An Introduction to Multisets

TL;DR

This paper introduces multisets, explores their mathematical properties and generalizations, and demonstrates their potential applications in signal processing, pattern recognition, and deep learning.

Contribution

It extends traditional multisets to include vectors, matrices, and continuous structures, and proposes new operations like the common product for advanced applications.

Findings

Defined complement operation and extended De Morgan laws for multisets.

Proposed the common product for multisets and mfunctions, analogous to inner product.

Showed potential applications in signal processing, pattern recognition, and deep learning.

Abstract

Multisets are sets that allow repetition of elements. As such, multisets pave the way to a number of interesting possibilities of theoretical and applied nature. In the present work, after revising the main aspects of traditional sets, we introduce some of the main concepts and characteristics of multisets, followed by their generalization to take into account vectors and matrices. An approach is also proposed in which the real, negative multiplicities are allowed, implying the multiset universe to become finite and well-defined, corresponding to the multiset with null multiplicities. The complement operation in multisets is then defined, which allows properties involving complement -- including the De Morgan theorem -- to be recovered in multisets. In addition, it becomes possible to extend multisets to functions (which become multifunctions), scalar fields and other continuous mathematical structure, therefore achieving an enhanced space endowed with all algebraic operations plus set theoretical operations including union, intersection, and complementation. The possibility to define a set operation between mfunctions, namely the common product, that is analogous to the traditional inner product is also proposed, paving the way to obtaining respective mfunction transformations, and it is argued that the Walsh functions provide an orthogonal basis for the mfunctions space under the common product. This result also allowed the proposal of performing integrated signal processing operations on mset mfunctions, including filtering and enhanced template matching. Relationships between the cosine similarity index and the Jaccard index are also identified, including the presentation of an intersection-based variation of the cosine index. The potential of multisets in pattern recognition and deep learning is also briefly characterized and illustrated.