Δημοσιεύσεις

Project Acronym: EFEST
Title: Efficient estimation of matrix functions with applications to statistics, networks and machine learning
Affiliation: university of athens
Pi: Marilena Mitrouli
Research Field: mathematics and computer sciences

Aitken's method for estimating bilinear forms arising in applications
by Fika, Paraskevi and Mitrouli, Marilena
Abstract:
In the present work we study how the Aitken's method can be applied to the sequence of moments $}{\$\backslash\c_k=(x, A^\k\x),\textasciitildek \backslashin \\backslashmathbb \Z\\\backslash\,$}{\$ \ c k = ( x , A k x ) , k ∈ Z \ , of a given symmetric positive definite matrix $}{\$A \backslashin \\backslashmathbb \R\\^\p \backslashtimes p\,$}{\$ A ∈ R p \texttimes p , for the prediction of possible unknown terms of this sequence. Direct estimation of $}{\$c_k$}{\$ c k leads to approximating bilinear quantities i.e. $}{\$(y, A^\k\x)$}{\$ ( y , A k x ) as well. By employing Taylor series expansion for appropriate f,  prediction of $}{\$y^Tf(A)x$}{\$ y T f ( A ) x can be achieved. Estimates for many useful linear algebra quantities can be derived by appropriately selecting the vectors y and x. Numerical examples concerning such applications are presented. The estimates are illustrated through numerical examples executed on the high-performance computing system ARIS.
Reference:
Aitken's method for estimating bilinear forms arising in applications (Fika, Paraskevi and Mitrouli, Marilena), In Calcolo, volume 54, 2017.
Bibtex Entry:
@article{Fika2017,
 author = {Fika, Paraskevi
		and Mitrouli, Marilena},
 title = {Aitken's method for estimating bilinear forms arising in applications},
 journal = {Calcolo},
 year = {2017},
 bibyear = {2017},
 month = {Mar},
 day = {01},
 volume = {54},
 number = {1},
 pages = {455--470},
 abstract = {In the present work we study how the Aitken's method can be applied to the sequence of moments                                                                           {\$}{\$}{\backslash}{\{}c{\_}k=(x, A^{\{}k{\}}x),{\textasciitilde}k {\backslash}in {\{}{\backslash}mathbb {\{}Z{\}}{\}}{\backslash}{\}},{\$}{\$}                                                                                    {\{}                                                  c                          k                                                =                                                  (                          x                          ,                                                      A                            k                                                    x                          )                                                ,                                                k                        ∈                        Z                        {\}}                        ,                                                                             of a given symmetric positive definite matrix                                                                           {\$}{\$}A {\backslash}in {\{}{\backslash}mathbb {\{}R{\}}{\}}^{\{}p {\backslash}times p{\}},{\$}{\$}                                                                                    A                        ∈                                                                              R                                                                                p                            {\texttimes}                            p                                                                          ,                                                                             for the prediction of possible unknown terms of this sequence. Direct estimation of                                                                           {\$}{\$}c{\_}k{\$}{\$}                                                                                    c                        k                                                                             leads to approximating bilinear quantities i.e.                                                                           {\$}{\$}(y, A^{\{}k{\}}x){\$}{\$}                                                                                    (                        y                        ,                                                  A                          k                                                x                        )                                                                             as well. By employing Taylor series expansion for appropriate f,  prediction of                                                                           {\$}{\$}y^Tf(A)x{\$}{\$}                                                                                                              y                          T                                                f                                                  (                          A                          )                                                x                                                                             can be achieved. Estimates for many useful linear algebra quantities can be derived by appropriately selecting the vectors y and x. Numerical examples concerning such applications are presented. The estimates are illustrated through numerical examples executed on the high-performance computing system ARIS.},
 issn = {1126-5434},
 doi = {10.1007/s10092-016-0193-0},
 url = {https://doi.org/10.1007/s10092-016-0193-0},
}