Draft:Inverse sets 1

• Comment: I am not sure what is going on with the duplicate submissions. Only one article should be submitted and it should be called "Draft:Inverse sets". Caleb Stanford (talk) 22:44, 30 July 2025 (UTC)

• Comment: Please do not submit multiple copies of drafts. It does not increase the likelihood that one of them will be accepted into article space, and annoys the reviewers. It is likely to be seen as an effort to game the system, and may result in the drafts being nominated for deletion. Robert McClenon (talk) 15:39, 30 July 2025 (UTC)

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Inverse sets

Inverse Sets are a mathematical abstraction that describes which sets contain a given element, effectively reversing the traditional view of sets as collections of elements. This concept is similar to structures like inverted indexes in information retrieval. Though not commonly formalized in mainstream mathematical literature, the idea has been explored in specific applications such as pattern recognition and artificial intelligence. In particular, some researchers have proposed using inverse sets to model data structures for rapid pattern learning and indexing in AI systems.

In mathematics and computer science, the concept of inverse sets arises when studying collections of sets and their relationships with individual elements. Typically, a set is a collection of elements—such as numbers, words, or objects. However, when dealing with multiple labeled or indexed sets, it becomes useful to reverse the perspective: instead of asking “What elements are in this set?”, we ask “Which sets contain this element?”. This reverse mapping is what defines an inverse set. Essentially, while a set groups elements together, an inverse set groups the names or indexes of sets that share a particular element.

This concept has practical importance in fields like machine learning, search engines, and big data analysis. For example, in a search engine, an inverted index is a data structure that stores a list of documents (or web pages) for each word. If you search for “sunrise”, the engine quickly finds all pages that include this term by looking it up in the inverse index. This is far more efficient than scanning every document one by one. Inverse sets provide a formal and flexible mathematical language to describe such inverse relationships in a structured and efficient way.

The concept has also been applied to artificial intelligence and pattern recognition. Inverse sets underpin systems such as Indextron, which performs pattern recognition using a direct inverse-representation method rather than time-consuming iterative training, as in neural networks. This shift in perspective—from elements grouped into sets to sets grouped by elements—enables more intuitive and computationally efficient processing in many modern technologies.

There exist many inverse entities in mathematics like inverse functions, inverse problems and even inverse images of sets. However, the definition of inverse sets was introduced only in 2011 and, independently, in 2012 , Although the two definitions are essentially equivalent, the used notations are quite different. This is because the first definition emerged in the context of pattern recognition and AI, whereas second definition appeared in the context of combinatorial problems. This article provides the first definition.

The notion of inverse sets is applicable to indexed sets only. Whereas sets contain elements, the inverse sets contain indexes of original sets. With the rise of AI, inverse sets, being indexing tools, have become fundamentally important because they formally describe inverse representations of patterns, - the representations that drastically cut down the computational load on search and training of AI .

The key distinguishing feature of inverse sets-based machine learning appears to be its “instant learning” capability that contrasts with the iterative training process typical of neural networks

Also, inverse sets are special in a sense that they underlie a biologically plausible formalism, which delivers the same pattern of memory activity as that in the inferotemporal region of the monkey cortex ,

An illustration of sets / inverse sets relation can be depicted as

Set	сontains	features
Z		{a,c}
Y		{b,c,d}
U		{a,b}
X		{a,d}
V		{d}

Feature	is contained in	sets
a		{Z,U,X}
b		{Y,U}
c		{Y,Z}
d		{Y,X,V}

“One can compare sets and inverse sets to other methods: patterns = features = descriptors, since they serve the same purpose. However, inverse sets contain representational advantages for machine learning and classification …” . In machine learning, inverse sets describe representations of patterns in an inverse index. The key distinguishing feature of the Indextron , which is an inverse representations-based inference method in AI appears to be its “instant learning” capability for pattern recognition that contrasts with the iterative training process typical of neural networks.

Let finite sets X_n = {x}_n, n = 1,2,...N, with integer elements be given. An inverse set {n}_x assigns to any element

$x\in \bigcup _{n=1}^{\mathbb {N} }X_{n}$ a subset of indexes

{n}x = {n:x∈ Xn}x

(1)

That is, the inverse set {n}_x is a collection of indexes of those sets, which the element x belongs to.

Let the sets X_n={x}_n, n=1,2,...N, and their inverse sets

{n}_x, $\forall x\in \bigcup _{n=1}^{\mathbb {N} }X_{n}$

be denoted as X_N and N_x, respectively. Then the rule (1) defines a one-to-one set transform T(X_N) = N_x, i.e.,

T^-1(T(X_N)) = X_N

The domain of the set transform T comprises all families of subsets of the union

$X=\bigcup _{n=1}^{\mathbb {N} }X_{n}$

Examples

The sets {a,b}₁, {b,c}₂ have the following three inverse sets and only these inverse sets {1}_a, {1,2}_b, {2}_c

Note that sets’ names can serve as sets’ indexes. Then the set A = {1,2} has the following two inverse sets and only these inverse sets {A}₁, {A}₂. The upper-case character A in curly brackets is a set name, rather than a set.

A back-of-the-book-index is a collection of inverse sets, where keywords serve as indexes of lists of page numbers.

Applications

Fully inverted files, alias inverse indexes, have been widely used in Google-type search engines . Such inverted files contain lists that store identities of documents. More recently, indexing techniques were applied to genomics problems, in particular, to genome alignment tasks . Inverse sets emerged as a concise mathematical description of indexing methods that deal efficiently with vast amounts of data. Inverse sets were mathematically defined in (2011) and, independently, in (2012). Potential applications of inverse sets include, for instance, artificial intelligence, pattern recognition, prediction problems, for example aircraft engine failure prediction , etc.

References

1. Alexei Mikhailov, “Indexing-based Pattern Recognition”, Advanced Materials Research Vols. 403-408 (2012), pp. 5254-5259. Online available since November 2011 at http://www.scientific.net.

2. Dhananjoy Biswas, “Inverse Sets - A Technique to Count in two Ways”, June 2012, http://www.artofproblemsolving.com/community/c6h394326p2704451

3. Sergey Brin, Larry Page, “The Anatomy of a large-scale hypotextual web search engine”, Stanford University, Stanford, CA, 94305, USA.

4. Scott Krig, Computer Vision Metrics: Survey, Taxonomy, and Analysis of Computer Vision, Visual Neuroscience, and Visual AI. 2nd Edition. Springer Nature Singapore, Pte Ltd. 2025. ISBN 978-981-99-3392-1, ISBN 978-981-99-3393-8 (eBook). https://link.springer.com/book/10.1007/978-981-99-3393-8

5. Alexei Mikhailov, Mikhail Karavai, Mais Farkhadov, “Inverse Sets in Big Data Processing”, Proc. of AICT2017, 11th IEEE Conf, 20-22 Sep. 2017.

6. Wang Liang, Zhao Kai Yong, “A new DNA Alignment Method based on Inverted Indexes”, 2013. https://doi.org/10.48550/arXiv.1307.0194

7. Alexei Mikhailov, Mikhail Karavay. Indextron. Proceedings of the 10th International Conference on Pattern Recognition Application and Methods, 4-6 Feb 2021, Vienna, vol. 1-978-989-758-486-2. Doi:10.5220/0010180301430149, pp. 143-149.