arXiv subjects

Xiaorui Sun

Publications and source records attributed to Xiaorui Sun.

26 records · Page 2Linked to original sources

arXiv2016

Structure and automorphisms of primitive coherent configurations

Coherent configurations (CCs) are highly regular colorings of the set of ordered pairs of a "vertex set"; each color represents a "constituent digraph." CCs arise in the study of permutation groups, combinatorial structures such as partially balanced designs, and the analysis of algorithms; their history goes back to Schur in the 1930s. A CC is primitive (PCC) if all its constituent digraphs are connected. We address the problem of classifying PCCs with large automorphism groups. This project was started in Babai's 1981 paper in which he showed that only the trivial PCC admits more than $\exp(\tilde{O}(n^{1/2}))$ automorphisms. (Here, $n$ is the number of vertices and the $\tilde{O}$ hides polylogarithmic factors.) In the present paper we classify all PCCs with more than $\exp(\tilde{O}(n^{1/3}))$ automorphisms, making the first progress on Babai's conjectured classification of all PCCs with more than $\exp(n^ε)$ automorphisms. A corollary to Babai's 1981 result solved a then 100-year-old problem on primitive but not doubly transitive permutation groups, giving an $\exp(\tilde{O}(n^{1/2}))$ bound on their order. In a similar vein, our result implies an $\exp(\tilde{O}(n^{1/3}))$ upper bound on the order of such groups, with known exceptions. This improvement of Babai's result was previously known only through the Classification of Finite Simple Groups (Cameron, 1981), while our proof, like Babai's, is elementary and almost purely combinatorial. Our analysis relies on a new combinatorial structure theory we develop for PCCs. In particular, we demonstrate the presence of "asymptotically uniform clique geometries" on PCCs in a certain range of the parameters.

math.CO↗

arXiv2014

Near-Optimal Density Estimation in Near-Linear Time Using Variable-Width Histograms

Let $p$ be an unknown and arbitrary probability distribution over $[0,1)$. We consider the problem of {\em density estimation}, in which a learning algorithm is given i.i.d. draws from $p$ and must (with high probability) output a hypothesis distribution that is close to $p$. The main contribution of this paper is a highly efficient density estimation algorithm for learning using a variable-width histogram, i.e., a hypothesis distribution with a piecewise constant probability density function. In more detail, for any $k$ and $ε$, we give an algorithm that makes $\tilde{O}(k/ε^2)$ draws from $p$, runs in $\tilde{O}(k/ε^2)$ time, and outputs a hypothesis distribution $h$ that is piecewise constant with $O(k \log^2(1/ε))$ pieces. With high probability the hypothesis $h$ satisfies $d_{\mathrm{TV}}(p,h) \leq C \cdot \mathrm{opt}_k(p) + ε$, where $d_{\mathrm{TV}}$ denotes the total variation distance (statistical distance), $C$ is a universal constant, and $\mathrm{opt}_k(p)$ is the smallest total variation distance between $p$ and any $k$-piecewise constant distribution. The sample size and running time of our algorithm are optimal up to logarithmic factors. The "approximation factor" $C$ in our result is inherent in the problem, as we prove that no algorithm with sample size bounded in terms of $k$ and $ε$ can achieve $C<2$ regardless of what kind of hypothesis distribution it uses.

cs.LG↗

arXiv2013

A composition theorem for parity kill number

In this work, we study the parity complexity measures ${\mathsf{C}^{\oplus}_{\min}}[f]$ and ${\mathsf{DT^{\oplus}}}[f]$. ${\mathsf{C}^{\oplus}_{\min}}[f]$ is the \emph{parity kill number} of $f$, the fewest number of parities on the input variables one has to fix in order to "kill" $f$, i.e. to make it constant. ${\mathsf{DT^{\oplus}}}[f]$ is the depth of the shortest \emph{parity decision tree} which computes $f$. These complexity measures have in recent years become increasingly important in the fields of communication complexity \cite{ZS09, MO09, ZS10, TWXZ13} and pseudorandomness \cite{BK12, Sha11, CT13}. Our main result is a composition theorem for ${\mathsf{C}^{\oplus}_{\min}}$. The $k$-th power of $f$, denoted $f^{\circ k}$, is the function which results from composing $f$ with itself $k$ times. We prove that if $f$ is not a parity function, then ${\mathsf{C}^{\oplus}_{\min}}[f^{\circ k}] \geq Ω({\mathsf{C}_{\min}}[f]^{k}).$ In other words, the parity kill number of $f$ is essentially supermultiplicative in the \emph{normal} kill number of $f$ (also known as the minimum certificate complexity). As an application of our composition theorem, we show lower bounds on the parity complexity measures of $\mathsf{Sort}^{\circ k}$ and $\mathsf{HI}^{\circ k}$. Here $\mathsf{Sort}$ is the sort function due to Ambainis \cite{Amb06}, and $\mathsf{HI}$ is Kushilevitz's hemi-icosahedron function \cite{NW95}. In doing so, we disprove a conjecture of Montanaro and Osborne \cite{MO09} which had applications to communication complexity and computational learning theory. In addition, we give new lower bounds for conjectures of \cite{MO09,ZS10} and \cite{TWXZ13}.

cs.CC↗

arXiv2013

The Complexity of Optimal Multidimensional Pricing

We resolve the complexity of revenue-optimal deterministic auctions in the unit-demand single-buyer Bayesian setting, i.e., the optimal item pricing problem, when the buyer's values for the items are independent. We show that the problem of computing a revenue-optimal pricing can be solved in polynomial time for distributions of support size 2, and its decision version is NP-complete for distributions of support size 3. We also show that the problem remains NP-complete for the case of identical distributions.

cs.GT↗

arXiv2013

Efficient Density Estimation via Piecewise Polynomial Approximation

We give a highly efficient "semi-agnostic" algorithm for learning univariate probability distributions that are well approximated by piecewise polynomial density functions. Let $p$ be an arbitrary distribution over an interval $I$ which is $τ$-close (in total variation distance) to an unknown probability distribution $q$ that is defined by an unknown partition of $I$ into $t$ intervals and $t$ unknown degree-$d$ polynomials specifying $q$ over each of the intervals. We give an algorithm that draws $\tilde{O}(t\new{(d+1)}/\eps^2)$ samples from $p$, runs in time $\poly(t,d,1/\eps)$, and with high probability outputs a piecewise polynomial hypothesis distribution $h$ that is $(O(τ)+\eps)$-close (in total variation distance) to $p$. This sample complexity is essentially optimal; we show that even for $τ=0$, any algorithm that learns an unknown $t$-piecewise degree-$d$ probability distribution over $I$ to accuracy $\eps$ must use $Ω({\frac {t(d+1)} {\poly(1 + \log(d+1))}} \cdot {\frac 1 {\eps^2}})$ samples from the distribution, regardless of its running time. Our algorithm combines tools from approximation theory, uniform convergence, linear programming, and dynamic programming. We apply this general algorithm to obtain a wide range of results for many natural problems in density estimation over both continuous and discrete domains. These include state-of-the-art results for learning mixtures of log-concave distributions; mixtures of $t$-modal distributions; mixtures of Monotone Hazard Rate distributions; mixtures of Poisson Binomial Distributions; mixtures of Gaussians; and mixtures of $k$-monotone densities. Our general technique yields computationally efficient algorithms for all these problems, in many cases with provably optimal sample complexities (up to logarithmic factors) in all parameters.

cs.LG↗

arXiv2012

Learning mixtures of structured distributions over discrete domains

Let $\mathfrak{C}$ be a class of probability distributions over the discrete domain $[n] = \{1,...,n\}.$ We show that if $\mathfrak{C}$ satisfies a rather general condition -- essentially, that each distribution in $\mathfrak{C}$ can be well-approximated by a variable-width histogram with few bins -- then there is a highly efficient (both in terms of running time and sample complexity) algorithm that can learn any mixture of $k$ unknown distributions from $\mathfrak{C}.$ We analyze several natural types of distributions over $[n]$, including log-concave, monotone hazard rate and unimodal distributions, and show that they have the required structural property of being well-approximated by a histogram with few bins. Applying our general algorithm, we obtain near-optimally efficient algorithms for all these mixture learning problems.

cs.LG↗

arXiv2011

Information Dissemination via Random Walks in d-Dimensional Space

We study a natural information dissemination problem for multiple mobile agents in a bounded Euclidean space. Agents are placed uniformly at random in the $d$-dimensional space $\{-n, ..., n\}^d$ at time zero, and one of the agents holds a piece of information to be disseminated. All the agents then perform independent random walks over the space, and the information is transmitted from one agent to another if the two agents are sufficiently close. We wish to bound the total time before all agents receive the information (with high probability). Our work extends Pettarin et al.'s work (Infectious random walks, arXiv:1007.1604v2, 2011), which solved the problem for $d \leq 2$. We present tight bounds up to polylogarithmic factors for the case $d = 3$. (While our results extend to higher dimensions, for space and readability considerations we provide only the case $d=3$ here.) Our results show the behavior when $d \geq 3$ is qualitatively different from the case $d \leq 2$. In particular, as the ratio between the volume of the space and the number of agents varies, we show an interesting phase transition for three dimensions that does not occur in one or two dimensions.

math.PR↗

arXiv2011

Optimal Pricing in Social Networks with Incomplete Information

In revenue maximization of selling a digital product in a social network, the utility of an agent is often considered to have two parts: a private valuation, and linearly additive influences from other agents. We study the incomplete information case where agents know a common distribution about others' private valuations, and make decisions simultaneously. The "rational behavior" of agents in this case is captured by the well-known Bayesian Nash equilibrium. Two challenging questions arise: how to compute an equilibrium and how to optimize a pricing strategy accordingly to maximize the revenue assuming agents follow the equilibrium? In this paper, we mainly focus on the natural model where the private valuation of each agent is sampled from a uniform distribution, which turns out to be already challenging. Our main result is a polynomial-time algorithm that can exactly compute the equilibrium and the optimal price, when pairwise influences are non-negative. If negative influences are allowed, computing any equilibrium even approximately is PPAD-hard. Our algorithm can also be used to design an FPTAS for optimizing discriminative price profile.

cs.GT↗