Let’s talk about hypertensors
Many people know a tensor which is deeply appreciated in deeplearning field.
But in mathematics, a hypertensor with infinite values is typically referred to as an "infinite hypertensor." It’s a very specific case of a tensor.
And many people will talk about them with new quantum computers appearing.
Well, the term "infinite" means that the hypertensor has an unbounded or unlimited number of dimensions or components, which can take on infinite values. The prefix "hyper" indicates that the hypertensor extends beyond the traditional notion of a tensor, which typically deals with multi-dimensional arrays.
It's important to note that hypertensors are not commonly encountered in mainstream mathematics, and the study of infinite hypertensors is still a topic of ongoing research and exploration in higher-dimensional algebraic structures. Therefore, there may not be a universally established or standard terminology specifically for hypertensors with infinite values. Different researchers or authors may use varying terminology depending on the specific context or framework they are working with.
In mathematics, a hypertensor is a generalization of tensors to higher-dimensional spaces. Tensors are multilinear maps that operate on vector spaces, while hypertensors extend this concept to operate on more than two vector spaces. Instead of just being a multi-dimensional array of numbers, hypertensors can have components that are themselves tensors, resulting in a hierarchical structure.
Formally, a hypertensor of rank (m, n) is an m-fold covariant and n-fold contravariant multilinear map. This means that it takes m vectors and n covectors as inputs and produces a scalar value as output. The rank of a hypertensor represents the number of indices needed to specify its components.
Hypertensors can be represented using index notation, similar to tensors. The components of a hypertensor are represented using indices, and the Einstein summation convention is often used for repeated indices. The components of a hypertensor can be expressed as a multidimensional array, where each index ranges over the dimensions of the corresponding vector space.
Hypertensors find applications in various areas of mathematics and physics, such as general relativity, quantum field theory, and algebraic geometry. They provide a powerful tool for representing and manipulating complex mathematical objects that involve multiple vector spaces and transformations.
While the concept of hypertensors is not as widely studied and formalized as tensors, there are a few references that discuss their properties and applications. Here are a few resources that you may find helpful:
"Geometric Algebra for Physicists" by Anthony Lasenby, Chris Doran, and Stephen Gull: This book introduces the concept of hypertensors within the framework of geometric algebra, which provides a powerful mathematical tool for representing and manipulating geometric objects.
"Tensor Analysis: Theory and Applications" by Sergei D. Silvestrov and Eugen Paal: This book explores the theory of tensors and includes a section on hypertensors, discussing their definition and properties.
Research papers and articles: Searching for research papers or articles on hypertensors in academic databases like arXiv or IEEE Xplore may provide more specific information and applications in various fields.
