On the sum-of-squares degree of symmetric quadratic functions

@inproceedings{Lee2016OnTS,
  title={On the sum-of-squares degree of symmetric quadratic functions},
  author={Troy Lee and Anupam Prakash and Ronald de Wolf and Henry S. Yuen},
  booktitle={Computational Complexity Conference},
  year={2016}
}
We study how well functions over the boolean hypercube of the form $f_k(x)=(|x|-k)(|x|-k-1)$ can be approximated by sums of squares of low-degree polynomials, obtaining good bounds for the case of approximation in $\ell_{\infty}$-norm as well as in $\ell_1$-norm. We describe three complexity-theoretic applications: (1) a proof that the recent breakthrough lower bound of Lee, Raghavendra, and Steurer on the positive semidefinite extension complexity of the correlation and TSP polytopes cannot be… 
Symmetric sums of squares over k-subset hypercubes
TLDR
A variant of the Gatermann–Parrilo symmetry-reduction method tailored to the authors' setting that allows for several simplifications and a connection to flag algebras is developed, and every symmetric polynomial that has a sos expression of a fixed degree also has a succinct sosexpression whose size depends only on the degree and not on the number of variables.
SoS Certification for Symmetric Quadratic Functions and Its Connection to Constrained Boolean Hypercube Optimization
TLDR
It is proved that the SoS rank for SQFs is at most $O(\sqrt{nk}\log(n)$ and connected to two constrained Boolean hypercube optimization problems.
Sum-Of-Squares Bounds via Boolean Function Analysis
TLDR
A method for proving bounds on the SoS rank based on Boolean Function Analysis and Approximation Theory is introduced and applied to improve upon existing results, thus making progress towards answering several open questions.
Harmonicity and invariance on slices of the Boolean cube
TLDR
It is shown that any real-valued harmonic multilinear polynomial on the slice whose degree is o(n) has approximately the same distribution under the slice and cube measures.
Sum of squares lower bounds from symmetry and a good story
TLDR
This paper proves a tight sum of squares lower bound for the following Turan type problem: Minimize the number of triangles in a graph $G$ with a fixed edge density.
Partial Boolean Functions With Exact Quantum Query Complexity One
TLDR
It is shown that the number of all n-bit partial Boolean functions that depend on k bits and can be computed exactly by a 1-query quantum algorithm is not bigger than an upper bound depending on n and k.
A ug 2 01 6 Exact quantum query complexity of EXACT nk , l
TLDR
The minimum number of queries for an exact quantum algorithm computing the function f is denoted by QE(f), and the following n bit function with 0 ≤ k ≤ l ≤ n is considered.
Optimal one-shot quantum algorithm for EQUALITY and AND
TLDR
It is shown that the lowest possible error probability for $AND_n$ and $EQUALITY_{n+1}$ is $1/2-n/(n^2+1)$.
Lower Bounds for Interactive Compression and Linear Programs
TLDR
A new technique, the notion of fooling distributions, is introduced to prove that information can be exponentially smaller than communication, and a tight non-negative rank lower bound is obtained for a family of matrices known as lopsided unique disjointness.
Exact Quantum Query Complexity of \text EXACT_k, l^n
TLDR
An optimal algorithm (for some cases), and a nontrivial general lower and upper bound on the minimum number of queries to the black box.
...
...

References

SHOWING 1-10 OF 59 REFERENCES
Sums of squares on the hypercube
Let X be a finite set of points in $${\mathbb {R}}^n$$Rn. A polynomial p nonnegative on X can be written as a sum of squares of rational functions modulo the vanishing ideal I(X). From the point of
Query Complexity in Expectation
TLDR
This work exactly characterize both the randomized and the quantum query complexity by two polynomial degrees, the nonnegative literal degree and the sum-of-squares degree, respectively, to derive some upper bounds on psd extension complexity by constructing efficient quantum query algorithms.
Quantum lower bounds by polynomials
TLDR
This work examines the number T of queries that a quantum network requires to compute several Boolean functions on {0,1}/sup N/ in the black-box model and gives asymptotically tight characterizations of T for all symmetric f in the exact, zero-error, and bounded-error settings.
Lower Bounds on the Size of Semidefinite Programming Relaxations
TLDR
It is proved that SDPs of polynomial-size are equivalent in power to those arising from degree-O(1) sum-of-squares relaxations, and this result yields the first super-polynomial lower bounds on the semidefinite extension complexity of any explicit family of polytopes.
New degree bounds for polynomial threshold functions
TLDR
The upper bounds for Boolean formulas yield the first known subexponential time learning algorithms for formulas of superconstant depth and the lower bounds for constant-depth circuits and intersections of halfspaces are the first new degree lower bounds since 1968.
Harmonicity and invariance on slices of the Boolean cube
TLDR
It is shown that any real-valued harmonic multilinear polynomial on the slice whose degree is o(n) has approximately the same distribution under the slice and cube measures.
A note on quantum algorithms and the minimal degree of ε-error polynomials for symmetric functions
  • R. D. Wolf
  • Computer Science
    Quantum Inf. Comput.
  • 2008
TLDR
This note shows how a tighter version of dege(f), without thelog-factors hidden in the Θ-notation, can be derived quite easily using the close connection between polynomials and quantum algorithms.
A note on quantum algorithms and the minimal degree of epsilon-error polynomials for symmetric functions
TLDR
A tighter version of the minimal degree deg_{\eps}(f) can be derived quite easily using the close connection between polynomials and quantum algorithms.
On Quantum Versions of the Yao Principle
The classical Yao principle states that the complexity R?(f) of an optimal randomized algorithm for a function f with success probability 1 - ? equals the complexity maxµ D?µ (f) of an optimal
Sum-of-squares hierarchy lower bounds for symmetric formulations
TLDR
The main technical theorem allows the study of the positive semidefiniteness to be reduced to the analysis of “well-behaved” univariate polynomial inequalities and gives a short elementary proof of Grigoriev/Laurent lower bound for finding the integer cut polytope of the complete graph.
...
...