Related papers: In Praise of Sequence (Co-)Algebra and its impleme…
We develop a new general framework for algebras and clones, called Universal Clone Algebra. Algebras and clones of finitary operations are to Universal Algebra what t-algebras and clone algebras are to Universal Clone Algebra. Clone…
We create a sequence version of calculus. First, we define equivalence, some fundamental operations, differential, and integral for sequences. Then, we propose sequence versions of identity function, power function, exponential function,…
Computer Algebra systems are widely spread because of some of their remarkable features such as their ease of use and performance. Nonetheless, this focus on performance sometimes leads to unwanted consequences: algorithms and computations…
We introduce a category of cluster algebras with fixed initial seeds. This category has countable coproducts, which can be constructed combinatorially, but no products. We characterise isomorphisms and monomorphisms in this category and…
Machine learning has a long collaborative tradition with several fields of mathematics, such as statistics, probability and linear algebra. We propose a new direction for machine learning research: $C^*$-algebraic ML $-$ a…
In this paper, we develop a generalized theory of convolutional signal processing and neural networks for Reproducing Kernel Hilbert Spaces (RKHS). Leveraging the theory of algebraic signal processing (ASP), we show that any RKHS allows the…
The success of kernel-based learning methods depend on the choice of kernel. Recently, kernel learning methods have been proposed that use data to select the most appropriate kernel, usually by combining a set of base kernels. We introduce…
This research started with an algebra for reasoning about rely/guarantee concurrency for a shared memory model. The approach taken led to a more abstract algebra of atomic steps, in which atomic steps synchronise (rather than interleave)…
We consider the problem of operator-valued kernel learning and investigate the possibility of going beyond the well-known separable kernels. Borrowing tools and concepts from the field of quantum computing, such as partial trace and…
Algebraic effects are computational effects that can be represented by an equational theory whose operations produce the effects at hand. The free model of this theory induces the expected computational monad for the corresponding effect.…
Many recursive functions can be defined elegantly as the unique homomorphisms, between two algebras, two coalgebras, or one each, that are induced by some universal property of a distinguished structure. Besides the well-known applications…
The database community lacks a unified relational query language for subset selection and optimisation queries, limiting both user expression and query optimiser reasoning about such problems. Decades of research (latterly under the rubric…
We provide a foundation for working with homological and homotopical methods in categorical algebra. This involves two mutually complementary components, namely (a) the strategic selection of suitable axiomatic frameworks, some well known…
Applying new computing paradigms like quantum computing to the field of machine learning has recently gained attention. However, as high-dimensional real-world applications are not yet feasible to be solved using purely quantum hardware,…
We provide a computational definition of the notions of vector space and bilinear functions. We use this result to introduce a minimal language combining higher-order computation and linear algebra. This language extends the Lambda-calculus…
Number sequences defined by a linear recursion relation are studied by means of generating functions. Indices of the terms in the recursion relation have arbitrary differenses. In addition to formulas for the nth term an algorithm is…
Motivation: Sequence mapping is the cornerstone of modern genomics. However, most existing sequence mapping algorithms are insufficiently general. Results: We introduce context schemes: a method that allows the unambiguous recognition of a…
Regular sequences generalize the extensively studied automatic sequences. Let $S$ be an abstract numeration system. When the numeration language $L$ is prefix-closed and regular, a sequence is said to be $S$-regular if the module generated…
Let W be a finite Coxeter group. We define its Hecke-group algebra by gluing together appropriately its group algebra and its 0-Hecke algebra. We describe in detail this algebra (dimension, several bases, conjectural presentation,…
Transformer architectures have facilitated the development of large-scale and general-purpose sequence models for prediction tasks in natural language processing and computer vision, e.g., GPT-3 and Swin Transformer. Although originally…