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We study the problems of learning and testing junta distributions on $\{-1,1\}^n$ with respect to the uniform distribution, where a distribution $p$ is a $k$-junta if its probability mass function $p(x)$ depends on a subset of at most $k$ variables. The main contribution is an algorithm for finding relevant coordinates in a $k$-junta distribution with subcube conditioning [BC18, CCKLW20]. We give two applications: 1. An algorithm for learning $k$-junta distributions with $\tilde{O}(k/\epsilon^2) \log n + O(2^k/\epsilon^2)$ subcube conditioning queries, and 2. An algorithm for testing $k$-junta distributions with $\tilde{O}((k + \sqrt{n})/\epsilon^2)$ subcube conditioning queries. All our algorithms are optimal up to poly-logarithmic factors. Our results show that subcube conditioning, as a natural model for accessing high-dimensional distributions, enables significant savings in learning and testing junta distributions compared to the standard sampling model. This addresses an open question posed by Aliakbarpour, Blais, and Rubinfeld [ABR17].

We show that there exist properties that are maximally hard for testing, while still admitting PCPPs with a proof size very close to linear. Specifically, for every fixed $\ell$, we construct a property $\mathcal{P}^{(\ell)}\subseteq\{0,1\}^n$ satisfying the following: Any testing algorithm for $\mathcal{P}^{(\ell)}$ requires $\Omega(n)$ many queries, and yet $\mathcal{P}^{(\ell)}$ has a constant query PCPP whose proof size is $O(n\cdot \log^{(\ell)}n)$, where $\log^{(\ell)}$ denotes the $\ell$ times iterated log function (e.g., $\log^{(2)}n = \log \log n$). The best previously known upper bound on the PCPP proof size for a maximally hard to test property was $O(n \cdot \mathrm{poly}\log{n})$. As an immediate application, we obtain stronger separations between the standard testing model and both the tolerant testing model and the erasure-resilient testing model: for every fixed $\ell$, we construct a property that has a constant-query tester, but requires $\Omega(n/\log^{(\ell)}(n))$ queries for every tolerant or erasure-resilient tester.

The emerging theory of graph limits exhibits an analytic perspective on graphs, showing that many important concepts and tools in graph theory and its applications can be described more naturally (and sometimes proved more easily) in analytic language. We extend the theory of graph limits to the ordered setting, presenting a limit object for dense vertex-ordered graphs, which we call an \emph{orderon}. As a special case, this yields limit objects for matrices whose rows and columns are ordered, and for dynamic graphs that expand (via vertex insertions) over time. Along the way, we devise an ordered locality-preserving variant of the cut distance between ordered graphs, showing that two graphs are close with respect to this distance if and only if they are similar in terms of their ordered subgraph frequencies. We show that the space of orderons is compact with respect to this distance notion, which is key to a successful analysis of combinatorial objects through their limits. We derive several applications of the ordered limit theory in extremal combinatorics, sampling, and property testing in ordered graphs. In particular, we prove a new ordered analogue of the well-known result by Alon and Stav [RS\&A'08] on the furthest graph from a hereditary property; this is the first known result of this type in the ordered setting. Unlike the unordered regime, here the random graph model $G(n, p)$ with an ordering over the vertices is \emph{not} always asymptotically the furthest from the property for some $p$. However, using our ordered limit theory, we show that random graphs generated by a stochastic block model, where the blocks are consecutive in the vertex ordering, are (approximately) the furthest. Additionally, we describe an alternative analytic proof of the ordered graph removal lemma [Alon et al., FOCS'17]., Added a new application: An Alon-Stav type result on the furthest ordered graph from a hereditary property; Fixed and extended proof sketch of the removal lemma application

In 1994, Talagrand showed a generalization of the celebrated KKL theorem. In this work, we prove that the converse of this generalization also holds. Namely, for any sequence of numbers $00$, it is possible to find a roughly balanced Boolean function $f$ such that $\textrm{Inf}_j[f] < a_j$ for every $1 \le j \le n$.