the product of the per-layer Mattis overlaps, so that it is minimised precisely when every layer retrieves the pattern of the same index; the stored association must be a surjective function of the cue, and we show why nothing else can be stored at all. A cavity/signal-to-noise analysis, made exact at leading order by a large-deviation evaluation of the noise, shows that the aligned hetero-associative state is a fixed point of the zero-temperature dynamics up to a number of stored patterns $P_c\sim e^{Nρ_L}$, exponential in the layer size, with an explicit rate $ρ_L$ that grows like $L\log 2$; enlarging the basins of attraction lowers the rate but never destroys its exponential character. Comparing the theory with structured data we find that the exponential capacity and the predicted basins survive correlated, many-to-one patterns: the network is a near-perfect content-addressable memory. The same closed forms describe, without refitting, a synthetic manifold, real T-cell-receptor/epitope triples and natural-language intent data, so the mechanism is domain-universal. Generalisation to unseen cues, though significantly above chance, stays below memorisation, and it is the geometry of the encoding, rather than the data domain, that sets how far above chance it reaches. In this family, exponential storage and strong generalisation are distinct capabilities.

Related papers

• Learn from your own latents and not from tokens: A sample-complexity theory [53.5821824211418]
  We study networks trained to predict their emphown latent representations of related views or masked regions.<n>We show that latent prediction achieves this with a number of samples emphconstant in $L$, up to logarithmic factors.<n>This suggests that explicit stacking such as H-JEPA is largely redundant.
  arXiv  Detail & Related papers  (2026-05-26T22:16:42Z)
• Neural Networks Learn Generic Multi-Index Models Near Information-Theoretic Limit [66.20349460098275]
  We study the gradient descent learning of a general Gaussian Multi-index model $f(boldsymbolx)=g(boldsymbolUboldsymbolx)$ with hidden subspace $boldsymbolUin mathbbRrtimes d$.<n>We prove that under generic non-degenerate assumptions on the link function, a standard two-layer neural network trained via layer-wise gradient descent can agnostically learn the target with $o_d(1)$ test error.
  arXiv  Detail & Related papers  (2025-11-19T04:46:47Z)
• An unsupervised tour through the hidden pathways of deep neural networks [6.063903439185316]
  This thesis focuses on characterizing semantic content of hidden representations with unsupervised learning tools.<n>In Chapter 3, we study the evolution of the probability density across the hidden layers in some state-of-the-art deep neural networks.<n>In Chapter 4, we study the problem of generalization in deep neural networks.
  arXiv  Detail & Related papers  (2025-10-24T15:50:31Z)
• Global Convergence and Rich Feature Learning in $L$-Layer Infinite-Width Neural Networks under $μ$P Parametrization [66.03821840425539]
  In this paper, we investigate the training dynamics of $L$-layer neural networks using the tensor gradient program (SGD) framework.<n>We show that SGD enables these networks to learn linearly independent features that substantially deviate from their initial values.<n>This rich feature space captures relevant data information and ensures that any convergent point of the training process is a global minimum.
  arXiv  Detail & Related papers  (2025-03-12T17:33:13Z)
• Multi-Valued Neural Networks I A Multi-Valued Associative Memory [0.0]
  We introduce the novel concept of such a network without numbers, investigate its properties, and give a learning algorithm in the multi-valued case. In the multi-valued neural network, all variables are not numbers, but elements or subsets of a lattice, i.e., they are all only partially-ordered.
  arXiv  Detail & Related papers  (2023-02-23T10:32:25Z)
• The Separation Capacity of Random Neural Networks [78.25060223808936]
  We show that a sufficiently large two-layer ReLU-network with standard Gaussian weights and uniformly distributed biases can solve this problem with high probability. We quantify the relevant structure of the data in terms of a novel notion of mutual complexity.
  arXiv  Detail & Related papers  (2021-07-31T10:25:26Z)
• Slope and generalization properties of neural networks [0.0]
  We show that the distribution of the slope of a well-trained neural network classifier is generally independent of the width of the layers in a fully connected network. The slope is of similar size throughout the relevant volume, and varies smoothly. It also behaves as predicted in rescaling examples. We discuss possible applications of the slope concept, such as using it as a part of the loss function or stopping criterion during network training, or ranking data sets in terms of their complexity.
  arXiv  Detail & Related papers  (2021-07-03T17:54:27Z)
• Redundant representations help generalization in wide neural networks [71.38860635025907]
  We study the last hidden layer representations of various state-of-the-art convolutional neural networks. We find that if the last hidden representation is wide enough, its neurons tend to split into groups that carry identical information, and differ from each other only by statistically independent noise.
  arXiv  Detail & Related papers  (2021-06-07T10:18:54Z)
• Towards an Understanding of Benign Overfitting in Neural Networks [104.2956323934544]
  Modern machine learning models often employ a huge number of parameters and are typically optimized to have zero training loss. We examine how these benign overfitting phenomena occur in a two-layer neural network setting. We show that it is possible for the two-layer ReLU network interpolator to achieve a near minimax-optimal learning rate.
  arXiv  Detail & Related papers  (2021-06-06T19:08:53Z)
• Binary autoencoder with random binary weights [0.0]
  It is shown that the sparse activation of the hidden layer arises naturally in order to preserve information between layers. With a large enough hidden layer, it is possible to get zero reconstruction error for any input just by varying the thresholds of neurons. The model is similar to an olfactory perception system of a fruit fly, and the presented theoretical results give useful insights toward understanding more complex neural networks.
  arXiv  Detail & Related papers  (2020-04-30T12:13:19Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.