Related papers: A Matrix Decomposition Method for Odd-Type Gaussia…
For a matrix ${\bf A}$ with linearly independent columns, this work studies to use its normalization $\bar{\bf A}$ and ${\bf A}$ itself to approximate its orthonormalization $\bf V$. We theoretically analyze the order of the approximation…
Matrix multiplication optimization remains a fundamental challenge in computational mathematics. This work introduces a novel approach that discovers matrix multiplication schemes whose coefficients are restricted to the set $\{-1, 0, 1\}$…
Isolated wide binary stars provide natural laboratories to directly probe gravity for Newtonian acceleration $g_{\rm{N}}\lesssim 10^{-9}$ m s$^{-2}$. Recent statistical analyses of wide binaries have been performed only with sky-projected…
We study two-stage stochastic optimization models with mixed-integer decision variables appearing in both stages. For these models, dual decomposition enables parallel computing implementation and can quickly provide a lower bound for the…
Galois field (GF) arithmetic is used to implement critical arithmetic components in communication and security-related hardware, and verification of such components is of prime importance. Current techniques for formally verifying such…
Learned image compression techniques have achieved considerable development in recent years. In this paper, we find that the performance bottleneck lies in the use of a single hyperprior decoder, in which case the ternary Gaussian model…
A spatially regularized Gaussian mixture model, LapGM, is proposed for the bias field correction and magnetic resonance normalization problem. The proposed spatial regularizer gives practitioners fine-tuned control between balancing bias…
It was demonstrated in earlier work that, by approximating its range kernel using shiftable functions, the non-linear bilateral filter can be computed using a series of fast convolutions. Previous approaches based on shiftable approximation…
We derive the Wu list-decoding algorithm for Generalised Reed-Solomon (GRS) codes by using Gr\"obner bases over modules and the Euclidean algorithm (EA) as the initial algorithm instead of the Berlekamp-Massey algorithm (BMA). We present a…
This paper presents a method to determine a set of basis polynomials from the extended Euclidean algorithm that allows Generalized Minimum Distance decoding of Reed-Solomon codes with a complexity of O(nd).
This paper is concerned with linear algebra based methods for solving exactly polynomial systems through so-called Gr\"obner bases, which allow one to compute modulo the polynomial ideal generated by the input equations. This is a topical…
We construct a normal form for the walled Brauer algebra, together with the reduction algorithm. We apply normal form to calculate the numbers of monomials in generators with minimal length. We further utilize normal form to give explicit…
Dimension reduction is often needed in the area of data mining. The goal of these methods is to map the given high-dimensional data into a low-dimensional space preserving certain properties of the initial data. There are two kinds of…
Jordan Normal Forms serve as excellent representatives of conjugacy classes of matrices over closed fields. Once we knows normal forms, we can compute functions of matrices, their main invariant, etc. The situation is much more complicated…
Although Buchberger's algorithm, in theory, allows us to compute Gr\"obner bases over any field, in practice, however, the computational efficiency depends on the arithmetic of the ground field. Consider a field $K = \mathbb{Q}(\alpha)$, a…
This paper presents a generalization of our earlier work in [19]. In this paper, the two concepts, generic regular decomposition (GRD) and regular-decomposition-unstable (RDU) variety introduced in [19] for generic zero-dimensional systems,…
Current techniques for formally verifying circuits implemented in Galois field (GF) arithmetic are limited to those with a known irreducible polynomial P(x). This paper presents a computer algebra based technique that extracts the…
Hashing methods have been widely investigated for fast approximate nearest neighbor searching in large data sets. Most existing methods use binary vectors in lower dimensional spaces to represent data points that are usually real vectors of…
The entropy computation of Gaussian mixture distributions with a large number of components has a prohibitive computational complexity. In this paper, we propose a novel approach exploiting the sphere decoding concept to bound and…
Methods for inference and simulation of linearly constrained Gaussian Markov Random Fields (GMRF) are computationally prohibitive when the number of constraints is large. In some cases, such as for intrinsic GMRFs, they may even be…