Geometric complexity theory, tensor rank, and Littlewood-Richardson coefficients

Diese Arbeit führt gründlich in die Geometrische Komplexitätstheorie ein, ein Ansatz, um untere Berechnungskomplexitätsschranken mittels Methoden aus der algebraischen Geometrie und Darstellungstheorie zu finden. Danach konzentrieren wir uns auf die relevanten darstellungstheoretischen Multiplizitäten, und zwar auf Plethysmenkoeffizienten, Kronecker-Koeffizienten und Littlewood-Richardson-Koeffizienten. Diese Multiplizitäten haben eine Beschreibung als Dimensionen von Höchstgewichtsvektorräumen, für welche konkrete Basen nur im Littlewood-Richardson-Fall bekannt sind.Durch explizite Konstruktion von Höchstgewichtsvektoren können wir zeigen, dass der Grenzrang der m x m Matrixmultiplikation mindestens 3 m^2 - 2 ist, und der Grenzrang der 2 x 2 Matrixmultiplikation genau sieben ist. Dies liefert einen neuen Beweis für ein Ergebnis von Landsberg (J. Amer. Math. Soc., 19:447-459, 2005).Desweiteren erhalten wir Nichtverschwindungsresultate für rechteckige Kronecker-Koeffizienten und wir beweisen eine Vermutung von Weintraub (J. Algebra, 129 (1): 103-114, 1990) uber das Nicht-Verschwinden von Plethysmen-koeffizienten von geraden Partitionen.Unsere eingehenden Untersuchungen zu Littlewood-Richardson-Koeffizienten c_We provide a thorough introduction to Geometric Complexity Theory, an approach towards computational complexity lower bounds via methods from algebraic geometry and representation theory. Then we focus on the relevant representation theoretic multiplicities, namely plethysm coefficients, Kronecker coefficients, and Littlewood-Richardson coefficients. These multiplicities can be described as dimensions of highest weight vector spaces for which explicit bases are known only in the Littlewood-Richardson case.By explicit construction of highest weight vectors we can show that the border rank of m x m matrix multiplication is a least 3 m^2 - 2 and the border rank of 2 x 2 matrix multiplication is exactly seven. The latter gives a new proof of a result by Landsberg (J. Amer. Math. Soc., 19:447-459, 2005).Moreover, we obtain new nonvanishing results for rectangular Kronecker coefficients and we prove a conjecture by Weintraub (J. Algebra, 129 (1): 103-114, 1990) about the nonvanishing of plethysm coefficients of even partitions.Our in-depth study of Littlewood-Richardson coefficients c_Tag der Verteidigung: 18.10.2012Paderborn, Univ., Diss., 201

Structure vs. Randomness for Bilinear Maps

We prove that the slice rank of a 3-tensor (a combinatorial notion introduced by Tao in the context of the cap-set problem), the analytic rank (a Fourier-theoretic notion introduced by Gowers and Wolf), and the geometric rank (a recently introduced algebro-geometric notion) are all equivalent up to an absolute constant. As a corollary, we obtain strong trade-offs on the arithmetic complexity of a biased bililnear map, and on the separation between computing a bilinear map exactly and on average. Our result settles open questions of Haramaty and Shpilka [STOC 2010], and of Lovett [Discrete Anal., 2019] for 3-tensors.Comment: Submitted on November 6, 2020 to the 53rd Annual ACM Symposium on Theory of Computing (STOC). Accepted on February 6, 202

Report on "Geometry and representation theory of tensors for computer science, statistics and other areas."

This is a technical report on the proceedings of the workshop held July 21 to July 25, 2008 at the American Institute of Mathematics, Palo Alto, California, organized by Joseph Landsberg, Lek-Heng Lim, Jason Morton, and Jerzy Weyman. We include a list of open problems coming from applications in 4 different areas: signal processing, the Mulmuley-Sohoni approach to P vs. NP, matchgates and holographic algorithms, and entanglement and quantum information theory. We emphasize the interactions between geometry and representation theory and these applied areas

No occurrence obstructions in geometric complexity theory

The permanent versus determinant conjecture is a major problem in complexity theory that is equivalent to the separation of the complexity classes VP_{ws} and VNP. Mulmuley and Sohoni (SIAM J. Comput., 2001) suggested to study a strengthened version of this conjecture over the complex numbers that amounts to separating the orbit closures of the determinant and padded permanent polynomials. In that paper it was also proposed to separate these orbit closures by exhibiting occurrence obstructions, which are irreducible representations of GL_{n^2}(C), which occur in one coordinate ring of the orbit closure, but not in the other. We prove that this approach is impossible. However, we do not rule out the general approach to the permanent versus determinant problem via multiplicity obstructions as proposed by Mulmuley and Sohoni.Comment: Substantial revision. This version contains an overview of the proof of the main result. Added material on the model of power sums. Theorem 4.14 in the old version, which had a complicated proof, became the easy Theorem 5.4. To appear in the Journal of the AM

Towards a Geometric Approach to Strassen's Asymptotic Rank Conjecture

We make a first geometric study of three varieties in $\mathbb{C}^m \otimes \mathbb{C}^m \otimes \mathbb{C}^m$ (for each $m$), including the Zariski closure of the set of tight tensors, the tensors with continuous regular symmetry. Our motivation is to develop a geometric framework for Strassen's Asymptotic Rank Conjecture that the asymptotic rank of any tight tensor is minimal. In particular, we determine the dimension of the set of tight tensors. We prove that this dimension equals the dimension of the set of oblique tensors, a less restrictive class introduced by Strassen.Comment: Final version. Revisions in Section 1 and Section