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Lasha Ephremidze

Publications and source records attributed to Lasha Ephremidze.

18 recordsLinked to original sources
arXiv2025

Exponential Speedup of the Janashia-Lagvilava Matrix Spectral Factorization Algorithm

Spectral factorization is a powerful mathematical tool with diverse applications in signal processing and beyond. The Janashia-Lagvilava method has emerged as a leading approach for matrix spectral factorization. In this paper, we extend a central equation of the method to the non-commutative case, enabling polynomial coefficients to be represented in block matrix form while preserving the equation's fundamental structure. This generalization results in an exponential speedup for high-dimensional matrices. Our approach addresses challenges in factorizing massive-dimensional matrices encountered in neural data analysis and other practical applications.

math.CV↗
arXiv2024

Stationary Processes, Wiener-Granger Causality, and Matrix Spectral Factorization

Granger causality has become an indispensable tool for analyzing causal relationships between time series. In this paper, we provide a detailed overview of its mathematical foundations, trace its historical development, and explore how recent computational advancements can enhance its application in various fields. We will not hesitate to present the proofs in full if they are simple and transparent. For more complex theorems on which we rely, we will provide supporting citations. We also discuss potential future directions for the method, particularly in the context of largescale data analysis.

math.CV↗
arXiv2023

On the exact spectral factorization of rational matrix functions with applications to paraunitary filter banks

In this paper, we enhance a recent algorithm for approximate spectral factorization of matrix functions, extending its capabilities to precisely factorize rational matrices when an exact lower-upper triangular factorization is available. This novel approach leverages a fundamental component of the improved algorithm for the precise design of rational paraunitary filter banks, allowing for the predetermined placement of zeros and poles. The introduced algorithm not only advances the state-of-the-art in spectral factorization but also opens new avenues for the tailored design of paraunitary filters with specific spectral properties, offering significant potential for applications in signal processing and beyond.

math.CV↗
arXiv2023

Random Generator of Orthogonal Matrices in Finite Fields

We propose a superfast method for constructing orthogonal matrices $M\in\mathcal{O}(n,q)$ in finite fields $GF(q)$. It can be used to construct $n\times n$ orthogonal matrices in $Z_p$ with very high values of $n$ and $p$, and also orthogonal matrices with a certain circulant structure. Equally well one can construct paraunitary filter banks or wavelet matrices over finite fields. The construction mechanism is highly efficient, allowing for the complete screening and selection of an orthogonal matrix that meets specific constraints. For instance, one can generate a complete list of orthogonal matrices with given $n$ and $q=p^m$ provided that the order of $\mathcal{O}(n,q)$ is not too large. Although the method is based on randomness, isolated cases of failure can be identified well in advance of the basic procedure's start. The proposed procedures are based on the Janashia-Lagvilava method which was developed for an entirely different task, therefore, it may seem somewhat unexpected.

cs.IT↗
arXiv2021

On multivariable matrix spectral factorization method

Spectral factorization is a prominent tool with several important applications in various areas of applied science. Wiener and Masani proved the existence of matrix spectral factorization. Their theorem has been extended to the multivariable case by Helson and Lowdenslager. Solving the problem numerically is challenging in both situations, and also important due to its practical applications. Therefore, several authors have developed algorithms for factorization. The Janashia-Lagvilava algorithm is a relatively new method for matrix spectral factorization which has proved to be useful in several applications. In this paper, we extend this method to the multivariable case. Consequently, a new numerical algorithm for multivariable matrix spectral factorization is constructed.

math.CV↗
arXiv2021

An algorithm for J-spectral factorization of certain matrix functions

The problems of matrix spectral factorization and J-spectral factorization appear to be important for practical use in many MIMO control systems. We propose a numerical algorithm for J-spectral factorization which extends Janashia-Lagvilava matrix spectral factorization method to the indefinite case. The algorithm can be applied to matrices that have constant signatures for all leading principle submatrices. A numerical example is presented for illustrative purposes.

math.NA↗
arXiv2018

Quantitative results on continuity of the spectral factorization mapping

The spectral factorization mapping $F\to F^+$ puts a positive definite integrable matrix function $F$ having an integrable logarithm of the determinant in correspondence with an outer analytic matrix function $F^+$ such that $F = F^+(F^+)^*$ almost everywhere. The main question addressed here is to what extent $\|F^+ - G^+\|_{H_2}$ is controlled by $\|F-G\|_{L_1}$ and $\|\log \det F - \log\det G\|_{L_1}$.

math.CV↗
arXiv2016

Quantitative results on continuity of the spectral factorization mapping in the scalar case

In the scalar case, the spectral factorization mapping $f\to f^+$ puts a nonnegative integrable function $f$ having an integrable logarithm in correspondence with an outer analytic function $f^+$ such that $f = |f^+|^2$ almost everywhere. The main question addressed here is to what extent $\|f^+ - g^+\|_{H_2}$ is controlled by $\|f-g\|_{L_1}$ and $\|\log f - \log g\|_{L_1}$.

math.CV↗
arXiv2012

Numerical comparison of different algorithms for construction of wavelet matrices

Factorization of compact wavelet matrices into primitive ones has been known for more than 20 years. This method makes it possible to generate wavelet matrix coefficients and also to specify them by their first row. Recently, a new parametrization of compact wavelet matrices of the same order and degree has been introduced by the last author. This method also enables us to fulfill the above mentioned tasks of matrix constructions. In the present paper, we briefly describe the corresponding algorithms based on two different methods, and numerically compare their performance

math.NA↗
arXiv2011

On compact wavelet matrices of rank m and of order and degree N

A new parametrization (one-to-one onto map) of compact wavelet matrices of rank $m$ and of order and degree $N$ is proposed in terms of coordinates in the Euclidian space $R^{(m-1)N}$. The developed method depends on Wiener-Hopf factorization of corresponding unitary matrix functions and allows to construct compact wavelet matrices efficiently. Some applications of the proposed method are discussed.

math.NA↗
arXiv2009

A New Method of Matrix Spectral Factorization

A new method of matrix spectral factorization is proposed which reliably computes an approximate spectral factor of any matrix spectral density that admits spectral factorization

math.CV↗
arXiv2008

On Parametrization of Compact Wavelet Matrices

It is given an efficient complete parametrization of wavelet matrices of rank $m$, genus $g+1$, and degree $g$, which are naturally identified with corresponding polynomial paraunitary matrix-functions. The parametrization depends on Wiener-Hopf factorization of unitary matrix-functions with constant determinant given in the unit circle. This method allows us to construct in real time the coefficients of wavelet matrices from the above class.

math.CA↗
arXiv2008

Fractional, Maximal and Singular Operators in Variable Exponent Lorentz Spaces

We introduce the Lorentz space $\mathcal{L}^{p(\cdot), q(\cdot)}$ with variable exponents $p(t),q(t)$ and prove the boundedness of singular integral and fractional type operators, and corresponding ergodic operators in these spaces. The main goal of the paper is to show that the boundedness of these operators in the spaces $\mathcal{L}^{p(\cdot), q(\cdot)}$ is possible without the local log-condition on the exponents, typical for the variable exponent Lebesgue spaces; instead the exponents $p(s)$ and $q(s)$ should only satisfy decay conditions of log-type as $s\to 0$ and $s\to\infty$. To prove this, we base ourselves on the recent progress in the problem of the validity of Hardy inequalities in variable exponent Lebesgue spaces.

math.FA↗