1 research outputs found
Unifying and generalizing known lower bounds via geometric complexity theory
We show that most arithmetic circuit lower bounds and relations between lower
bounds naturally fit into the representation-theoretic framework suggested by
geometric complexity theory (GCT), including: the partial derivatives technique
(Nisan-Wigderson), the results of Razborov and Smolensky on ,
multilinear formula and circuit size lower bounds (Raz et al.), the degree
bound (Strassen, Baur-Strassen), the connected components technique (Ben-Or),
depth 3 arithmetic circuit lower bounds over finite fields
(Grigoriev-Karpinski), lower bounds on permanent versus determinant
(Mignon-Ressayre, Landsberg-Manivel-Ressayre), lower bounds on matrix
multiplication (B\"{u}rgisser-Ikenmeyer) (these last two were already known to
fit into GCT), the chasms at depth 3 and 4 (Gupta-Kayal-Kamath-Saptharishi;
Agrawal-Vinay; Koiran), matrix rigidity (Valiant) and others. That is, the
original proofs, with what is often just a little extra work, already provide
representation-theoretic obstructions in the sense of GCT for their respective
lower bounds. This enables us to expose a new viewpoint on GCT, whereby it is a
natural unification and broad generalization of known results. It also shows
that the framework of GCT is at least as powerful as known methods, and gives
many new proofs-of-concept that GCT can indeed provide significant asymptotic
lower bounds. This new viewpoint also opens up the possibility of fruitful
two-way interactions between previous results and the new methods of GCT; we
provide several concrete suggestions of such interactions. For example, the
representation-theoretic viewpoint of GCT naturally provides new properties to
consider in the search for new lower bounds