This is Part II of the two-part comprehensive survey devoted to a computing framework most commonly known under the names Hyperdimensional Computing and Vector Symbolic Architectures (HDC/VSA). Both ...names refer to a family of computational models that use high-dimensional distributed representations and rely on the algebraic properties of their key operations to incorporate the advantages of structured symbolic representations and vector distributed representations. Holographic Reduced Representations 321, 326 is an influential HDC/VSA model that is well known in the machine learning domain and often used to refer to the whole family. However, for the sake of consistency, we use HDC/VSA to refer to the field.Part I of this survey 222 covered foundational aspects of the field, such as the historical context leading to the development of HDC/VSA, key elements of any HDC/VSA model, known HDC/VSA models, and the transformation of input data of various types into high-dimensional vectors suitable for HDC/VSA. This second part surveys existing applications, the role of HDC/VSA in cognitive computing and architectures, as well as directions for future work. Most of the applications lie within the Machine Learning/Artificial Intelligence domain; however, we also cover other applications to provide a complete picture. The survey is written to be useful for both newcomers and practitioners.
This two-part comprehensive survey is devoted to a computing framework most commonly known under the names Hyperdimensional Computing and Vector Symbolic Architectures (HDC/VSA). Both names refer to ...a family of computational models that use high-dimensional distributed representations and rely on the algebraic properties of their key operations to incorporate the advantages of structured symbolic representations and distributed vector representations. Notable models in the HDC/VSA family are Tensor Product Representations, Holographic Reduced Representations, Multiply-Add-Permute, Binary Spatter Codes, and Sparse Binary Distributed Representations but there are other models too. HDC/VSA is a highly interdisciplinary field with connections to computer science, electrical engineering, artificial intelligence, mathematics, and cognitive science. This fact makes it challenging to create a thorough overview of the field. However, due to a surge of new researchers joining the field in recent years, the necessity for a comprehensive survey of the field has become extremely important. Therefore, amongst other aspects of the field, this Part I surveys important aspects such as: known computational models of HDC/VSA and transformations of various input data types to high-dimensional distributed representations. Part II of this survey 84 is devoted to applications, cognitive computing and architectures, as well as directions for future work. The survey is written to be useful for both newcomers and practitioners.
A MINIMAL CONGRUENCE LATTICE REPRESENTATION FOR BUNN, ROGER; GROW, DAVID; INSALL, MATT ...
Journal of the Australian Mathematical Society (2001),
06/2020, Volume:
108, Issue:
3
Journal Article
Peer reviewed
Let
$p$
be an odd prime. The unary algebra consisting of the dihedral group of order
$2p$
, acting on itself by left translation, is a minimal congruence lattice representation of
$\mathbb{M}_{p+1}$
....
We determine the irreducible 2-modular representations of the group G=GLn+1(2) in which a Singer cycle has eigenvalue 1, and show that in these representations every element g∈G has eigenvalue 1.
Since Otto Frank first introduced the Windkessel (0D) model in 1899 to reproduce the arterial network, various versions were constructed which differ through their number of elements and their ...disposal. The purpose of this paper is to couple the left heart model given by M. Danielsen and J.T. Ottesen to a classic three-element 0D representation of the arterial network and compare the results with the output of a 13-elements arterial model coupled with the heart. This is done in order to proof that no great benefit is gained when increasing the number of elements of the Windkessel model if the input satisfies normal physiological conditions. After confirming that the model works properly, a simulation is performed with parameters specific for a hypertensive person. The output is assessed against results specific for a normal person and the differences are further analysed.
A positive representation for a set of complex densities is constructed. In particular, complex measures on a direct product of
U
(1) groups are studied. After identifying general conditions which ...such representations should satisfy, several concrete realizations are proposed. Their utility is illustrated in few concrete examples representing problems in abelian lattice gauge theories.
A new section of databases and programs devoted to double crystallographic groups (point and space groups) has been implemented in the Bilbao Crystallographic Server (http://www.cryst.ehu.es). The ...double crystallographic groups are required in the study of physical systems whose Hamiltonian includes spin‐dependent terms. In the symmetry analysis of such systems, instead of the irreducible representations of the space groups, it is necessary to consider the single‐ and double‐valued irreducible representations of the double space groups. The new section includes databases of symmetry operations (DGENPOS) and of irreducible representations of the double (point and space) groups (REPRESENTATIONS DPG and REPRESENTATIONS DSG). The tool DCOMPREL provides compatibility relations between the irreducible representations of double space groups at different k vectors of the Brillouin zone when there is a group–subgroup relation between the corresponding little groups. The program DSITESYM implements the so‐called site‐symmetry approach, which establishes symmetry relations between localized and extended crystal states, using representations of the double groups. As an application of this approach, the program BANDREP calculates the band representations and the elementary band representations induced from any Wyckoff position of any of the 230 double space groups, giving information about the properties of these bands. Recently, the results of BANDREP have been extensively applied in the description of and the search for topological insulators.
A new section of computer tools devoted to the double crystallographic groups has been implemented in the Bilbao Crystallographic Server (http://www.cryst.ehu.es). The section includes databases of symmetry operations and irreducible representations of the double point and space groups and programs that compute the compatibility relations, generate relevant information related to the site‐symmetry approach, and calculate band representations and elementary band representations induced from any Wyckoff position of any double space group.
It is an old idea to replace averages of observables with respect to a complex weight by expectation values with respect to a genuine probability measure on complexified space. This is precisely what ...one would like to get from complex Langevin simulations. Unfortunately, these fail in many cases of physical interest. We will describe method of deriving positive representations by matching of moments and show simple examples of successful constructions. It will be seen that the problem is greatly underdetermined.
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