An Introduction to Multisets
a day ago
- #multisets
- #mathematics
- #pattern recognition
- Multisets are sets that allow repetition of elements, offering theoretical and applied possibilities.
- The work revises traditional sets, introduces multisets, and generalizes them to vectors and matrices.
- Proposes allowing real, negative multiplicities, making the multiset universe finite and well-defined.
- Defines complement operation in multisets, recovering properties like the De Morgan theorem.
- Extends multisets to functions (multifunctions), scalar fields, and other continuous structures.
- Introduces a set operation analogous to the inner product for mfunctions, enabling transformations.
- Identifies Walsh functions as an orthogonal basis for mfunctions under the common product.
- Proposes integrated signal processing operations on mset mfunctions, including filtering and template matching.
- Explores relationships between cosine similarity and Jaccard index, presenting an intersection-based variation.
- Briefly characterizes and illustrates the potential of multisets in pattern recognition and deep learning.