Explicit lower bounds via geometric complexity theory

Abstract

We prove the lower bound R(Mm) ≥ 3 2 m 2 − 2 on the border rank of m × m matrix multiplication by exhibiting explicit representation theoretic (occurence) obstructions in the sense the geometric complexity theory (GCT) program. While this bound is weaker than the one recently obtained by Landsberg and Ottaviani, these are the first significant lower bounds obtained within the GCT program. Behind the proof is an explicit description of the highest weight vectors in Sym d 3 (C n ) * in terms of combinatorial objects, called obstruction designs. This description results from analyzing the process of polarization and Schur-Weyl duality.

Fundamental invariants of orbit closures

“…Remark 5.10. The fact k n 2 (n) = 1 also immediately follows combinatorially from the upper and lower bounds on the Kronecker coefficient in [32] and [6], see also [22,Lemma 2.3].…”

Geometric aspects of Iterated Matrix Multiplication

“…In particular, sequences of polynomials that are complete for some complexity class and that are characterized by their stabilizer can be considered best representatives of their class. This approach was used in the past to provide representation theoretic obstructions proving lower bounds on the complexity of certain polynomials (see [9] and [7]) and will be part of future work in the setting of IM M n q . In Section 2, we determine the symmetry group of the polynomial IM M n q (Theorem 2.9).…”

Unifying Known Lower Bounds via Geometric Complexity Theory

“…Both Question IV.3 and this one seem within reach, especially given the recent occurrence obstructions constructed by Bürgisser and Ikenmeyer [15] in the context of matrix multiplication.…”

“…• Lower bounds on restricted depth 3 algebraic circuits in characteristic zero [3] • Lower bounds on (unrestricted) depth 3 algebraic circuits over finite fields [4] • The recent lower bounds on depth 4 algebraic circuits with bottom fan-in O( √ n) [5] • Lower bounds on multilinear formula size [6] • The degree bound of Strassen [7] and BaurStrassen [8] (see below) • Lower bounds on real (semi-)algebraic decision trees [9], [10] • Lower bounds on bounded depth Boolean circuits [11], [12] • The best known lower bounds (n 2 /2) on permanent versus determinant [13] (already shown to use a separating module [14]) • Many lower bounds on matrix multiplication (already shown to use a separating module [15], [16], [17]) We expect that results which use similar techniques can be shown to use separating modules as well. We also observe that many relations between lower bounds yield relations between separating modules.…”

“…Recall that the radical of an ideal I is the ideal √ I def = {f : f k ∈ I for some k ≥ 1} 4. Separating modules are nearly equivalent to the "HWV obstructions" of Bürgisser and Ikenmeyer[15]. For a discussion of the exact relationship and choice of terminology see Appendix C FULL .…”