MATLAB tensor classes for fast algorithm prototyping.

The C++ Tensor Toolbox is a software package for computing tensor decompositions. It is based on the Matlab Tensor Toolbox, and is particularly optimized for sparse data sets. This user manual briefly overviews tensor decomposition mathematics, software capabilities, and installation of the package. Tensors (also known as multidimensional arrays or N-way arrays) are used in a variety of applications ranging from chemometrics to network analysis. The Tensor Toolbox provides classes for manipulating dense, sparse, and structured tensors in C++. The Toolbox compiles into libraries and is intended for use with custom applications written by users.

Large-scale datasets in computational chemistry typically require distributed-memory parallel methods to perform a special operation known as tensor contraction. Tensors are multidimensional arrays, and a tensor contraction is akin to matrix multiplication with special types of permutations. Creating an efficient algorithm and optimized im- plementation in this domain is complex, tedious, and error-prone. To address this, we develop a notation to express data distributions so that we can apply use automated methods to find optimized implementations for tensor contractions. We consider the spin-adapted coupled cluster singles and doubles method from computational chemistry and use our methodology to produce an efficient implementation. Experiments per- formed on the IBM Blue Gene/Q and Cray XC30 demonstrate impact both improved performance and reduced memory consumption.

A multivariate biorthogonal wavelet system can be obtained from a pair of multivariate biorthogonal refinement masks in Multiresolution Analysis setup. Some multivariate refinement masks may be decomposed into lower dimensional refinement masks. Tensor product is a popular way to construct a decomposable multivariate refinement mask from lower dimensional refinement masks. We present an alternative method, which we call coset sum, for constructing multivariate refinement masks from univariate refinement masks. The coset sum shares many essential features of the tensor product that make it attractive in practice: (1) it preserves the biorthogonality of univariate refinement masks, (2) it preserves the accuracy number of the univariate refinement mask, and (3) the wavelet system associated with it has fast algorithms for computing and inverting the wavelet coefficients. The coset sum can even provide a wavelet system with faster algorithms in certain cases than the tensor product. These features of the coset sum suggest that it is worthwhile to develop and practice alternative methods to the tensor product for constructing multivariate wavelet systems. Some experimental results using 2-D images are presented to illustrate our findings.

We present an object-oriented open-source library for developing tensor network algorithms written in C++ called Uni10. With Uni10, users can build a symmetric tensor from a collection of bonds, while the bonds are constructed from a list of quantum numbers associated with different quantum states. It is easy to label and permute the indices of the tensors and access a block associated with a particular quantum number. Furthermore a network class is used to describe arbitrary tensor network structure and to perform network contractions efficiently. We give an overview of the basic structure of the library and the hierarchy of the classes. We present examples of the construction of a spin-1 Heisenberg Hamiltonian and the implementation of the tensor renormalization group algorithm to illustrate the basic usage of the library. The library described here is particularly well suited to explore and fast prototype novel tensor network algorithms and to implement highly efficient codes for existing algorithms.

Recently, principal component analysis (PCA) has made remarkable progress in some research areas including computer vision and pattern recognition. The objects handled by PCA have been extended from vectors to higher-order tensors, also known as multidimensional or N-way arrays including vectors (first order) and matrices (second order). A number of methods for tensor PCA have been proposed so far. In this chapter, we proposed a generalized formulation for dimensionality reduction of tensors and derive several state-of-the-art techniques for tensor PCA and low-rank approximation of tensors from the proposed general formulation. We also show experimental results of the applications of the derived methods to image analysis such as outliers removal, image compression, and face recognition.

Radix-3 fast algorithms for polynomial time frequency transforms

In this chapter, two types of local cubic splines on non-uniform grids are described: 1. The simplest variation-diminishing splines and 2. The quasi-interpolating splines. The splines are computed by a simple fast computational algorithms that utilizes a relation between the splines and cubic interpolation polynomials. Those splines can serve as an efficient tool for real-time signal processing. As an input, they use either clean or noised arbitrarily-spaced samples. On the other hand, the capability to adapt the grid to the structure of an object and minimal requirements to the operating memory are great advantages for off-line processing of signals and multidimensional data arrays.

There are (in general) three common ways to arrive at a wavelet basis for R2. The first (separable) way takes the tensor product of two one-dimensional wavelets and their associated scaling functions. One wavelet (psi) (x) and scaling function (phi) (y) pair; one wavelet (psi) (x) wavelet (psi) (y) pair and one scaling function (phi) (x) and wavelet (psi) (y) pair leading to three `detail' terms per scale and translate. This leads to the familiar Mallat representation of the image. The second, less frequently used basis, is formed by taking all dilations of the product of two one dimensional wavelets, leading to one detail term per single scale and translate, but far more scales since the dilate changes between the two wavelets. In the third way, the basis is formed not by a tensor product, rather by dilation of a set of functions which is supported on a lattice, the translates of which complete the integer lattice. This technique leads to k detail terms corresponding to the k wavelets in the basis. In this paper, non-separable two dimensional Haar wavelets are considered to represent images for compression purposes. The irregular (self-similar) support of non-separable wavelets make them a natural candidate for image compression. A fast algorithm for decomposition and reconstruction of images in terms of the non-separable Haar wavelets is discussed, and the connection between the `best basis' representation and Quad (n-ary) Trees is discussed.© (1993) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

ANOVA approximation with mixed tensor product basis on scattered points

Greedy low-rank approximation in Tucker format of solutions of tensor linear systems

As constructing multi-D wavelets remains a challenging problem, we propose a new method called prime coset sum to construct multi-D wavelets. Our method provides a systematic way to construct multi-D non-separable wavelet filter banks from two 1-D low-pass filters, with one of which being interpolatory. Our method has many important features including the following: 1) it works for any spatial dimension, and any prime scalar dilation; 2) the vanishing moments of the multi-D wavelet filter banks are guaranteed by certain properties of the initial 1-D low-pass filters, and furthermore; 3) the resulting multi-D wavelet filter banks are associated with fast algorithms that are faster than the existing fast tensor product algorithms.

The connection between the recoupling scheme of four copies of $\mathfrak{su}(1,1)$, the generic superintegrable system on the 3 sphere, and bivariate Racah polynomials is identified. The Racah polynomials are presented as connection coefficients between eigenfunctions separated in different spherical coordinate systems and equivalently as different irreducible decompositions of the tensor product representations. As a consequence of the model, an extension of the quadratic algebra ${\rm QR}(3)$ is given. It is shown that this algebra closes only with the inclusion of an additional shift operator, beyond the eigenvalue operators for the bivariate Racah polynomials, whose polynomial eigenfunctions are determined. The duality between the variables and the degrees, and hence the bispectrality of the polynomials, is interpreted in terms of expansion coefficients of the separated solutions.

The unit cost model is both convenient and largely realistic for describing integer decision algorithms over + ,×. Additional operations like division with remainder or bitwise conjunction, although equally supported by computing hardware, may lead to a considerable drop in complexity. We show a variety of concrete problems to benefit from such non-arithmetic primitives by presenting and analyzing corresponding fast algorithms.

In this paper, a fast algorithm is proposed to calculate kth power of an n × n Boolean matrix that requires O(kn3p) addition operations, where p is the probability that an entry of the matrix is 1. The algorithm generates a single set of inference rules at the beginning. It then selects entries (specified by the same inference rule) from any matrix Ak-1 and adds them up for calculating corresponding entries of Ak. No multiplication operation is required. A modification of the proposed algorithm can compute the diameter of any graph and for a massive random graph, it requires only O(n2(1-p)E[q]) operations, where q is the number of attempts required to find the first occurrence of 1 in a column in a linear search. The performance comparisons say that the proposed algorithms outperform the existing ones.

In this paper, we research the operation rules and scalar multiplication of Twisted Edwards elliptic curve. Based on the properties of point addition and double point operations, we propose the fast continuous double point algorithm, continuous point addition algorithm and continuous double point-point addition algorithm on Twisted Edwards elliptic curve. These algorithms reduce the amount of operations significantly by reducing the inverse operations. For comparison purposes, the top-level operations of the new algorithm use the non-adjacent form(NAF) and double base number system(DBNS) scalar multiplication algorithms. In contrast, it effectively improves the computational efficiency of the Twisted Edwards elliptic curve scalar multiplication algorithm, in which it is more than 50% faster than the underlying operation using Horner's rule, continuous multiplier algorithm, and about 10% to 35% or more faster than the base operation using the 4-NNAF algorithm.