License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2022.53
URN: urn:nbn:de:0030-drops-158636
URL: https://drops.dagstuhl.de/opus/volltexte/2022/15863/
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Ramya, C. ; Tengse, Anamay

On Finer Separations Between Subclasses of Read-Once Oblivious ABPs

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LIPIcs-STACS-2022-53.pdf (0.9 MB)


Abstract

Read-once Oblivious Algebraic Branching Programs (ROABPs) compute polynomials as products of univariate polynomials that have matrices as coefficients. In an attempt to understand the landscape of algebraic complexity classes surrounding ROABPs, we study classes of ROABPs based on the algebraic structure of these coefficient matrices. We study connections between polynomials computed by these structured variants of ROABPs and other well-known classes of polynomials (such as depth-three powering circuits, tensor-rank and Waring rank of polynomials).
Our main result concerns commutative ROABPs, where all coefficient matrices commute with each other, and diagonal ROABPs, where all the coefficient matrices are just diagonal matrices. In particular, we show a somewhat surprising connection between these models and the model of depth-three powering circuits that is related to the Waring rank of polynomials. We show that if the dimension of partial derivatives captures Waring rank up to polynomial factors, then the model of diagonal ROABPs efficiently simulates the seemingly more expressive model of commutative ROABPs. Further, a commutative ROABP that cannot be efficiently simulated by a diagonal ROABP will give an explicit polynomial that gives a super-polynomial separation between dimension of partial derivatives and Waring rank.
Our proof of the above result builds on the results of Marinari, Möller and Mora (1993), and Möller and Stetter (1995), that characterise rings of commuting matrices in terms of polynomials that have small dimension of partial derivatives. The algebraic structure of the coefficient matrices of these ROABPs plays a crucial role in our proofs.

BibTeX - Entry

@InProceedings{ramya_et_al:LIPIcs.STACS.2022.53,
  author =	{Ramya, C. and Tengse, Anamay},
  title =	{{On Finer Separations Between Subclasses of Read-Once Oblivious ABPs}},
  booktitle =	{39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
  pages =	{53:1--53:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-222-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{219},
  editor =	{Berenbrink, Petra and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/15863},
  URN =		{urn:nbn:de:0030-drops-158636},
  doi =		{10.4230/LIPIcs.STACS.2022.53},
  annote =	{Keywords: Algebraic Complexity Theory, Algebraic Branching Programs, Commutative Matrices}
}

Keywords: Algebraic Complexity Theory, Algebraic Branching Programs, Commutative Matrices
Collection: 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)
Issue Date: 2022
Date of publication: 09.03.2022


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