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Proceedings Article, Paper
@InProceedings
Beitrag in Tagungsband, Workshop

Author, Editor
Author(s):
Agrawal, Manindra
Saha, Chandan
Saptharishi, Ramprasad
Saxena, Nitin
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Not MPG Author(s):
Agrawal, Manindra
Saptharishi, Ramprasad
Saxena, Nitin
Editor(s):
BibTeX cite key*:
ASSS12
Title, Booktitle
Title*:
Jacobian hits circuits: hitting-sets, lower bounds for depth-D occur-k formulas & depth-3 transcendence degree-k circuits
Booktitle*:
STOC'12 : Proceedings of the 2012 ACM Symposium on Theory of Computing
Event, URLs
Conference URL::
http://cs.nyu.edu/~stoc2012/default.htm
Downloading URL:
http://www.mpi-inf.mpg.de/~csaha/jacobian.pdf
Event Address*:
New York, NY
Language:
English
Event Date*
(no longer used):
Organization:
Event Start Date:
19 May 2012
Event End Date:
22 May 2012
Publisher
Name*:
ACM
URL:
http://www.acm.org/
Address*:
New York, NY
Type:
Vol, No, Year, pp.
Series:
Volume:
Number:
Month:
Pages:
599-614
Year*:
2012
VG Wort Pages:
ISBN/ISSN:
978-1-4503-1245-5
Sequence Number:
DOI:
10.1145/2213977.2214033
Note, Abstract, ©
(LaTeX) Abstract:
We present a single common tool to strictly subsume \emph{all} known cases of polynomial time blackbox polynomial identity testing (PIT), that have been hitherto solved using diverse tools and techniques, over fields of zero or large characteristic. In particular, we show that polynomial time hitting-set generators for identity testing of the two seemingly different and
well studied models - depth-$3$ circuits with bounded top fanin, and constant-depth constant-read multilinear formulas - can be constructed using one common algebraic-geometry theme: \emph{Jacobian} captures algebraic independence. By exploiting the Jacobian, we design the {\em first} efficient hitting-set generators for broad generalizations of the bove-mentioned models, namely: \begin{itemize} \item depth-$3$ ($\Sigma \Pi \Sigma$) circuits with constant \emph{transcendence degree} of the polynomials computed by the
product gates (\emph{no} bounded top fanin restriction), and \item constant-depth constant-\emph{occur} formulas (\emph{no} multilinear restriction). \end{itemize}
Constant-\emph{occur} of a variable, as we define it, is a much more general concept than constant-read. Also, earlier work on the latter model assumed that the formula is multilinear. Thus, our work goes further beyond the related results obtained by Saxena \& Seshadhri (STOC 2011), Saraf \& Volkovich (STOC 2011), Anderson et al.\ (CCC 2011), Beecken et al.\ (ICALP 2011) and Grenet et al.\ (FSTTCS 2011), and brings them under one unifying technique.
In addition, using the same Jacobian based approach, we prove exponential lower bounds for the immanant (which includes permanent and determinant) on the \emph{same} depth-$3$ and depth-$4$ models for which we give efficient PIT algorithms. Our results reinforce the intimate
connection between identity testing and lower bounds by exhibiting a concrete mathematical tool - the Jacobian - that is equally effective in solving both the problems on certain interesting and previously well-investigated (but not well understood) models of computation.
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Access Level:
Internal

Correlation
MPG Unit:
Max-Planck-Institut für Informatik
MPG Subunit:
Algorithms and Complexity Group
Appearance:
MPII WWW Server, MPII FTP Server, MPG publications list, university publications list, working group publication list, Fachbeirat, VG Wort



BibTeX Entry:

@INPROCEEDINGS{ASSS12,
AUTHOR = {Agrawal, Manindra and Saha, Chandan and Saptharishi, Ramprasad and Saxena, Nitin},
TITLE = {Jacobian hits circuits: hitting-sets, lower bounds for {depth-D} occur-k formulas & depth-3 transcendence degree-k circuits},
BOOKTITLE = {STOC'12 : Proceedings of the 2012 ACM Symposium on Theory of Computing},
PUBLISHER = {ACM},
YEAR = {2012},
PAGES = {599--614},
ADDRESS = {New York, NY},
ISBN = {978-1-4503-1245-5},
DOI = {10.1145/2213977.2214033},
}


Entry last modified by Uwe Brahm, 02/01/2013
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Editor(s)
[Library]
Created
01/17/2013 08:42:25 AM
Revisions
3.
2.
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Editor(s)
Uwe Brahm
Anja Becker
Chandan Saha
Chandan Saha
Edit Dates
01-02-2013 02:52:34 PM
25.01.2013 11:00:03
01/18/2013 05:19:27 AM
01/17/2013 08:42:25 AM