Showing papers by "Joseph M. Landsberg published in 2018"

Polynomials and the exponent of matrix multiplication

Abstract: We define tensors, corresponding to cubic polynomials, which have the same exponent $\omega$ as the matrix multiplication tensor. In particular, we study the symmetrized matrix multiplication tensor $sM_n$ defined on an $n\times n$ matrix $A$ by $sM_n(A)=trace(A^3)$. The use of polynomials enables the introduction of additional techniques from algebraic geometry in the study of the matrix multiplication exponent $\omega$.

Matrix product states and the quantum max-flow/min-cut conjectures

Abstract: In this note, we discuss the geometry of matrix product states with periodic boundary conditions and provide three infinite sequences of examples where the quantum max-flow is strictly less than the quantum min-cut. In the first, we fix the underlying graph to be a 4-cycle and verify a prediction of Hastings that inequality occurs for infinitely many bond dimensions. In the second, we generalize this result to a 2d-cycle. In the third, we show that the 2d-cycle with periodic boundary conditions gives inequality for all d when all bond dimensions equal two, namely, a gap of at least 2d−2 between the quantum max-flow and the quantum min-cut.

Matrix product states and the quantum max-flow/min-cut conjectures

Abstract: In this note we discuss the geometry of matrix product states with periodic boundary conditions and provide three infinite sequences of examples where the quantum max-flow is strictly less than the quantum min-cut In the first we fix the underlying graph to be a 4-cycle and verify a prediction of Hastings that inequality occurs for infinitely many bond dimensions In the second we generalize this result to a 2d-cycle In the third we show that the 2d-cycle with periodic boundary conditions gives inequality for all d when all bond dimensions equal two, namely a gap of at least 2^{d-2} between the quantum max-flow and the quantum min-cut

A geometric study of Strassen's asymptotic rank conjecture and its variants.

Abstract: We establish basic information about the set of tight tensors, the tensors with continuous regular symmetry. Our motivation is Strassen's astounding 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. Surprisingly we prove this dimension equals the dimension of the set of oblique tensors, a less restrictive class of tensors that Strassen identified as useful for his laser method.

The geometry of rank decompositions of matrix multiplication II: $3\times 3$ matrices

Abstract: This is the second in a series of papers on rank decompositions of the matrix multiplication tensor. We present new rank $23$ decompositions for the $3\times 3$ matrix multiplication tensor $M_{\langle 3\rangle}$. All our decompositions have symmetry groups that include the standard cyclic permutation of factors but otherwise exhibit a range of behavior. One of them has 11 cubes as summands and admits an unexpected symmetry group of order 12. We establish basic information regarding symmetry groups of decompositions and outline two approaches for finding new rank decompositions of $M_{\langle n\rangle}$ for larger $n$.

Towards a Geometric Approach to Strassen's Asymptotic Rank Conjecture

Abstract: 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.

On the Geometry of Matrix Multiplication

Abstract: Our story begins with a spectacular failure: The standard algorithm to multiply two nxn matrices uses n multiplications. In 1969, while attempting to show that the standard algorithm was optimal, V. Strassen discovered an explicit algorithm to multiply 2x2 matrices using 7 multiplications rather than 8 = 2. It is a central question to determine just how efficiently one can multiply nxn matrices, both practically and asymptotically. In this talk, I will present a history of the problem, both of upper and lower complexity bounds. I will discuss how geometry, more precisely algebraic geometry and representation theory, are used in the study of matrix multiplication. In particular, I will explain how, had someone asked him 100 years ago, the algebraic geometer Terracini could have predicted Strassen’s algorithm. The talk will conclude with the recent use of representation theory to construct algorithms, more precisely, rank decompositions. For those who can’t wait for the talk, a detailed history and the state of the art appears in Landsberg, J. (2017). Geometry and Complexity Theory (Cambridge Studies in Advanced Mathematics 169).

The complexity of matrix multiplication: developments since 2014. Extended abstract of 2018 Oberwolfach Complexity meeting plenary lecture.

Abstract: This is an overview of recent developments regarding the complexity of matrix multiplication, with an emphasis on the uses of algebraic geometry and representation theory in complexity theory.