Related papers: From vectors to mnesors
Since its introduction by Gauss, Matrix Algebra has facilitated understanding of scientific problems, hiding distracting details and finding more elegant and efficient ways of computational solving. Today's largest problems, which often…
Multitask learning (MTL) can utilize the relatedness between multiple tasks for performance improvement. The advent of multimodal data allows tasks to be referenced by multiple indices. High-order tensors are capable of providing efficient…
This chapter gives an overview of the core concepts of machine learning (ML) -- the use of algorithms that learn from data, identify patterns, and make predictions or decisions without being explicitly programmed -- that are relevant to…
Once developed for quantum theory, tensor networks have been established as a successful machine learning paradigm. Now, they have been ported back to the quantum realm in the emerging field of quantum machine learning to assess problems…
Sequential data such as time series, video, or text can be challenging to analyse as the ordered structure gives rise to complex dependencies. At the heart of this is non-commutativity, in the sense that reordering the elements of a…
We undertook a study of the use of a memristor network for music generation, making use of the memristor's memory to go beyond the Markov hypothesis. Seed transition matrices are created and populated using memristor equations, and which…
Machine learning models are trained with relatively simple objectives, such as next token prediction. However, on deployment, they appear to capture a more fundamental representation of their input data. It is of interest to understand the…
Embedding spaces contain interpretable dimensions indicating gender, formality in style, or even object properties. This has been observed multiple times. Such interpretable dimensions are becoming valuable tools in different areas of…
Bayesian models are a powerful tool for studying complex data, allowing the analyst to encode rich hierarchical dependencies and leverage prior information. Most importantly, they facilitate a complete characterization of uncertainty…
In this article we introduce theory and algorithms for learning discrete representations that take on a lattice that is embedded in an Euclidean space. Lattice representations possess an interesting combination of properties: a) they can be…
We discuss various old and new definitions of the notion of a vector field on a convenient manifold that can be proved to give rise to Lie algebras, and are in finite dimensions equivalent to the standard notion of a vector field.
Combining neural networks with continuous logic and multicriteria decision making tools can reduce the black box nature of neural models. In this study, we show that nilpotent logical systems offer an appropriate mathematical framework for…
Tensor network methods are a class of numerical tools and algorithms to study many-body quantum systems in and out of equilibrium, based on tailored variational wave functions. They have found significant applications in simulating lattice…
In this paper we give a comparison between the formulation of the concept of metric for a real vector space of finite dimension in terms of \emph{tensors} and \emph{extensors}. A nice property of metric extensors is that they have inverses…
The advancement in generative AI could be boosted with more accessible mathematics. Beyond human-AI chat, large language models (LLMs) are emerging in programming, algorithm discovery, and theorem proving, yet their genomics application is…
Vector-space models, from word embeddings to neural network parsers, have many advantages for NLP. But how to generalise from fixed-length word vectors to a vector space for arbitrary linguistic structures is still unclear. In this paper we…
Complex systems may morph between structures with different dimensionality and degrees of freedom. As a tool for their modelling, nonlinear embeddings are introduced that encompass objects with different dimensionality as a continuous…
Decision trees are renowned for their ability to achieve high predictive performance while remaining interpretable, especially on tabular data. Traditionally, they are constructed through recursive algorithms, where they partition the data…
Coupled oscillators are being increasingly used as the basis of machine learning (ML) architectures, for instance in sequence modeling, graph representation learning and in physical neural networks that are used in analog ML devices. We…
A finite semifield is a division algebra over a finite field where multiplication is not necessarily associative. We consider here the complexity of the multiplication in small semifields and finite field extensions. For this operation, the…