Higher-order singular value decomposition (HOSVD) is an extension of the traditional singular value decomposition (SVD) to tensor data, which are multi-dimensional generalizations of matrices. While a matrix is a two-dimensional array (with rows and columns), a tensor can have three or more dimensions, commonly referred to as modes.
The HOSVD (Higher-Order Singular Value Decomposition) is a mathematical tool used in tensor decomposition, which is particularly useful in the fields of control theory, signal processing, and machine learning for tasks involving multi-way data or tensor representations. In the context of Tensor Product (TP) functions and quasi-linear parameter-varying (qLPV) models, the HOSVD can be applied to represent these complex systems in a more compact and interpretable form.
A glossary of tensor theory typically includes definitions and explanations of key terms and concepts related to tensors and their applications in fields such as mathematics, physics, and engineering. Here are some important terms that are often included: ### A - **Alignment**: The relationship between two tensors that involve certain conditions of their components in relation to each other and the coordinate systems used.
Gama's Theorem, often spelled as Gamas Theorem, is a concept in the field of computational geometry, particularly related to the study of convex polytopes and their properties. It states that in a convex polytope, the number of facets (or faces) of a particular dimension is related to the vertices and edges of the polytope, following certain combinatorial relationships.
Exterior algebra is a mathematical framework used primarily in the fields of linear algebra, differential geometry, and algebraic topology. It provides a way to construct and manipulate multi-linear forms and generalized notions of vectors in a vector space. The key components of exterior algebra are: 1. **Vector Spaces**: Exterior algebra begins with a vector space \( V \) over a field (usually the real or complex numbers).
Einstein notation, also known as Einstein summation convention, is a notational scheme used primarily in the fields of mathematics and physics to simplify expressions involving tensors and multi-indexed arrays. It was introduced by the physicist Albert Einstein in the context of his work on the theory of relativity. The key principle of Einstein notation is that when an index variable appears twice in a single term, it implies a summation over that index.
Discrete exterior calculus (DEC) is a mathematical framework that extends concepts from traditional differential geometry and exterior calculus to discrete settings. It is particularly useful in computational applications, especially in numerical simulations, computer graphics, and finite element methods. In traditional exterior calculus, one deals with smooth manifolds and differential forms, which allow the formulation of concepts like integration over manifolds, differential operators like the exterior derivative, and cohomology.
Cubic form typically refers to the mathematical representation of a cubic equation or polynomial, which is a polynomial of degree three.
Cryptographic multilinear maps are advanced mathematical constructs used in cryptography, particularly in advanced protocols and schemes such as functional encryption, attribute-based encryption, and other complex cryptographic primitives. They generalize linear maps (or bilinear maps) to higher dimensions, allowing for operations involving multiple inputs that can be combined in sophisticated ways.
A bivector is a geometric object that arises in the context of multivector algebra, particularly within the framework of geometric algebra. It represents an oriented area element and can be thought of as being associated with the plane spanned by two vectors in a vector space. In more formal terms, a bivector is defined as the exterior (or wedge) product of two vectors.
The Binet–Cauchy identity is a result in combinatorics and linear algebra that relates the determinants of matrices and their block structures. It provides a way to compute the determinant of a block matrix in terms of the determinants of its components.
A bilinear map is a mathematical function defined on two vector spaces (or modules) that is linear in each of its arguments when the other is held fixed.
The Berezin integral is a concept from mathematical physics, particularly in the field of supersymmetry and quantum mechanics. It is a type of integral used for integrating functions of Grassmann variables, which are anticommuting variables that the Berezin calculus is built around. Grassmann variables play a fundamental role in the formulation of supersymmetry, where they are used to describe fermionic fields or states.
An alternating multilinear map is a special type of multilinear function that takes several input arguments from a vector space and has the property of being alternating. Here's a more detailed breakdown of what this means: 1. **Multilinear Map**: A function \( f: V_1 \times V_2 \times \ldots \times V_n \to W \) is called multilinear if it is linear in each of its \( n \) arguments.
Tensors are mathematical objects that generalize scalars, vectors, and matrices to higher dimensions. They are fundamental in various fields, including physics, engineering, and machine learning, particularly in deep learning. Here’s a brief overview of what tensors are: 1. **Definition**: A tensor is essentially a multi-dimensional array that can be used to represent data. Tensors can have any number of dimensions. - A **scalar** (a single number) is a 0-dimensional tensor.
Monoidal categories are a fundamental concept in category theory, providing a framework that captures notions of multiplicative structures in a categorical setting. A monoidal category consists of a category equipped with a tensor product (which can be thought of as a kind of "multiplication" between objects), an identity object, and certain coherence conditions that ensure the structure behaves well.
Invariant theory is a branch of mathematics, particularly in the fields of algebra, geometry, and representation theory, that studies properties of mathematical objects that remain unchanged (or invariant) under transformations from a certain group. The most common transformations considered are linear transformations, but the theory can also apply to more general transformation groups. Historically, invariant theory originated in the 19th century, with significant contributions from mathematicians such as David Hilbert and Hermann Weyl.
Clifford algebras are a type of algebra associated with a quadratic form on a vector space. They arise in various areas of mathematics and physics, particularly in geometry, algebra, and the theory of spinors. The concept was introduced by the mathematician William Kingdon Clifford in the late 19th century.
Tim Paterson is an American computer programmer and software developer best known for his work on the MS-DOS operating system. In the early 1980s, he created a system called QDOS (Quick and Dirty Operating System), which was designed to be compatible with DOS and was eventually purchased by Microsoft. This acquisition played a significant role in Microsoft's rise in the personal computing market, as QDOS was later rebranded as MS-DOS.
Microsoft MACRO-80 is an assembly language macro assembler developed by Microsoft for the Intel 8080 microprocessor. Released in the late 1970s, MACRO-80 allows programmers to write assembly language programs using macros, which are essentially sequences of code that can be used to simplify repetitive tasks or generate code dynamically. This tool was particularly valuable for developing software for early personal computers that utilized the Intel 8080 CPU architecture, such as the Altair 8800.