ǂA ǂblock coordinate descent-based projected gradient algorithm for orthogonal non-negative matrix factorization

Soodabeh Asadi (Author), Janez Povh (Author)

Abstract

This article uses the projected gradient method (PG) for a non-negative matrix factorization problem (NMF), where one or both matrix factors must have orthonormal columns or rows. We penalize the orthonormality constraints and apply the PG method via a block coordinate descent approach. This means that at a certain time one matrix factor is fixed and the other is updated by moving along the steepest descent direction computed from the penalized objective function and projecting onto the space of non-negative matrices. Our method is tested on two sets of synthetic data for various values of penalty parameters. The performance is compared to the well-known multiplicative update (MU) method from Ding (2006), and with a modified global convergent variant of the MU algorithm recently proposed by Mirzal (2014). We provide extensive numerical results coupled with appropriate visualizations, which demonstrate that our method is very competitive and usually outperforms the other two methods.

Keywords

nenegativna matrična faktorizacija;pogoji pravokotnosti;metoda projiciranega gradienta;multiplikativni algoritem posodabljanja;koordinatni spust;non-negative matrix factorization;orthogonality conditions;projected gradient method;multiplicative update algorithm;block coordinate descent;

Data

Language:	English
Year of publishing:	2021
Typology:	1.01 - Original Scientific Article
Organization:	UL FS - Faculty of Mechanical Engineering
UDC:	519.61(045)
COBISS:	56467971
ISSN:	2227-7390
Parent publication:	Mathematics
Views:	331
Downloads:	129
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	Slovenian
Secondary abstract:	V članku predstavimo, kako z metodo projiciranega gradienta (PG) rešiti problem negativne matrične faktorizacije (NMF), kjer mora eden ali oba matrična faktorja zadoščati pogoju ortonormalnosti. Najprej ta pogoj prenesemo v kriterijsko funkcijo z metodo natančnega kaznovanja, nato pa uporabimo metodo PG v kombinaciji s koordinatnim spustom. Slednje pomeni, da je vsakem koraku najprej en matrični faktor določen, drugega pa izračunamo s premikanjem v smeri najbolj strmega spusta ter končnim projiciranjem na prostor nenegativnih matrik. Nato pa obratno. Metodo preizkusimo z različnimi vhodnimi parametri na dveh sklopih sintetičnih podatkov, kjer tudi testiramo različne vrednosti vhodnih parametrov. Naš pristop primerjamo z znano metodo, ki temelji na multiplikativnih posodobitvah (Ding, 2006) in z metodo, ki uporablja prilagojeno izvedbo multiplikativnih posodobitev in ima globalne konvergenčne lastnosti (Mirzal, 2014). Članek vsebuje obsežne numerične primerjave skupaj z ustreznimi vizualizacijami, ki kažejo, da je naša metoda zelo konkurenčna in običajno prekaša drugi dve metodi.
Secondary keywords:	nenegativna matrična faktorizacija;pogoji pravokotnosti;metoda projiciranega gradienta;multiplikativni algoritem posodabljanja;koordinatni spust;
Type (COBISS):	Article
Pages:	str. 1-22
Volume:	ǂVol. ǂ9
Issue:	ǂiss. ǂ5
Chronology:	Mar. 2021
DOI:	10.3390/math9050540
ID:	12664770