Related papers: The Programming of Algebra
Numerical data structures for positive dimensional solution sets of polynomial systems are sets of generic points cut out by random planes of complimentary dimension. We may represent the linear spaces defined by those planes either by…
Recurrence equations lie at the heart of many computational paradigms including dynamic programming, graph analysis, and linear solvers. These equations are often expensive to compute and much work has gone into optimizing them for…
Modern analytical workloads increasingly combine relational data with array-valued attributes. While columnar database systems efficiently process such workloads, their ability to optimize queries that interleave relational operators with…
Analytical queries often require a mixture of relational and linear algebra operations applied to the same data. This poses a challenge to analytic systems that must bridge the gap between relations and matrices. Previous work has mainly…
Data processing systems roughly group into families such as relational, array, graph, and key-value. Many data processing tasks exceed the capabilities of any one family, require data stored across families, or run faster when partitioned…
We assign a relational structure to any finite algebra in a canonical way, using solution sets of equations, and we prove that this relational structure is polymorphism-homogeneous if and only if the algebra itself is…
Text analytical tasks like word embedding, phrase mining, and topic modeling, are placing increasing demands as well as challenges to existing database management systems. In this paper, we provide a novel algebraic approach based on…
We propose a new approach for metric learning by framing it as learning a sparse combination of locally discriminative metrics that are inexpensive to generate from the training data. This flexible framework allows us to naturally derive…
Modular Decomposition focuses on repeatedly identifying a module M (a collection of vertices that shares exactly the same neighbourhood outside of M) and collapsing it into a single vertex. This notion of exactitude of neighbourhood is very…
Quasi-trees generalize trees in that the unique "path" between two nodes may be infinite and have any countable order type. They are used to define the rank-width of a countable graph in such a way that it is equal to the least upper-bound…
It has by now become a standard approach to use the theory of sparse (or toric) elimination, based on the Newton polytope of a polynomial, in order to reveal and exploit the structure of algebraic systems. This talk surveys compact…
Many current challenges involve understanding the complex dynamical interplay between the constituents of systems. Typically, the number of such constituents is high, but only limited data sources on them are available. Conventional…
We develop a linear-algebraic framework for dimensional analysis in systems with constraints, particularly when variables are numerous or related by implicit relations so that direct elimination is impractical. By expressing both…
An alternative proof of the completeness of relational algebra with respect to allowed formulas of first-order logic is presented. The proof relies on the well-known embedding of relational algebra into cylindric algebra, which makes it…
We initiate the study of computable presentations of real and complex C*-algebras under the program of effective metric structure theory. With the group situation as a model, we develop corresponding notions of recursive presentations and…
Polynomial reduction is one of the main tools in computational algebra with innumerable applications in many areas, both pure and applied. Since many years both the theory and an efficient design of the related algorithm have been solidly…
This paper develops new combinatorial approaches to analyze and compute special set partitions, called complementary set partitions, which are fundamental in the study of generalized cumulants. Moving away from traditional graph-based and…
Information algebra is algebraic structure for local computation and inference. Given an initial universe set and a parameter set, we show that a soft set system over them is an information algebra. Moreover, in a soft set system, the…
The principle behind algebraic language theory for various kinds of structures, such as words or trees, is to use a compositional function from the structures into a finite set. To talk about compositionality, one needs some way of…
In this paper we explore algebraic and geometric structures that arise on parallelizable manifolds. Given a parallelizable manifold $\mathbb{L}$, there exists a global trivialization of the tangent bundle, which defines a map…