Related papers: Categorical Hopfield Networks
Game comonads provide a categorical syntax-free approach to finite model theory, and their Eilenberg-Moore coalgebras typically encode important combinatorial parameters of structures. In this paper, we develop a framework whereby the…
For groups of a topological origin, such as braid groups and mapping class groups, an important source of interesting and highly non-trivial representations is given by their actions on the twisted homology of associated spaces; these are…
We consider the Hopfield neural network as a model of associative memory and we define its neuronal interaction matrix $\mathbf{J}$ as a function of a set of $K \times M$ binary vectors $\{\mathbf{\xi}^{\mu, A} \}_{\mu=1,...,K}^{A=1,...,M}$…
In this paper we introduce and exploit the real replica approach for a minimal generalization of the Hopfield model, by assuming the learned patterns to be distributed accordingly to a standard unit Gaussian. We consider the high storage…
The paper examines the problem of accessing a vector memory from a single neuron in a Hebbian neural network. It begins with the review of the author's earlier method, which is different from the Hopfield model in that it recruits…
This paper describes how realistic neuromorphic networks can have their connectivity fully characterized in analytical fashion. By assuming that all neurons have the same shape and are regularly distributed along the two-dimensional…
Tree-child networks are a recently-described class of directed acyclic graphs that have risen to prominence in phylogenetics (the study of evolutionary trees and networks). Although these networks have a number of attractive mathematical…
Conceptual and mathematical models of neurons have lagged behind empirical understanding for decades. Here we extend previous work in modeling biological systems with fully scale-independent quantum information-theoretic tools to develop a…
We revisit the problem of characterising the mean-field limit of a network of Hopfield-like neurons. Building on the previous works of Ben Arous and Guionnet we establish for a large class of networks of Hopfield-like neurons, i.e. rate…
We consider the class of Hopfield models of associative memory with activation function $F$ and state space $\{-1,1\}^N$, where each vertex of the cube describes a configuration of $N$ binary neurons. $M$ randomly chosen configurations,…
It has been recently shown that a learning transition happens when a Hopfield Network stores examples generated as superpositions of random features, where new attractors corresponding to such features appear in the model. In this work we…
Modern Hopfield Networks (MHNs) have emerged as powerful components in deep learning, serving as effective replacements for pooling layers, LSTMs, and attention mechanisms. While recent advancements have significantly improved their storage…
We present a novel deep graphical representation that seamlessly merges principles of game theory with laws of statistical mechanics. It performs feature extraction, dimensionality reduction, and pattern classification within a single…
Modelling compositionality has been a longstanding area of research in the field of vector space semantics. The categorical approach to compositionality maps grammar onto vector spaces in a principled way, but comes under fire for requiring…
When designing plans in engineering, it is often necessary to consider attributes associated to objects, e.g. the location of a robot. Our aim in this paper is to incorporate attributes into existing categorical formalisms for planning,…
In this paper we provide an abstract model theory for the untyped differential lambda-calculus and the resource calculus. In particular we propose a general definition of model of these calculi, namely the notion of linear reflexive object…
We give a presentation of Feynman categories from a representation--theoretical viewpoint. Feynman categories are a special type of monoidal categories and their representations are monoidal functors. They can be viewed as a far reaching…
Neural networks are widely used, yet their analysis and verification remain challenging. In this work, we present a Lean 4 formalization of neural networks, covering both deterministic and stochastic models. We first formalize Hopfield…
This is a report on aspects of the theory and use of monoidal categories. The first section introduces the main concepts through the example of the category of vector spaces. String notation is explained and shown to lead naturally to a…
We associate learning and adaptation in living systems with the shaping of the velocity vector field in the respective dynamical systems in response to external, generally random, stimuli. With this, a mathematical concept of self-shaping…