Related papers: An Optimal Tester for $k$-Linear
We study quantum algorithms for verifying properties of the output probability distribution of a classical or quantum circuit, given access to the source code that generates the distribution. We consider the basic task of uniformity…
We consider the task of testing properties of Boolean functions that are invariant under linear transformations of the Boolean cube. Previous work in property testing, including the linearity test and the test for Reed-Muller codes, has…
We prove a lower bound of $\tilde{\Omega}(n^{1/3})$ for the query complexity of any two-sided and adaptive algorithm that tests whether an unknown Boolean function $f:\{0,1\}^n\rightarrow \{0,1\}$ is monotone or far from monotone. This…
We give a unateness tester for functions of the form $f:[n]^d\rightarrow R$, where $n,d\in \mathbb{N}$ and $R\subseteq \mathbb{R}$ with query complexity $O(\frac{d\log (\max(d,n))}{\epsilon})$. Previously known unateness testers work only…
We study testing $\pi$-freeness of functions $f:[n]^d\to\mathbb{R}$, where $f$ is $\pi$-free if there there are no $k$ indices $x_1\prec\cdots\prec x_k\in [n]^d$ such that $f(x_i)<f(x_j)$ and $\pi(i) < \pi(j)$ for all $i,j \in [k]$, where…
It is known that a positive Boolean function f depending on n variables has at least n + 1 extremal points, i.e. minimal ones and maximal zeros. We show that f has exactly n + 1 extremal points if and only if it is linear read-once. The…
We consider the problem of testing whether an unknown and arbitrary set $S \subseteq \mathbb{R}^n$ (given as a black-box membership oracle) is convex, versus $\varepsilon$-far from every convex set, under the standard Gaussian distribution.…
A natural problem in high-dimensional inference is to decide if a classifier $f:\mathbb{R}^n \rightarrow \{-1,1\}$ depends on a small number of linear directions of its input data. Call a function $g: \mathbb{R}^n \rightarrow \{-1,1\}$, a…
In this paper, we study classes of Boolean functions that are testable with $O(\psi+1/\epsilon)$ queries, where $\psi$ depends on the parameters of the class (e.g., the number of terms, the number of relevant variables, etc.) but not on the…
We show that every algorithm for testing $n$-variate Boolean functions for monotonicity must have query complexity $\tilde{\Omega}(n^{1/4})$. All previous lower bounds for this problem were designed for non-adaptive algorithms and, as a…
Monotonicity testing of Boolean functions on the hypergrid, $f:[n]^d \to \{0,1\}$, is a classic topic in property testing. Determining the non-adaptive complexity of this problem is an important open question. For arbitrary $n$,…
The (low soundness) linearity testing problem for the middle slice of the Boolean cube is as follows. Let $\varepsilon>0$ and $f$ be a function on the middle slice on the Boolean cube, such that when choosing a uniformly random quadruple…
A $k$-modal probability distribution over the discrete domain $\{1,...,n\}$ is one whose histogram has at most $k$ "peaks" and "valleys." Such distributions are natural generalizations of monotone ($k=0$) and unimodal ($k=1$) probability…
We study testing of local properties in one-dimensional and multi-dimensional arrays. A property of $d$-dimensional arrays $f:[n]^d \to \Sigma$ is $k$-local if it can be defined by a family of $k \times \ldots \times k$ forbidden…
Let $d_k(n) = \sum_{n_1 \cdots n_k = n}1$ be the $k$-fold divisor function. We call a function $f:\mathbb{N} \to \mathbb{C}$ a $d_k$-bounded multiplicative function, if $f$ is multiplicative and $|f(n)| \leq d_k(n)$ for every $n \in…
We consider one-sided error property testing of $\mathcal{F}$-minor freeness in bounded-degree graphs for any finite family of graphs $\mathcal{F}$ that contains a minor of $K_{2,k}$, the $k$-circus graph, or the $(k\times 2)$-grid for any…
We study monotonicity testing of functions $f \colon \{0,1\}^d \to \{0,1\}$ using sample-based algorithms, which are only allowed to observe the value of $f$ on points drawn independently from the uniform distribution. A classic result by…
We prove a lower bound of $\Omega(n^{1/2 - c})$, for all $c>0$, on the query complexity of (two-sided error) non-adaptive algorithms for testing whether an $n$-variable Boolean function is monotone versus constant-far from monotone. This…
A Boolean function is called read-once over a basis B if it can be expressed by a formula over B where no variable appears more than once. A checking test for a read-once function f over B depending on all its variables is a set of input…
We study property testing of directed acyclicity in the unidirectional bounded-degree oracle model, where a query to a vertex reveals its outgoing neighbors. We prove that there exist absolute constants $d_0\in\mathbb{N}$ and…