Algebraic Branching Programs, Border Complexity, and Tangent Spaces

03/10/2020
by   Markus Bläser, et al.
0

Nisan showed in 1991 that the width of a smallest noncommutative single-(source,sink) algebraic branching program (ABP) to compute a noncommutative polynomial is given by the ranks of specific matrices. This means that the set of noncommutative polynomials with ABP width complexity at most k is Zariski-closed, an important property in geometric complexity theory. It follows that approximations cannot help to reduce the required ABP width. It was mentioned by Forbes that this result would probably break when going from single-(source,sink) ABPs to trace ABPs. We prove that this is correct. Moreover, we study the commutative monotone setting and prove a result similar to Nisan, but concerning the analytic closure. We observe the same behavior here: The set of polynomials with ABP width complexity at most k is closed for single-(source,sink) ABPs and not closed for trace ABPs. The proofs reveal an intriguing connection between tangent spaces and the vector space of flows on the ABP. We close with additional observations on VQP and the closure of VNP which allows us to establish a separation between the two classes.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
11/23/2018

On polyhedral approximations of the positive semidefinite cone

Let D be the set of n× n positive semidefinite matrices of trace equal t...
research
02/26/2022

Transparency Beyond VNP in the Monotone Setting

Recently Hrubes and Yehudayoff (2021) showed a connection between the mo...
research
08/25/2017

Network Essence: PageRank Completion and Centrality-Conforming Markov Chains

Jiří Matoušek (1963-2015) had many breakthrough contributions in mathema...
research
11/10/2019

Implementing geometric complexity theory: On the separation of orbit closures via symmetries

Understanding the difference between group orbits and their closures is ...
research
05/17/2023

Border Complexity of Symbolic Determinant under Rank One Restriction

VBP is the class of polynomial families that can be computed by the dete...
research
06/03/2021

On the Computation of the Algebraic Closure of Finitely Generated Groups of Matrices

We investigate the complexity of computing the Zariski closure of a fini...
research
07/23/2020

Detecting and Enumerating Small Induced Subgraphs in c-Closed Graphs

Fox et al. [SIAM J. Comp. 2020] introduced a new parameter, called c-clo...

Please sign up or login with your details

Forgot password? Click here to reset