Related papers: Log-rank and lifting for AND-functions
For an integer n>2 we define a polylogarithm, which is a holomorphic function on the universal abelian cover of C-{0,1} defined modulo (2 pi i)^n/(n-1)!. We use the formal properties of its functional relations to define groups lifting…
We prove that if a degree-$d$ homogeneous polynomial $f$ has border Waring rank $\underline{\mathrm{WR}}({f}) = r$, then its Waring rank is bounded by \[ {\mathrm{WR}}({f}) \leq d \cdot r^{O(\sqrt{r})}. \] This result significantly improves…
We prove a complexity dichotomy for a class of counting problems expressible as bipartite 3-regular Holant problems. For every problem of the form $\operatorname{Holant}\left(f\mid =_3 \right)$, where $f$ is any integer-valued ternary…
The Weighted First-Order Model Counting Problem (WFOMC) asks to compute the weighted sum of models of a given first-order logic sentence over a given domain. Conditioning WFOMC on evidence -- fixing the truth values of a set of ground…
In this note we compare two measures of the complexity of a class $\mathcal F$ of Boolean functions studied in (unconditional) pseudorandomness: $\mathcal F$'s ability to distinguish between biased and uniform coins (the coin problem), and…
In this paper we turn the spotlight on a class of lexicographic ranking functions introduced by Bradley, Manna and Sipma in a seminal CAV 2005 paper, and establish for the first time the complexity of some problems involving the inference…
Let the randomized query complexity of a relation for error probability $\epsilon$ be denoted by $R_\epsilon(\cdot)$. We prove that for any relation $f \subseteq \{0,1\}^n \times \mathcal{R}$ and Boolean function $g:\{0,1\}^m \rightarrow…
We introduce partial differential encodings of Boolean functions as a way of measuring the complexity of Boolean functions. These encodings enable us to derive from group actions non-trivial bounds on the Chow-Rank of polynomials used to…
We present a constructive SAT-based algorithm to determine the multiplicative complexity of a Boolean function, i.e., the smallest number of AND gates in any logic network that consists of 2-input AND gates, 2-input XOR gates, and…
In this paper we study quantum nondeterminism in multiparty communication. There are three (possibly) different types of nondeterminism in quantum computation: i) strong, ii) weak with classical proofs, and iii) weak with quantum proofs.…
In the standard model of computing multi-output functions in logspace ($\mathsf{FL}$), we are given a read-only tape holding $x$ and a logarithmic length worktape, and must print $f(x)$ to a dedicated write-only tape. However, there has…
$\newcommand{\F}{\mathbb{F}}$We study the Boolean function parameters sensitivity ($s$), block sensitivity ($bs$), and alternation ($alt$) under specially designed affine transforms. For a function $f:\F_2^n\to \{0,1\}$, and $A=Mx+b$ for $M…
We give a complexity dichotomy for the problem of computing the partition function of a weighted Boolean constraint satisfaction problem. Such a problem is parameterized by a set of rational-valued functions, which generalize constraints.…
The notion of non-deterministic logical matrix (where connectives are interpreted as multi-functions) preserves many good properties of traditional semantics based on logical matrices (where connectives are interpreted as functions) whilst…
We give the first {\sl reconstruction algorithm} for decision trees: given queries to a function $f$ that is $\mathrm{opt}$-close to a size-$s$ decision tree, our algorithm provides query access to a decision tree $T$ where: $\circ$ $T$ has…
The study of complexity measures of Boolean functions led Nisan and Szegedy to state the sensitivity conjecture in 1994, claiming a polynomial relation between degree and sensitivity. This problem remained unsolved until 2019, when Huang…
We give a deterministic polynomial time $2^{O(r)}$-approximation algorithm for the number of bases of a given matroid of rank $r$ and the number of common bases of any two matroids of rank $r$. To the best of our knowledge, this is the…
We study distributed low rank approximation in which the matrix to be approximated is only implicitly represented across the different servers. For example, each of $s$ servers may have an $n \times d$ matrix $A^t$, and we may be interested…
We consider solving the low rank matrix sensing problem with Factorized Gradient Descend (FGD) method when the true rank is unknown and over-specified, which we refer to as over-parameterized matrix sensing. If the ground truth signal…
In this paper we show that inference in 2-variable Markov logic networks (MLNs) with cardinality and function constraints is domain-liftable. To obtain this result we use existing domain-lifted algorithms for weighted first-order model…