arXiv subjects

Rocco A. Servedio

Publications and source records attributed to Rocco A. Servedio.

At least 55 records · Page 3Linked to original sources

arXiv2020

Quantitative Correlation Inequalities via Semigroup Interpolation

Most correlation inequalities for high-dimensional functions in the literature, such as the Fortuin-Kasteleyn-Ginibre (FKG) inequality and the celebrated Gaussian Correlation Inequality of Royen, are qualitative statements which establish that any two functions of a certain type have non-negative correlation. In this work we give a general approach that can be used to bootstrap many qualitative correlation inequalities for functions over product spaces into quantitative statements. The approach combines a new extremal result about power series, proved using complex analysis, with harmonic analysis of functions over product spaces. We instantiate this general approach in several different concrete settings to obtain a range of new and near-optimal quantitative correlation inequalities, including: $\bullet$ A quantitative version of Royen's celebrated Gaussian Correlation Inequality. Royen (2014) confirmed a conjecture, open for 40 years, stating that any two symmetric, convex sets must be non-negatively correlated under any centered Gaussian distribution. We give a lower bound on the correlation in terms of the vector of degree-2 Hermite coefficients of the two convex sets, analogous to the correlation bound for monotone Boolean functions over $\{0,1\}^n$ obtained by Talagrand (1996). $\bullet$ A quantitative version of the well-known FKG inequality for monotone functions over any finite product probability space, generalizing the quantitative correlation bound for monotone Boolean functions over $\{0,1\}^n$ obtained by Talagrand (1996). The only prior generalization of which we are aware is due to Keller (2008, 2009, 2012), which extended Talagrand's result to product distributions over $\{0,1\}^n$. We also give two different quantitative versions of the FKG inequality for monotone functions over the continuous domain $[0,1]^n$, answering a question of Keller (2009).

math.PR↗

arXiv2020

Polynomial-time trace reconstruction in the low deletion rate regime

In the \emph{trace reconstruction problem}, an unknown source string $x \in \{0,1\}^n$ is transmitted through a probabilistic \emph{deletion channel} which independently deletes each bit with some fixed probability $δ$ and concatenates the surviving bits, resulting in a \emph{trace} of $x$. The problem is to reconstruct $x$ given access to independent traces. Trace reconstruction of arbitrary (worst-case) strings is a challenging problem, with the current state of the art for poly$(n)$-time algorithms being the 2004 algorithm of Batu et al. \cite{BKKM04}. This algorithm can reconstruct an arbitrary source string $x \in \{0,1\}^n$ in poly$(n)$ time provided that the deletion rate $δ$ satisfies $δ\leq n^{-(1/2 + \varepsilon)}$ for some $\varepsilon > 0$. In this work we improve on the result of \cite{BKKM04} by giving a poly$(n)$-time algorithm for trace reconstruction for any deletion rate $δ\leq n^{-(1/3 + \varepsilon)}$. Our algorithm works by alternating an alignment-based procedure, which we show effectively reconstructs portions of the source string that are not "highly repetitive", with a novel procedure that efficiently determines the length of highly repetitive subwords of the source string.

cs.DS↗

arXiv2020

Learning Sums of Independent Random Variables with Sparse Collective Support

We study the learnability of sums of independent integer random variables given a bound on the size of the union of their supports. For $\mathcal{A} \subset \mathbf{Z}_{+}$, a sum of independent random variables with collective support $\mathcal{A}$} (called an $\mathcal{A}$-sum in this paper) is a distribution $\mathbf{S} = \mathbf{X}_1 + \cdots + \mathbf{X}_N$ where the $\mathbf{X}_i$'s are mutually independent (but not necessarily identically distributed) integer random variables with $\cup_i \mathsf{supp}(\mathbf{X}_i) \subseteq \mathcal{A}.$ We give two main algorithmic results for learning such distributions: 1. For the case $| \mathcal{A} | = 3$, we give an algorithm for learning $\mathcal{A}$-sums to accuracy $ε$ that uses $\mathsf{poly}(1/ε)$ samples and runs in time $\mathsf{poly}(1/ε)$, independent of $N$ and of the elements of $\mathcal{A}$. 2. For an arbitrary constant $k \geq 4$, if $\mathcal{A} = \{ a_1,...,a_k\}$ with $0 \leq a_1 < ... < a_k$, we give an algorithm that uses $\mathsf{poly}(1/ε) \cdot \log \log a_k$ samples (independent of $N$) and runs in time $\mathsf{poly}(1/ε, \log a_k).$ We prove an essentially matching lower bound: if $|\mathcal{A}| = 4$, then any algorithm must use $Ω(\log \log a_4) $ samples even for learning to constant accuracy. We also give similar-in-spirit (but quantitatively very different) algorithmic results, and essentially matching lower bounds, for the case in which $\mathcal{A}$ is not known to the learner.

cs.DS↗

arXiv2020

Polynomial-time trace reconstruction in the smoothed complexity model

In the \emph{trace reconstruction problem}, an unknown source string $x \in \{0,1\}^n$ is sent through a probabilistic \emph{deletion channel} which independently deletes each bit with probability $δ$ and concatenates the surviving bits, yielding a \emph{trace} of $x$. The problem is to reconstruct $x$ given independent traces. This problem has received much attention in recent years both in the worst-case setting where $x$ may be an arbitrary string in $\{0,1\}^n$ \cite{DOS17,NazarovPeres17,HHP18,HL18,Chase19} and in the average-case setting where $x$ is drawn uniformly at random from $\{0,1\}^n$ \cite{PeresZhai17,HPP18,HL18,Chase19}. This paper studies trace reconstruction in the \emph{smoothed analysis} setting, in which a ``worst-case'' string $x^{\worst}$ is chosen arbitrarily from $\{0,1\}^n$, and then a perturbed version $\bx$ of $x^{\worst}$ is formed by independently replacing each coordinate by a uniform random bit with probability $σ$. The problem is to reconstruct $\bx$ given independent traces from it. Our main result is an algorithm which, for any constant perturbation rate $0<σ< 1$ and any constant deletion rate $0 < δ< 1$, uses $\poly(n)$ running time and traces and succeeds with high probability in reconstructing the string $\bx$. This stands in contrast with the worst-case version of the problem, for which $\text{exp}(O(n^{1/3}))$ is the best known time and sample complexity \cite{DOS17,NazarovPeres17}. Our approach is based on reconstructing $\bx$ from the multiset of its short subwords and is quite different from previous algorithms for either the worst-case or average-case versions of the problem. The heart of our work is a new $\poly(n)$-time procedure for reconstructing the multiset of all $O(\log n)$-length subwords of any source string $x\in \{0,1\}^n$ given access to traces of $x$.

cs.DS↗

arXiv2020

Reconstructing weighted voting schemes from partial information about their power indices

A number of recent works [Goldberg 2006; O'Donnell and Servedio 2011; De, Diakonikolas, and Servedio 2017; De, Diakonikolas, Feldman, and Servedio 2014] have considered the problem of approximately reconstructing an unknown weighted voting scheme given information about various sorts of ``power indices'' that characterize the level of control that individual voters have over the final outcome. In the language of theoretical computer science, this is the problem of approximating an unknown linear threshold function (LTF) over $\{-1, 1\}^n$ given some numerical measure (such as the function's $n$ ``Chow parameters,'' a.k.a. its degree-1 Fourier coefficients, or the vector of its $n$ Shapley indices) of how much each of the $n$ individual input variables affects the outcome of the function. In this paper we consider the problem of reconstructing an LTF given only partial information about its Chow parameters or Shapley indices; i.e. we are given only the Chow parameters or the Shapley indices corresponding to a subset $S \subseteq [n]$ of the $n$ input variables. A natural goal in this partial information setting is to find an LTF whose Chow parameters or Shapley indices corresponding to indices in $S$ accurately match the given Chow parameters or Shapley indices of the unknown LTF. We refer to this as the Partial Inverse Power Index Problem. Our main results are a polynomial time algorithm for the ($\varepsilon$-approximate) Chow Parameters Partial Inverse Power Index Problem and a quasi-polynomial time algorithm for the ($\varepsilon$-approximate) Shapley Indices Partial Inverse Power Index Problem.

cs.GT↗

arXiv2019

Testing noisy linear functions for sparsity

We consider the following basic inference problem: there is an unknown high-dimensional vector $w \in \mathbb{R}^n$, and an algorithm is given access to labeled pairs $(x,y)$ where $x \in \mathbb{R}^n$ is a measurement and $y = w \cdot x + \mathrm{noise}$. What is the complexity of deciding whether the target vector $w$ is (approximately) $k$-sparse? The recovery analogue of this problem --- given the promise that $w$ is sparse, find or approximate the vector $w$ --- is the famous sparse recovery problem, with a rich body of work in signal processing, statistics, and computer science. We study the decision version of this problem (i.e.~deciding whether the unknown $w$ is $k$-sparse) from the vantage point of property testing. Our focus is on answering the following high-level question: when is it possible to efficiently test whether the unknown target vector $w$ is sparse versus far-from-sparse using a number of samples which is completely independent of the dimension $n$? We consider the natural setting in which $x$ is drawn from a i.i.d.~product distribution $\mathcal{D}$ over $\mathbb{R}^n$ and the $\mathrm{noise}$ process is independent of the input $x$. As our main result, we give a general algorithm which solves the above-described testing problem using a number of samples which is completely independent of the ambient dimension $n$, as long as $\mathcal{D}$ is not a Gaussian. In fact, our algorithm is fully noise tolerant, in the sense that for an arbitrary $w$, it approximately computes the distance of $w$ to the closest $k$-sparse vector. To complement this algorithmic result, we show that weakening any of our condition makes it information-theoretically impossible for any algorithm to solve the testing problem with fewer than essentially $\log n$ samples.

cs.CC↗

arXiv2019

Kruskal-Katona for convex sets, with applications

The well-known Kruskal-Katona theorem in combinatorics says that (under mild conditions) every monotone Boolean function $f: \{0,1\}^n \to \{0,1\}$ has a nontrivial "density increment." This means that the fraction of inputs of Hamming weight $k+1$ for which $f=1$ is significantly larger than the fraction of inputs of Hamming weight $k$ for which $f=1.$ We prove an analogous statement for convex sets. Informally, our main result says that (under mild conditions) every convex set $K \subset \mathbb{R}^n$ has a nontrivial density increment. This means that the fraction of the radius-$r$ sphere that lies within $K$ is significantly larger than the fraction of the radius-$r'$ sphere that lies within $K$, for $r'$ suitably larger than $r$. For centrally symmetric convex sets we show that our density increment result is essentially optimal. As a consequence of our Kruskal-Katona type theorem, we obtain the first efficient weak learning algorithm for convex sets under the Gaussian distribution. We show that any convex set can be weak learned to advantage $Ω(1/n)$ in $\mathsf{poly}(n)$ time under any Gaussian distribution and that any centrally symmetric convex set can be weak learned to advantage $Ω(1/\sqrt{n})$ in $\mathsf{poly}(n)$ time. We also give an information-theoretic lower bound showing that the latter advantage is essentially optimal for $\mathsf{poly}(n)$ time weak learning algorithms. As another consequence of our Kruskal-Katona theorem, we give the first nontrivial Gaussian noise stability bounds for convex sets at high noise rates. Our results extend the known correspondence between monotone Boolean functions over $ \{0,1\}^n$ and convex bodies in Gaussian space.

cs.CC↗

arXiv2019

A Lower Bound on Cycle-Finding in Sparse Digraphs

We consider the problem of finding a cycle in a sparse directed graph $G$ that is promised to be far from acyclic, meaning that the smallest feedback arc set in $G$ is large. We prove an information-theoretic lower bound, showing that for $N$-vertex graphs with constant outdegree any algorithm for this problem must make $\tildeΩ(N^{5/9})$ queries to an adjacency list representation of $G$. In the language of property testing, our result is an $\tildeΩ(N^{5/9})$ lower bound on the query complexity of one-sided algorithms for testing whether sparse digraphs with constant outdegree are far from acyclic. This is the first improvement on the $Ω(\sqrt{N})$ lower bound, implicit in Bender and Ron, which follows from a simple birthday paradox argument.

cs.DS↗

arXiv2019

Efficient average-case population recovery in the presence of insertions and deletions

Several recent works have considered the \emph{trace reconstruction problem}, in which an unknown source string $x\in\{0,1\}^n$ is transmitted through a probabilistic channel which may randomly delete coordinates or insert random bits, resulting in a \emph{trace} of $x$. The goal is to reconstruct the original string~$x$ from independent traces of $x$. While the best algorithms known for worst-case strings use $\exp(O(n^{1/3}))$ traces \cite{DOS17,NazarovPeres17}, highly efficient algorithms are known \cite{PZ17,HPP18} for the \emph{average-case} version, in which $x$ is uniformly random. We consider a generalization of this average-case trace reconstruction problem, which we call \emph{average-case population recovery in the presence of insertions and deletions}. In this problem, there is an unknown distribution $\cal{D}$ over $s$ unknown source strings $x^1,\dots,x^s \in \{0,1\}^n$, and each sample is independently generated by drawing some $x^i$ from $\cal{D}$ and returning an independent trace of $x^i$. Building on \cite{PZ17} and \cite{HPP18}, we give an efficient algorithm for this problem. For any support size $s \leq \smash{\exp(Θ(n^{1/3}))}$, for a $1-o(1)$ fraction of all $s$-element support sets $\{x^1,\dots,x^s\} \subset \{0,1\}^n$, for every distribution $\cal{D}$ supported on $\{x^1,\dots,x^s\}$, our algorithm efficiently recovers ${\cal D}$ up to total variation distance $ε$ with high probability, given access to independent traces of independent draws from $\cal{D}$. The algorithm runs in time poly$(n,s,1/ε)$ and its sample complexity is poly$(s,1/ε,\exp(\log^{1/3}n)).$ This polynomial dependence on the support size $s$ is in sharp contrast with the \emph{worst-case} version (when $x^1,\dots,x^s$ may be any strings in $\{0,1\}^n$), in which the sample complexity of the most efficient known algorithm \cite{BCFSS19} is doubly exponential in $s$.

cs.DS↗

arXiv2019

Learning from satisfying assignments under continuous distributions

What kinds of functions are learnable from their satisfying assignments? Motivated by this simple question, we extend the framework of De, Diakonikolas, and Servedio [DDS15], which studied the learnability of probability distributions over $\{0,1\}^n$ defined by the set of satisfying assignments to "low-complexity" Boolean functions, to Boolean-valued functions defined over continuous domains. In our learning scenario there is a known "background distribution" $\mathcal{D}$ over $\mathbb{R}^n$ (such as a known normal distribution or a known log-concave distribution) and the learner is given i.i.d. samples drawn from a target distribution $\mathcal{D}_f$, where $\mathcal{D}_f$ is $\mathcal{D}$ restricted to the satisfying assignments of an unknown low-complexity Boolean-valued function $f$. The problem is to learn an approximation $\mathcal{D}'$ of the target distribution $\mathcal{D}_f$ which has small error as measured in total variation distance. We give a range of efficient algorithms and hardness results for this problem, focusing on the case when $f$ is a low-degree polynomial threshold function (PTF). When the background distribution $\mathcal{D}$ is log-concave, we show that this learning problem is efficiently solvable for degree-1 PTFs (i.e.,~linear threshold functions) but not for degree-2 PTFs. In contrast, when $\mathcal{D}$ is a normal distribution, we show that this learning problem is efficiently solvable for degree-2 PTFs but not for degree-4 PTFs. Our hardness results rely on standard assumptions about secure signature schemes.

cs.DS↗

arXiv2019

Beyond trace reconstruction: Population recovery from the deletion channel

\emph{Population recovery} is the problem of learning an unknown distribution over an unknown set of $n$-bit strings, given access to independent draws from the distribution that have been independently corrupted according to some noise channel. Recent work has intensively studied such problems both for the bit-flip and erasure noise channels. We initiate the study of population recovery under the \emph{deletion channel}, in which each bit is independently \emph{deleted} with some fixed probability and the surviving bits are concatenated and transmitted. This is a far more challenging noise model than bit-flip~noise or erasure noise; indeed, even the simplest case in which the population size is 1 (corresponding to a trivial distribution supported on a single string) corresponds to the \emph{trace reconstruction} problem, a challenging problem that has received much recent attention (see e.g.~\cite{DOS17,NP17,PZ17,HPP18,HHP18}). We give algorithms and lower bounds for population recovery under the deletion channel when the population size is some $\ell>1$. As our main sample complexity upper bound, we show that for any $\ell=o(\log n/\log \log n)$, a population of $\ell$ strings from $\{0,1\}^n$ can be learned under deletion channel noise using $\smash{2^{n^{1/2+o(1)}}}$ samples. On the lower bound side, we show that $n^{Ω(\ell)}$ samples are required to perform population recovery under the deletion channel, for all $\ell \leq n^{1/2-ε}$. Our upper bounds are obtained via a robust multivariate generalization of a polynomial-based analysis, due to Krasikov and Roddity \cite{KR97}, of how the $k$-deck of a bit-string uniquely identifies the string; this is a very different approach from recent algorithms for trace reconstruction (the $\ell=1$ case). Our lower bounds build on moment-matching results of Roos~\cite{Roo00} and Daskalakis and Papadimitriou~\cite{DP15}.

cs.DS↗

arXiv2018

Density estimation for shift-invariant multidimensional distributions

We study density estimation for classes of shift-invariant distributions over $\mathbb{R}^d$. A multidimensional distribution is "shift-invariant" if, roughly speaking, it is close in total variation distance to a small shift of it in any direction. Shift-invariance relaxes smoothness assumptions commonly used in non-parametric density estimation to allow jump discontinuities. The different classes of distributions that we consider correspond to different rates of tail decay. For each such class we give an efficient algorithm that learns any distribution in the class from independent samples with respect to total variation distance. As a special case of our general result, we show that $d$-dimensional shift-invariant distributions which satisfy an exponential tail bound can be learned to total variation distance error $ε$ using $\tilde{O}_d(1/ ε^{d+2})$ examples and $\tilde{O}_d(1/ ε^{2d+2})$ time. This implies that, for constant $d$, multivariate log-concave distributions can be learned in $\tilde{O}_d(1/ε^{2d+2})$ time using $\tilde{O}_d(1/ε^{d+2})$ samples, answering a question of [Diakonikolas, Kane and Stewart, 2016] All of our results extend to a model of noise-tolerant density estimation using Huber's contamination model, in which the target distribution to be learned is a $(1-ε,ε)$ mixture of some unknown distribution in the class with some other arbitrary and unknown distribution, and the learning algorithm must output a hypothesis distribution with total variation distance error $O(ε)$ from the target distribution. We show that our general results are close to best possible by proving a simple $Ω\left(1/ε^d\right)$ information-theoretic lower bound on sample complexity even for learning bounded distributions that are shift-invariant.

cs.LG↗

arXiv2018

Fooling Polytopes

We give a pseudorandom generator that fools $m$-facet polytopes over $\{0,1\}^n$ with seed length $\mathrm{polylog}(m) \cdot \log n$. The previous best seed length had superlinear dependence on $m$. An immediate consequence is a deterministic quasipolynomial time algorithm for approximating the number of solutions to any $\{0,1\}$-integer program.

cs.CC↗

arXiv2018

Luby--Veličković--Wigderson revisited: Improved correlation bounds and pseudorandom generators for depth-two circuits

We study correlation bounds and pseudorandom generators for depth-two circuits that consist of a $\mathsf{SYM}$-gate (computing an arbitrary symmetric function) or $\mathsf{THR}$-gate (computing an arbitrary linear threshold function) that is fed by $S$ $\mathsf{AND}$ gates. Such circuits were considered in early influential work on unconditional derandomization of Luby, Veličković, and Wigderson [LVW93], who gave the first non-trivial PRG with seed length $2^{O(\sqrt{\log(S/\varepsilon)})}$ that $\varepsilon$-fools these circuits. In this work we obtain the first strict improvement of [LVW93]'s seed length: we construct a PRG that $\varepsilon$-fools size-$S$ $\{\mathsf{SYM},\mathsf{THR}\} \circ\mathsf{AND}$ circuits over $\{0,1\}^n$ with seed length \[ 2^{O(\sqrt{\log S })} + \mathrm{polylog}(1/\varepsilon), \] an exponential (and near-optimal) improvement of the $\varepsilon$-dependence of [LVW93]. The above PRG is actually a special case of a more general PRG which we establish for constant-depth circuits containing multiple $\mathsf{SYM}$ or $\mathsf{THR}$ gates, including as a special case $\{\mathsf{SYM},\mathsf{THR}\} \circ \mathsf{AC^0}$ circuits. These more general results strengthen previous results of Viola [Vio06] and essentially strengthen more recent results of Lovett and Srinivasan [LS11]. Our improved PRGs follow from improved correlation bounds, which are transformed into PRGs via the Nisan--Wigderson "hardness versus randomness" paradigm [NW94]. The key to our improved correlation bounds is the use of a recent powerful \emph{multi-switching} lemma due to Håstad [Hås14].

cs.CC↗

arXiv2018

Distribution-free Junta Testing

We study the problem of testing whether an unknown $n$-variable Boolean function is a $k$-junta in the distribution-free property testing model, where the distance between functions is measured with respect to an arbitrary and unknown probability distribution over $\{0,1\}^n$. Our first main result is that distribution-free $k$-junta testing can be performed, with one-sided error, by an adaptive algorithm that uses $\tilde{O}(k^2)/ε$ queries (independent of $n$). Complementing this, our second main result is a lower bound showing that any non-adaptive distribution-free $k$-junta testing algorithm must make $Ω(2^{k/3})$ queries even to test to accuracy $ε=1/3$. These bounds establish that while the optimal query complexity of non-adaptive $k$-junta testing is $2^{Θ(k)}$, for adaptive testing it is $\text{poly}(k)$, and thus show that adaptivity provides an exponential improvement in the distribution-free query complexity of testing juntas.

cs.CC↗

arXiv2018

Deterministic search for CNF satisfying assignments in almost polynomial time

We consider the fundamental derandomization problem of deterministically finding a satisfying assignment to a CNF formula that has many satisfying assignments. We give a deterministic algorithm which, given an $n$-variable $\mathrm{poly}(n)$-clause CNF formula $F$ that has at least $\varepsilon 2^n$ satisfying assignments, runs in time \[ n^{\tilde{O}(\log\log n)^2} \] for $\varepsilon \ge 1/\mathrm{polylog}(n)$ and outputs a satisfying assignment of $F$. Prior to our work the fastest known algorithm for this problem was simply to enumerate over all seeds of a pseudorandom generator for CNFs; using the best known PRGs for CNFs [DETT10], this takes time $n^{\tildeΩ(\log n)}$ even for constant $\varepsilon$. Our approach is based on a new general framework relating deterministic search and deterministic approximate counting, which we believe may find further applications.

cs.CC↗

arXiv2018

Improved pseudorandom generators from pseudorandom multi-switching lemmas

We give the best known pseudorandom generators for two touchstone classes in unconditional derandomization: an $\varepsilon$-PRG for the class of size-$M$ depth-$d$ $\mathsf{AC}^0$ circuits with seed length $\log(M)^{d+O(1)}\cdot \log(1/\varepsilon)$, and an $\varepsilon$-PRG for the class of $S$-sparse $\mathbb{F}_2$ polynomials with seed length $2^{O(\sqrt{\log S})}\cdot \log(1/\varepsilon)$. These results bring the state of the art for unconditional derandomization of these classes into sharp alignment with the state of the art for computational hardness for all parameter settings: improving on the seed lengths of either PRG would require breakthrough progress on longstanding and notorious circuit lower bounds. The key enabling ingredient in our approach is a new \emph{pseudorandom multi-switching lemma}. We derandomize recently-developed \emph{multi}-switching lemmas, which are powerful generalizations of Håstad's switching lemma that deal with \emph{families} of depth-two circuits. Our pseudorandom multi-switching lemma---a randomness-efficient algorithm for sampling restrictions that simultaneously simplify all circuits in a family---achieves the parameters obtained by the (full randomness) multi-switching lemmas of Impagliazzo, Matthews, and Paturi [IMP12] and Håstad [Hås14]. This optimality of our derandomization translates into the optimality (given current circuit lower bounds) of our PRGs for $\mathsf{AC}^0$ and sparse $\mathbb{F}_2$ polynomials.

cs.CC↗

arXiv2017

Sample-based high-dimensional convexity testing

In the problem of high-dimensional convexity testing, there is an unknown set $S \subseteq \mathbb{R}^n$ which is promised to be either convex or $\varepsilon$-far from every convex body with respect to the standard multivariate normal distribution $\mathcal{N}(0, 1)^n$. The job of a testing algorithm is then to distinguish between these two cases while making as few inspections of the set $S$ as possible. In this work we consider sample-based testing algorithms, in which the testing algorithm only has access to labeled samples $(\boldsymbol{x},S(\boldsymbol{x}))$ where each $\boldsymbol{x}$ is independently drawn from $\mathcal{N}(0, 1)^n$. We give nearly matching sample complexity upper and lower bounds for both one-sided and two-sided convexity testing algorithms in this framework. For constant $\varepsilon$, our results show that the sample complexity of one-sided convexity testing is $2^{\tildeΘ(n)}$ samples, while for two-sided convexity testing it is $2^{\tildeΘ(\sqrt{n})}$.

cs.CC↗