Demystifying the border of depth-3 algebraic circuits

P Dutta, P Dwivedi, N Saxena - 2021 IEEE 62nd Annual …, 2022‏ - ieeexplore.ieee.org
Border complexity of polynomials plays an integral role in GCT (Geometric Complexity
Theory) approach to P versus NP. It tries to formalize the notion of 'approximating a …

Hitting sets and reconstruction for dense orbits in vp_ {e} and ΣΠΣ circuits

D Medini, A Shpilka - 36th Computational Complexity …, 2021‏ - drops.dagstuhl.de
In this paper we study polynomials in VP_ {e}(polynomial-sized formulas) and in ΣΠΣ
(polynomial-size depth-3 circuits) whose orbits, under the action of the affine group GL^{aff} …

[PDF][PDF] Completeness classes in algebraic complexity theory

P Bürgisser - arxiv preprint arxiv:2406.06217, 2024‏ - arxiv.org
arxiv:2406.06217v1 [cs.CC] 10 Jun 2024 Page 1 COMPLETENESS CLASSES IN
ALGEBRAIC COMPLEXITY THEORY PETER BÜRGISSER Abstract. The purpose of this …

Hitting sets for orbits of circuit classes and polynomial families

C Saha, B Thankey - ACM Transactions on Computation Theory, 2024‏ - dl.acm.org
The orbit of an n-variate polynomial over a field is the set. The orbit of a polynomial f is a
geometrically interesting subset of the set of affine projections of f. Affine projections of …

Fixed-parameter debordering of Waring rank

P Dutta, F Gesmundo, C Ikenmeyer, G **dal… - arxiv preprint arxiv …, 2024‏ - arxiv.org
Border complexity measures are defined via limits (or topological closures), so that any
function which can approximated arbitrarily closely by low complexity functions itself has low …

[كتاب][B] Algebraic complexity, asymptotic spectra and entanglement polytopes

J Zuiddam - 2018‏ - eprints.illc.uva.nl
Matrix rank is well-known to be multiplicative under the Kronecker product, additive under
the direct sum, normalised on identity matrices and non-increasing under multiplying from …

Succinct hitting sets and barriers to proving lower bounds for algebraic circuits

MA Forbes, A Shpilka, BL Volk - Theory of Computing, 2018‏ - theoryofcomputing.org
We formalize a framework of algebraically natural lower bounds for algebraic circuits. Just
as with the natural proofs notion of Razborov and Rudich (1997) for Boolean circuit lower …

Discovering the roots: Uniform closure results for algebraic classes under factoring

P Dutta, N Saxena, A Sinhababu - Proceedings of the 50th Annual ACM …, 2018‏ - dl.acm.org
Newton iteration (NI) is an almost 350 years old recursive formula that approximates a
simple root of a polynomial quite rapidly. We generalize it to a matrix recurrence (allRootsNI) …

On the power of border of depth-3 arithmetic circuits

M Kumar - ACM Transactions on Computation Theory (TOCT), 2020‏ - dl.acm.org
We show that over the field of complex numbers, every homogeneous polynomial of degree
d can be approximated (in the border complexity sense) by a depth-3 arithmetic circuit of top …

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

C Ikenmeyer, U Kandasamy - Proceedings of the 52nd Annual ACM …, 2020‏ - dl.acm.org
Understanding the difference between group orbits and their closures is a key difficulty in
geometric complexity theory (GCT): While the GCT program is set up to separate certain …