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Lower bounds for bounded depth arithmetic circuits

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Title
Lower bounds for bounded depth arithmetic circuits
Name (type = personal)
NamePart (type = family)
Kumar
NamePart (type = given)
Mrinal
NamePart (type = date)
1990-
DisplayForm
Mrinal Kumar
Role
RoleTerm (authority = RULIB)
author
Name (type = personal)
NamePart (type = family)
Saraf
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Shubhangi
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Shubhangi Saraf
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Advisory Committee
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chair
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Kopparty
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Swastik
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Swastik Kopparty
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Advisory Committee
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co-chair
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Allender
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Eric
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Eric Allender
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Advisory Committee
Role
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internal member
Name (type = personal)
NamePart (type = family)
Raz
NamePart (type = given)
Ran
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Ran Raz
Affiliation
Advisory Committee
Role
RoleTerm (authority = RULIB)
outside member
Name (type = corporate)
NamePart
Rutgers University
Role
RoleTerm (authority = RULIB)
degree grantor
Name (type = corporate)
NamePart
Graduate School - New Brunswick
Role
RoleTerm (authority = RULIB)
school
TypeOfResource
Text
Genre (authority = marcgt)
theses
OriginInfo
DateCreated (qualifier = exact)
2017
DateOther (qualifier = exact); (type = degree)
2017-05
CopyrightDate (encoding = w3cdtf); (qualifier = exact)
2017
Place
PlaceTerm (type = code)
xx
Language
LanguageTerm (authority = ISO639-2b); (type = code)
eng
Subject (authority = RUETD)
Topic
Computer Science
RelatedItem (type = host)
TitleInfo
Title
Rutgers University Electronic Theses and Dissertations
Identifier (type = RULIB)
ETD
Identifier
ETD_7915
PhysicalDescription
Form (authority = gmd)
electronic resource
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application/pdf
InternetMediaType
text/xml
Extent
1 online resource (viii, 201 p. : ill.)
Note (type = degree)
Ph.D.
Note (type = bibliography)
Includes bibliographical references
Abstract (type = abstract)
Proving lower bounds for arithmetic circuits is a problem of fundamental importance in
theoretical computer science. In recent years, an approach to this problem has emerged
via the depth reduction results of Agrawal and Vinay [AV08], which show that strong
enough lower bounds for extremely structured bounded depth circuits (even homogeneous
depth-4 circuits) suffice for general arithmetic circuits lower bounds. In this dissertation,
we study homogeneous depth-4 and homogeneous depth-5 arithmetic circuits
with a view towards proving strong lower bounds, and understanding the optimality of
the depth reduction results. Some of our main results are as follows.
• We show a hierarchy theorem for bottom fan-in for homogeneous depth-4 circuits
with bounded bottom fan-in. More formally, we show that there for a wide range
of choice of parameter t, there is a homogeneous polynomial in n variables of
degree d = nΘ(1) which can be computed by a homogeneous depth-4 circuit of
bottom fan-in t, but any homogeneous depth-4 circuit of bottom fan-in at most
t/20 must have top fan-in nΩ(d/t)
• We show that there is an explicit polynomial family such that any homogeneous
depth-4 arithmetic circuit computing it must have super-polynomial size. These
were the first superpolynomial lower bounds for homogeneous depth-4 circuits
with no restriction on top or bottom fan-in. Simultaneously and independently,
a similar lower bound was also proved by Kayal et al [KLSS14b].
• We show that any homogeneous depth-4 circuit computing the iterated matrix
multiplication polynomial in n variables and degree d = n
Θ(1) must have size at
least nΩ(√d). This shows that the upper bounds of depth reduction from general
arithmetic circuits to homogeneous depth-4 circuits are almost optimal, up to a
constant in the exponent. Moreover, these were the first nΩ(√d) lower bounds
for homogeneous depth-4 circuits over all fields. Prior to our work, Kayal et
al. [KLSS14a] had shown such a lower bound over the fields of characteristic zero.
• We show that there is a family of polynomials in n variables and degree d =
O(log2 n) which can be computed by linear size homogeneous depth-5 circuits
and polynomial size non-homogeneous depth-3 circuits but require homogeneous
depth-4 circuits of size nΩ(√d). In addition to indicating the power of increased
depth, and non-homogeneity, these results also show that for the range of parameters
considered here, the upper bounds for the depth reduction results [AV08,
Koi12, Tav15] are close to optimal in a very strong sense : a general depth reduction
to homogeneous depth-4 circuits of size nΩ(√d) is not possible even for
homogeneous depth-5 circuits of linear size.
• We show an exponential lower bound for homogeneous depth-5 circuits computing
an explicit polynomial over all finite fields of constant size. For any non-binary
field, these were the first such super-polynomial lower bounds, and prior to our
work, even cubic lower bounds were not known for homogeneous depth-5 circuits.
On the way to our proofs, we study the complexity of some natural polynomial families
(for instance, homogeneous depth-4, depth-5 circuits, iterated matrix multiplication)
with respect to many existing partial derivative based complexity measures, and also
define and analyze some new variants of these measures [KS14, KS15b].
Subject (authority = ETD-LCSH)
Topic
Computer arithmetic and logic units
Note (type = statement of responsibility)
by Mrinal Kumar
RelatedItem (type = host)
TitleInfo
Title
Graduate School - New Brunswick Electronic Theses and Dissertations
Identifier (type = local)
rucore19991600001
Location
PhysicalLocation (authority = marcorg); (displayLabel = Rutgers, The State University of New Jersey)
NjNbRU
Identifier (type = doi)
doi:10.7282/T3V98C10
Genre (authority = ExL-Esploro)
ETD doctoral
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The author owns the copyright to this work.
RightsHolder (type = personal)
Name
FamilyName
Kumar
GivenName
Mrinal
Role
Copyright Holder
RightsEvent
Type
Permission or license
DateTime (encoding = w3cdtf); (qualifier = exact); (point = start)
2017-03-30 09:29:05
AssociatedEntity
Name
Mrinal Kumar
Role
Copyright holder
Affiliation
Rutgers University. Graduate School - New Brunswick
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Author Agreement License
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I hereby grant to the Rutgers University Libraries and to my school the non-exclusive right to archive, reproduce and distribute my thesis or dissertation, in whole or in part, and/or my abstract, in whole or in part, in and from an electronic format, subject to the release date subsequently stipulated in this submittal form and approved by my school. I represent and stipulate that the thesis or dissertation and its abstract are my original work, that they do not infringe or violate any rights of others, and that I make these grants as the sole owner of the rights to my thesis or dissertation and its abstract. I represent that I have obtained written permissions, when necessary, from the owner(s) of each third party copyrighted matter to be included in my thesis or dissertation and will supply copies of such upon request by my school. I acknowledge that RU ETD and my school will not distribute my thesis or dissertation or its abstract if, in their reasonable judgment, they believe all such rights have not been secured. I acknowledge that I retain ownership rights to the copyright of my work. I also retain the right to use all or part of this thesis or dissertation in future works, such as articles or books.
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Copyright protected
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Open
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Permission or license
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