The summer gets off to a flying start, with three property testing papers, spanning differential privacy, distribution testing, and juntas in Gaussian space!

**On closeness to \(k\)-wise uniformity**, by Ryan O’Donnell and Yu Zhao (arXiv)

In this paper, the authors consider the following structural question about probability distributions over the Boolean hypercube \(\{-1,1\}^n\): ” what is the relation between total variation distance \(\delta\) to \(k\)-wise independence, and bound \(\varepsilon\) on the Fourier coefficients of the distribution on degrees up to \(k\)?”

While this question might seem a bit esoteric at first glance, it has direct and natural applications to derandomization, and of course to distribution testing (namely, to test \(k\)-wise independence and its generalization, \((\varepsilon, k)\)-wise independence of distributions over the hypercube).

The main contribution here is to improve (by a \((\log n)^{O(k)}\) factor) the bounds on \(\delta (n,k,\varepsilon)\) over the previous work by Alon et al. [AAK+07], making them either tight (for \(k\) even) or near-tight. To do so, the authors introduce a new hammer to the game, using linear programming duality in the proof of both their upper and lower bounds.

**Property Testing for Differential Privacy**, by Anna Gilbert and Audra McMillan (arXiv)

Differential privacy, as introduced by Dwork et al., needs no introduction. Property testing, especially on this website, needs even less. *What about a combination of the two?* Namely, given black-box access to an algorithm claiming to perform a differentially private computation, how to test whether this is indeed the case?

Introducing and considering this quite natural question for the first time, this work shows, roughly speaking, that testing differential privacy is *hard*. Specifically, they show that for many notions of differential privacy (pure, approximate, and their distributional counterparts), testing is either impossible or possible but not with a sublinear number of queries (even when the tester is provided with side information about the black-box). In other terms, as the authors put it: trusting the privacy of an algorithm “requires compromise by either the verifier or algorithm owner” (and, in the latter case, even then it’s not a simple matter).

**Is your data low-dimensional?**, by Anindya De, Elchanan Mossel, and Joe Neeman (arXiv)

*(Well, is it?)* To state it upfront, I am biased here, as it is a problem I was very eager to see investigated to begin with. To recap, the question is as follows: “given query access to some unknown Boolean-valued function \(f\colon \mathbb{R}^n \to \{-1,1\}\) over the high-dimensional space \(\mathbb{R}^n\) endowed with the Gaussian measure, how can one check whether \(f\) only depends on “few” (i.e., \(k \ll n\)) variables?”

This is the continuous, Gaussian version of the (quite famous) junta testing problem, which has gathered significant attention over the past years *(the Gaussian version has, to the best of my knowledge, never been investigated).* Now, the above formulation has a major flaw: specifically, it is uninteresting. In Gaussian space*, who cares about the particular basis I expressed my input vector in? So a more relevant question, and that that the authors tackle, is the more robust and natural one: “given query access to some unknown Boolean-valued function \(f\colon \mathbb{R}^n \to \{-1,1\}\) over the high-dimensional space \(\mathbb{R}^n\) endowed with the Gaussian measure, how can one check whether \(f\) only depends on a low-dimensional linear combination of the variables?” Or, put differently, does all the relevant information for \(f\) live in a low-dimensional subspace?

De, Mossel, and Neeman show how can do this, non-adaptively, with a query complexity independent of the dimension \(n\) (hurray!), but instead polynomial in \(k\), the distance parameter \(\varepsilon\), and the * surface area \(s\)* of \(f\). And since this last parameter may seem quite arbitrary, they also proceed to show that a polynomial dependence in this \(s\) is indeed required.

**”In Gaussian space, no one can hear you change basis?”*