# News for December 2017

December 2017 concluded the year in style, with seven property testing papers spanning quite the range. Let’s hope 2018 will keep up on that trend! (And, of course, if we missed any, please point it out in the comments.The blame would be on our selves from the past year.)

We begin with graphs:

High Dimensional Expanders, by Alexander Lubotzky (arXiv). This paper surveys the recent developments in studying high dimensional expander graphs, a recent generalization of expanders which has become quite active in the past years and has intimate connections to property testing.

Generalized Turán problems for even cycles, by Dániel Gerbner, Ervin Győri, Abhishek Methuku, and Máté Vizer (arXiv).
A Generalized Turán Problem and its Applications, by Lior Gishboliner and Asaf Shapira (arXiv).
In these two independent works, the authors study questions of a following flavor: two subgraphs (patterns) $$H,H’$$, what is the maximum number of copies of $$H$$ which can exist in a graph $$G$$ promised to be $$H’$$-free? They consider the case where the said patterns are cycles on $$\ell,k$$ vertices respectively, and obtain asymptotic bounds on the above quantity (the two papers obtain somewhat incomparable bounds, and the first focuses on the case where both $$\ell,k$$ are even). These estimates, in turn, have applications to graph removal lemmata, as discussed in the second work (Section 1.2): specifically, it implies the existence of a removal lemma with a
tight super-polynomial bound, a question which was previously open.

Approximating the Spectrum of a Graph, by David Cohen-Steiner, Weihao Kong, Christian Sohler, and Gregory Valiant (arXiv). The authors obtain constant-time and query algorithm for the task of approximating (in $$\ell_1$$ norm) the spectrum of a graph $$G$$, i.e. the eigenvalues of its Laplacian, given random query access to the nodes of $$G$$ and to the neighbors of any given node. They also study the applications of this result to property testing in the bounded-degree model, showing that a large class of spectral properties of high-girth graphs is testable.

Then, we go quantum:

Schur-Weyl Duality for the Clifford Group with Applications: Property Testing, a Robust Hudson Theorem, and de Finetti Representations, by David Gross, Sepehr Nezami, and Michael Walter (arXiv).
Introducing and studying a duality theory for the Clifford group, the authors are able (among other results) to resolve an open question in quantum property testing, establishing a constant-query tester (indeed, making 6 queries) for testing whether an unknown quantum state is a stabilizer state. The previous best upper bound was linear in the number of qubits, as it proceeded by learning the state (“testing by learning”).

Quantum Lower Bound for a Tripartite Version of the Hidden Shift Problem, by Aleksandrs Belovs (arXiv). This work introduces and studies a generalization of (both) the hidden shift and 3-sum problems, and shows an $$\Omega(n^{1/3})$$ lower bound on its quantum query complexity. The author also considers a property testing version of the problem, for which he proves a similar lower bound—interestingly, this polynomial lower bound is shown using the adversary method, evading the “property testing barrier” which states that (a restricted version of) this method cannot yield better than a constant-query lower bound.

And to conclude, a distribution testing paper:

Approximate Profile Maximum Likelihood, by Dmitri S. Pavlichin, Jiantao Jiao, and Tsachy Weissman (arXiv) This paper proposes an efficient (linear-time) algorithm to approximate the profile maximum likelihood of a sequence of i.i.d. samples from an unknown distribution, i.e. the probability distribution which, ignoring the labels of the samples and keeping only the collision counts, maximizes the likelihood of the sequence observed. This provides a candidate solution to an open problem suggested by Orlitsky in a FOCS’17 workshop (see also Open problem 84), and one which would have direct implications to tolerant testing and estimation of symmetric properties of distributions.