FRINK Linked Data Fragments server

Matches in SemOpenAlex for { <https://semopenalex.org/work/W2964049658> ?p ?o ?g. }

• W2964049658 startingPage "19" @default.
• W2964049658 abstract "Arithmetic complexity, the study of the cost of computing polynomials via additions and multiplications, is considered (for many good reasons) simpler to understand than Boolean complexity, namely computing Boolean functions via logical gates. And indeed, we seem to have significantly more lower bound techniques and in arithmetic complexity than in Boolean complexity. Despite many successes and rapid progress, however, foundational challenges, like proving super-polynomial lower bounds on circuit or formula size for explicit polynomials, or super-linear lower bounds on explicit 3-dimensional tensors, remain elusive.At the same time (and possibly for similar reasons), we have plenty more excuses, in the form of results for failing to prove basic lower bounds in Boolean complexity than in arithmetic complexity. Efforts to find barriers to arithmetic lower bound techniques seem harder, and despite some attempts we have no excuses of similar quality for these failures in arithmetic complexity. This paper aims to add to this study.In this paper we address rank methods, which were long recognized as encompassing and abstracting almost all known arithmetic lower bounds to-date, including the most recent impressive successes. Rank methods (under the name of flattenings) are also in wide use in algebraic geometry for proving tensor rank and symmetric tensor rank lower bounds. Our main are barriers to these methods. In particular, 1. Rank methods cannot prove better than (2^d)*n^(d/2) lower bound on the tensor rank of any d-dimensional tensor of side n. (In particular, they cannot prove super-linear, indeed even >8n tensor rank lower bounds for any 3-dimensional tensors.)2. Rank methods cannot prove (d+1)n^(d/2) on the Waring rank of any n-variate polynomial of degree d. (In particular, they cannot prove such lower bounds on stronger models, including depth-3 circuits.)The of these bounds use simple linear-algebraic arguments, leveraging connections between the symbolic rank of matrix polynomials and the usual rank of their evaluations. These techniques can perhaps be extended to barriers for other arithmetic models on which progress has halted. To see how these barrier directly inform the state-of-art in arithmetic complexity we note the following.First, the bounds above nearly match the best explicit bounds we know for these models, hence offer an explanations why the rank methods got stuck there. Second, the bounds above are a far cry (quadratically away) from the true complexity (e.g. of random polynomials) in these models, which if achieved (by any methods), are known to imply super-polynomial formula lower bounds.We also explain the relation of our barrier to other attempts, and in particular how they significantly differ from the recent attempts to find analogues of proofs for arithmetic complexity. Finally, we discuss the few arithmetic lower bound approaches which fall outside rank methods, and some natural directions our barriers suggest." @default.
• W2964049658 created "2019-07-30" @default.
• W2964049658 creator A5027551258 @default.
• W2964049658 creator A5039043184 @default.
• W2964049658 creator A5069625742 @default.
• W2964049658 creator A5083692440 @default.
• W2964049658 date "2018-01-01" @default.
• W2964049658 modified "2023-09-27" @default.
• W2964049658 title "Barriers for Rank Methods in Arithmetic Complexity" @default.
• W2964049658 doi "https://doi.org/10.4230/lipics.itcs.2018.1" @default.
• W2964049658 hasPublicationYear "2018" @default.
• W2964049658 type Work @default.
• W2964049658 sameAs 2964049658 @default.
• W2964049658 citedByCount "10" @default.
• W2964049658 countsByYear W29640496582018 @default.
• W2964049658 countsByYear W29640496582019 @default.
• W2964049658 countsByYear W29640496582020 @default.
• W2964049658 countsByYear W29640496582021 @default.
• W2964049658 crossrefType "proceedings-article" @default.
• W2964049658 hasAuthorship W2964049658A5027551258 @default.
• W2964049658 hasAuthorship W2964049658A5039043184 @default.
• W2964049658 hasAuthorship W2964049658A5069625742 @default.
• W2964049658 hasAuthorship W2964049658A5083692440 @default.
• W2964049658 hasConcept C11413529 @default.
• W2964049658 hasConcept C114614502 @default.
• W2964049658 hasConcept C118615104 @default.
• W2964049658 hasConcept C134306372 @default.
• W2964049658 hasConcept C136119220 @default.
• W2964049658 hasConcept C155281189 @default.
• W2964049658 hasConcept C155920423 @default.
• W2964049658 hasConcept C164226766 @default.
• W2964049658 hasConcept C179799912 @default.
• W2964049658 hasConcept C187455244 @default.
• W2964049658 hasConcept C202444582 @default.
• W2964049658 hasConcept C311688 @default.
• W2964049658 hasConcept C33923547 @default.
• W2964049658 hasConcept C77553402 @default.
• W2964049658 hasConcept C83581934 @default.
• W2964049658 hasConcept C90119067 @default.
• W2964049658 hasConcept C9376300 @default.
• W2964049658 hasConcept C94375191 @default.
• W2964049658 hasConceptScore W2964049658C11413529 @default.
• W2964049658 hasConceptScore W2964049658C114614502 @default.
• W2964049658 hasConceptScore W2964049658C118615104 @default.
• W2964049658 hasConceptScore W2964049658C134306372 @default.
• W2964049658 hasConceptScore W2964049658C136119220 @default.
• W2964049658 hasConceptScore W2964049658C155281189 @default.
• W2964049658 hasConceptScore W2964049658C155920423 @default.
• W2964049658 hasConceptScore W2964049658C164226766 @default.
• W2964049658 hasConceptScore W2964049658C179799912 @default.
• W2964049658 hasConceptScore W2964049658C187455244 @default.
• W2964049658 hasConceptScore W2964049658C202444582 @default.
• W2964049658 hasConceptScore W2964049658C311688 @default.
• W2964049658 hasConceptScore W2964049658C33923547 @default.
• W2964049658 hasConceptScore W2964049658C77553402 @default.
• W2964049658 hasConceptScore W2964049658C83581934 @default.
• W2964049658 hasConceptScore W2964049658C90119067 @default.
• W2964049658 hasConceptScore W2964049658C9376300 @default.
• W2964049658 hasConceptScore W2964049658C94375191 @default.
• W2964049658 hasLocation W29640496581 @default.
• W2964049658 hasOpenAccess W2964049658 @default.
• W2964049658 hasPrimaryLocation W29640496581 @default.
• W2964049658 hasRelatedWork W1557604257 @default.
• W2964049658 hasRelatedWork W1561337879 @default.
• W2964049658 hasRelatedWork W1619519317 @default.
• W2964049658 hasRelatedWork W1922438893 @default.
• W2964049658 hasRelatedWork W1982983168 @default.
• W2964049658 hasRelatedWork W1986127216 @default.
• W2964049658 hasRelatedWork W1996146326 @default.
• W2964049658 hasRelatedWork W2019603005 @default.
• W2964049658 hasRelatedWork W2044352141 @default.
• W2964049658 hasRelatedWork W2084313768 @default.
• W2964049658 hasRelatedWork W2102892751 @default.
• W2964049658 hasRelatedWork W2165292791 @default.
• W2964049658 hasRelatedWork W2399133390 @default.
• W2964049658 hasRelatedWork W2401945280 @default.
• W2964049658 hasRelatedWork W2556081478 @default.
• W2964049658 hasRelatedWork W2575542057 @default.
• W2964049658 hasRelatedWork W2597007794 @default.
• W2964049658 hasRelatedWork W2952397747 @default.
• W2964049658 hasRelatedWork W2964226640 @default.
• W2964049658 hasRelatedWork W84137593 @default.
• W2964049658 hasVolume "94" @default.
• W2964049658 isParatext "false" @default.
• W2964049658 isRetracted "false" @default.
• W2964049658 magId "2964049658" @default.
• W2964049658 workType "article" @default.