# News for November 2015

(Updating post with an additional paper that we missed in our first posting. Sorry! Feel free to email us at little.oh.of.n@gmail.com to inform us of papers we should mention.)

We’ve got exciting new in November! Optimal results for testing of monotone conjunctions, a new lower bound for monotonicity testing (yay!), and new lower bounds for Locally Testable Codes.

Tight Bounds for the Distribution-Free Testing of Monotone Conjunctions by Xi Chen and Jinyu Xie (arxiv). Consider functions $$f: \{0,1\}^n \rightarrow \{0,1\}$$ (amen), and the basic property of being a monotone conjunction. Our notion of distance is with respect to a distribution $$\mathcal{D}$$, so the distance between functions $$f$$ and $$g$$ is $$\mathop{Pr}_{x \sim \mathcal{D}} [f(x) \neq g(x)]$$. In the distribution-free testing model, the tester does not know the distribution $$\mathcal{D}$$, but is allowed samples from $$\mathcal{D}$$. When $$\mathcal{D}$$ is uniform, this coincides with standard property testing. There can be significant gaps between standard and distribution-free testing, evidenced by conjunctions. Parnas, Ron, and Samorodnitsky proved that monotone conjunctions can be tested in the vanilla setting in time independent of $$\mathcal{n}$$, while Glasner-Servedio prove a $$\Omega(n^{1/5})$$ lower bound for distribution-free testing. This paper provides a one-sided, adaptive distribution-free tester that makes $$\widetilde{O}(n^{1/3})$$ queries, and they also prove a matching (up to polylog terms and $$\epsilon$$-dependencies) two-sided, adaptive lower bound. This is a significant improvement on the previous upper bound of $$\widetilde{O}(\sqrt{n})$$ of Dolev-Ron, as well as over the previous lower bound. Furthermore, the results of this paper hold for general conjunctions.

A Polynomial Lower Bound for Testing Monotonicity by Aleksandrs Belovs and Eric Blais (arxiv). Surely the reader needs little introduction to testing monotonicity of Boolean functions, and a previous Open Problem post should help. A quick summary: we want to test monotonicity of $$f:\{0,1\}^n \rightarrow \{0,1\}$$. The best upper bound is the $$\widetilde{O}(\sqrt{n})$$ non-adaptive, one-sided tester of Khot et al. There is a matching, non-adaptive lower bound (up to polylog terms) by Chen et al, implying the best-known adaptive lower bound of $$\Omega(\log n)$$. Can adaptivity help? This paper proves an adaptive lower bound of $$\Omega(n^{1/4})$$. Exciting! The approach of Chen et al is to focus on monotonicity of linear threshold functions (technically, regular LTFS, where the weights are bounded). The authors show that this property can be tested in $$\mathop{poly}(\log n)$$ time, shooting down hopes of extending the LTF approach for adaptive lower bounds. The key insight is to work with the distribution of Talagrand’s random DNF instead, which is the most noise sensitive DNF. (Talagrand strikes again. He helps the upper bound, he helps the lower bound.) Perturbations of this DNF lead to non-monotone functions, the paper proves that these distributions cannot be distinguished in $$o(n^{1/4})$$ queries.

Lower bounds for constant query affine-invariant LCCs and LTCs by Arnab Bhattacharyya and Sivakanth Gopi (arxiv). Think of any code as a set/property of codewords in the domain $$\Sigma^N$$. Such a code is an $$r$$-LTC if it has an $$r$$-query property tester. An $$r$$-LCC is has the property that, given any $$x \in \Sigma^n$$ sufficiently close to a codeword, one can determine any coordinate of the codeword using $$r$$-queries. A fundamental question is to understand the length of LCCs and LTCS, or alternately (fixing $$\Sigma^N$$) determining the largest possible set of codewords. Existing constructions typically have much symmetry, either linear or affine invariance. (Check out Arnab Bhattacharyya’s survey on affine invariant properties for more details.) It is convenient to think of any codeword as a function $$f: [N] \rightarrow \Sigma$$, and furthermore, think of $$[N]$$ as a vector space $$\mathbb{K}^n$$. The best known construction of Guo, Kopparty, and Sudan yields (affine-invariant) LCCs of size $$\exp(\Theta(n^{r-1}))$$ and LTCs of size $$\exp(\Theta(n^{r-2}))$$ (many dependences on $$\mathbb{K}$$, the rate, etc. are hidden in the big-Oh). This paper show that these bounds are actually the best achievable by any affine-invariant code. (Previous lower bounds of Ben-Sasson and Sudan only held for linear codes.) The intriguing and wide-open question is to prove such lower bounds without affine invariance.