This paper gives new results for synchronization strings, a powerful combinatorial object introduced by [Haeupler, Shahrasbi; STOC’17] that allows to efficiently deal with insertions and deletions in various communication problems: - We give a deterministic, linear time synchronization string construction, improving over an O(n5) time randomized construction. Independently of this work, a deterministic O(n log2 logn) time construction was proposed by Cheng, Li, and Wu. - We give a deterministic construction of an infinite synchronization string which outputs the first n symbols in O(n) time. Previously it was not known whether such a string was computable. - Both synchronization string constructions are highly explicit, i.e., the ith symbol can be deterministically computed in O(logi) time. - This paper also introduces a generalized notion we call long-distance synchronization strings. Such strings allow for local and very fast decoding. In particular only O(log3 n) time and access to logarithmically many symbols is required to decode any index. The paper also provides several applications for these improved synchronization strings: - For any δ < 1 and є > 0 we provide an insdel error correcting block code with rate 1 − δ − є which can correct any δ/3 fraction of insertion and deletion errors in O(n log3 n) time. This near linear computational efficiency is surprising given that we do not even know how to compute the (edit) distance between the decoding input and output in sub-quadratic time. - We show that local decodability implies that error correcting codes constructed with long-distance synchronization strings can not only efficiently recover from δ fraction of insdel errors but, similar to [Schulman, Zuckerman; TransInf’99], also from any O(δ / logn) fraction of block transpositions and block replications. These block corruptions allow arbitrarily long substrings to be swapped or replicated anywhere. - We show that highly explicitness and local decoding allow for infinite channel simulations with exponentially smaller memory and decoding time requirements. These simulations can then be used to give the first near linear time interactive coding scheme for insdel errors, similar to the result of [Brakerski, Naor; SODA’13] for Hamming errors.
{"title":"Synchronization strings: explicit constructions, local decoding, and applications","authors":"Bernhard Haeupler, Amirbehshad Shahrasbi","doi":"10.1145/3188745.3188940","DOIUrl":"https://doi.org/10.1145/3188745.3188940","url":null,"abstract":"This paper gives new results for synchronization strings, a powerful combinatorial object introduced by [Haeupler, Shahrasbi; STOC’17] that allows to efficiently deal with insertions and deletions in various communication problems: - We give a deterministic, linear time synchronization string construction, improving over an O(n5) time randomized construction. Independently of this work, a deterministic O(n log2 logn) time construction was proposed by Cheng, Li, and Wu. - We give a deterministic construction of an infinite synchronization string which outputs the first n symbols in O(n) time. Previously it was not known whether such a string was computable. - Both synchronization string constructions are highly explicit, i.e., the ith symbol can be deterministically computed in O(logi) time. - This paper also introduces a generalized notion we call long-distance synchronization strings. Such strings allow for local and very fast decoding. In particular only O(log3 n) time and access to logarithmically many symbols is required to decode any index. The paper also provides several applications for these improved synchronization strings: - For any δ < 1 and є > 0 we provide an insdel error correcting block code with rate 1 − δ − є which can correct any δ/3 fraction of insertion and deletion errors in O(n log3 n) time. This near linear computational efficiency is surprising given that we do not even know how to compute the (edit) distance between the decoding input and output in sub-quadratic time. - We show that local decodability implies that error correcting codes constructed with long-distance synchronization strings can not only efficiently recover from δ fraction of insdel errors but, similar to [Schulman, Zuckerman; TransInf’99], also from any O(δ / logn) fraction of block transpositions and block replications. These block corruptions allow arbitrarily long substrings to be swapped or replicated anywhere. - We show that highly explicitness and local decoding allow for infinite channel simulations with exponentially smaller memory and decoding time requirements. These simulations can then be used to give the first near linear time interactive coding scheme for insdel errors, similar to the result of [Brakerski, Naor; SODA’13] for Hamming errors.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80869385","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The approximate degree of a Boolean function f is the least degree of a real polynomial that approximates f pointwise to error at most 1/3. The approximate degree of f is known to be a lower bound on the quantum query complexity of f (Beals et al., FOCS 1998 and J. ACM 2001). We resolve or nearly resolve the approximate degree and quantum query complexities of several basic functions. Specifically, we show the following: *k-distinctness: For any constant k, the approximate degree and quantum query complexity of the k-distinctness function is Ω(n3/4−1/(2k)). This is nearly tight for large k, as Belovs (FOCS 2012) has shown that for any constant k, the approximate degree and quantum query complexity of k-distinctness is O(n3/4−1/(2k+2−4)). *Image Size Testing: The approximate degree and quantum query complexity of testing the size of the image of a function [n] → [n] is Ω(n1/2). This proves a conjecture of Ambainis et al. (SODA 2016), and it implies tight lower bounds on the approximate degree and quantum query complexity of the following natural problems. **k-junta testing: A tight Ω(k1/2) lower bound for k-junta testing, answering the main open question of Ambainis et al. (SODA 2016). **Statistical Distance from Uniform: A tight Ω(n1/2) lower bound for approximating the statistical distance from uniform of a distribution, answering the main question left open by Bravyi et al. (STACS 2010 and IEEE Trans. Inf. Theory 2011). **Shannon entropy: A tight Ω(n1/2) lower bound for approximating Shannon entropy up to a certain additive constant, answering a question of Li and Wu (2017). *Surjectivity: The approximate degree of the Surjectivity function is Ω(n3/4). The best prior lower bound was Ω(n2/3). Our result matches an upper bound of Õ(n3/4) due to Sherstov, which we reprove using different techniques. The quantum query complexity of this function is known to be Θ(n) (Beame and Machmouchi, Quantum Inf. Comput. 2012 and Sherstov, FOCS 2015). Our upper bound for Surjectivity introduces new techniques for approximating Boolean functions by low-degree polynomials. Our lower bounds are proved by significantly refining techniques recently introduced by Bun and Thaler (FOCS 2017).
布尔函数f的近似度数是实多项式的最小度数，它对f逐点逼近，误差不超过1/3。已知f的近似度是f的量子查询复杂度的下界(Beals et al.， FOCS 1998 and J. ACM 2001)。我们解决或接近解决了几个基本函数的近似度和量子查询复杂性。*k-distinctness:对于任意常数k, k-distinctness函数的近似度和量子查询复杂度为Ω(n3/4−1/(2k))。对于大k，这几乎是紧的，因为Belovs (FOCS 2012)已经表明，对于任意常数k, k-distinctness的近似程度和量子查询复杂度为O(n3/4−1/(2k+2−4))。*图像大小测试:测试函数[n]→[n]的图像大小的近似程度和量子查询复杂度为Ω(n1/2)。这证明了Ambainis et al. (SODA 2016)的一个猜想，它暗示了以下自然问题的近似程度和量子查询复杂性的紧密下界。**k-军政府测试:k-军政府测试的严格Ω(k1/2)下界，回答了Ambainis等人(SODA 2016)的主要开放问题。**离均匀的统计距离:近似分布离均匀的统计距离的一个紧密的Ω(n1/2)下界，回答了Bravyi等人(STACS 2010和IEEE Trans)留下的主要问题。Inf. Theory 2011)。**香农熵:一个紧密的Ω(n1/2)下界，用于逼近香农熵到某个附加常数，回答了Li和Wu(2017)的问题。*满射性:满射性函数的近似程度为Ω(n3/4)。最佳先验下界为Ω(n2/3)。由于Sherstov，我们的结果符合Õ(n3/4)的上界，我们使用不同的技术对其进行了修正。已知该函数的量子查询复杂度为Θ(n) (Beame和Machmouchi, quantum Inf. computer . 2012和Sherstov, FOCS 2015)。我们的满性上界引入了用低次多项式逼近布尔函数的新技术。我们的下限被Bun和Thaler最近引入的显著改进技术证明(FOCS 2017)。
{"title":"The polynomial method strikes back: tight quantum query bounds via dual polynomials","authors":"Mark Bun, Robin Kothari, J. Thaler","doi":"10.1145/3188745.3188784","DOIUrl":"https://doi.org/10.1145/3188745.3188784","url":null,"abstract":"The approximate degree of a Boolean function f is the least degree of a real polynomial that approximates f pointwise to error at most 1/3. The approximate degree of f is known to be a lower bound on the quantum query complexity of f (Beals et al., FOCS 1998 and J. ACM 2001). We resolve or nearly resolve the approximate degree and quantum query complexities of several basic functions. Specifically, we show the following: *k-distinctness: For any constant k, the approximate degree and quantum query complexity of the k-distinctness function is Ω(n3/4−1/(2k)). This is nearly tight for large k, as Belovs (FOCS 2012) has shown that for any constant k, the approximate degree and quantum query complexity of k-distinctness is O(n3/4−1/(2k+2−4)). *Image Size Testing: The approximate degree and quantum query complexity of testing the size of the image of a function [n] → [n] is Ω(n1/2). This proves a conjecture of Ambainis et al. (SODA 2016), and it implies tight lower bounds on the approximate degree and quantum query complexity of the following natural problems. **k-junta testing: A tight Ω(k1/2) lower bound for k-junta testing, answering the main open question of Ambainis et al. (SODA 2016). **Statistical Distance from Uniform: A tight Ω(n1/2) lower bound for approximating the statistical distance from uniform of a distribution, answering the main question left open by Bravyi et al. (STACS 2010 and IEEE Trans. Inf. Theory 2011). **Shannon entropy: A tight Ω(n1/2) lower bound for approximating Shannon entropy up to a certain additive constant, answering a question of Li and Wu (2017). *Surjectivity: The approximate degree of the Surjectivity function is Ω(n3/4). The best prior lower bound was Ω(n2/3). Our result matches an upper bound of Õ(n3/4) due to Sherstov, which we reprove using different techniques. The quantum query complexity of this function is known to be Θ(n) (Beame and Machmouchi, Quantum Inf. Comput. 2012 and Sherstov, FOCS 2015). Our upper bound for Surjectivity introduces new techniques for approximating Boolean functions by low-degree polynomials. Our lower bounds are proved by significantly refining techniques recently introduced by Bun and Thaler (FOCS 2017).","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81079610","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give the first rigorous proof of the convergence of Riemannian Hamiltonian Monte Carlo, a general (and practical) method for sampling Gibbs distributions. Our analysis shows that the rate of convergence is bounded in terms of natural smoothness parameters of an associated Riemannian manifold. We then apply the method with the manifold defined by the log barrier function to the problems of (1) uniformly sampling a polytope and (2) computing its volume, the latter by extending Gaussian cooling to the manifold setting. In both cases, the total number of steps needed is O*(mn2/3), improving the state of the art. A key ingredient of our analysis is a proof of an analog of the KLS conjecture for Gibbs distributions over manifolds.
{"title":"Convergence rate of Riemannian Hamiltonian Monte Carlo and faster polytope volume computation","authors":"Y. Lee, S. Vempala","doi":"10.1145/3188745.3188774","DOIUrl":"https://doi.org/10.1145/3188745.3188774","url":null,"abstract":"We give the first rigorous proof of the convergence of Riemannian Hamiltonian Monte Carlo, a general (and practical) method for sampling Gibbs distributions. Our analysis shows that the rate of convergence is bounded in terms of natural smoothness parameters of an associated Riemannian manifold. We then apply the method with the manifold defined by the log barrier function to the problems of (1) uniformly sampling a polytope and (2) computing its volume, the latter by extending Gaussian cooling to the manifold setting. In both cases, the total number of steps needed is O*(mn2/3), improving the state of the art. A key ingredient of our analysis is a proof of an analog of the KLS conjecture for Gibbs distributions over manifolds.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":"63 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80733146","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
High dimensional expanders is a vibrant emerging field of study. Nevertheless, the only known construction of bounded degree high dimensional expanders is based on Ramanujan complexes, whereas one dimensional bounded degree expanders are abundant. In this work we construct new families of bounded degree high dimensional expanders obeying the local spectral expansion property. A property that implies, geometric overlapping, fast mixing of high dimensional random walks, agreement testing and agreement expansion. The construction also yields new families of expander graphs. The construction is quite elementary and it is presented in a self contained manner; This is in contrary to the highly involved construction of the Ramanujan complexes. The construction is also strongly symmetric; The symmetry of the construction could be used, for example, to obtain good symmetric LDPC codes that were previously based on Ramanujan graphs.
{"title":"Construction of new local spectral high dimensional expanders","authors":"T. Kaufman, I. Oppenheim","doi":"10.1145/3188745.3188782","DOIUrl":"https://doi.org/10.1145/3188745.3188782","url":null,"abstract":"High dimensional expanders is a vibrant emerging field of study. Nevertheless, the only known construction of bounded degree high dimensional expanders is based on Ramanujan complexes, whereas one dimensional bounded degree expanders are abundant. In this work we construct new families of bounded degree high dimensional expanders obeying the local spectral expansion property. A property that implies, geometric overlapping, fast mixing of high dimensional random walks, agreement testing and agreement expansion. The construction also yields new families of expander graphs. The construction is quite elementary and it is presented in a self contained manner; This is in contrary to the highly involved construction of the Ramanujan complexes. The construction is also strongly symmetric; The symmetry of the construction could be used, for example, to obtain good symmetric LDPC codes that were previously based on Ramanujan graphs.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84396491","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Newton iteration (NI) is an almost 350 years old recursive formula that approximates a simple root of a polynomial quite rapidly. We generalize it to a matrix recurrence (allRootsNI) that approximates all the roots simultaneously. In this form, the process yields a better circuit complexity in the case when the number of roots r is small but the multiplicities are exponentially large. Our method sets up a linear system in r unknowns and iteratively builds the roots as formal power series. For an algebraic circuit f(x1,…,xn) of size s we prove that each factor has size at most a polynomial in: s and the degree of the squarefree part of f. Consequently, if f1 is a 2Ω(n)-hard polynomial then any nonzero multiple ∏i fiei is equally hard for arbitrary positive ei’s, assuming that ∑ideg(fi) is at most 2O(n). It is an old open question whether the class of poly(n)-sized formulas (resp. algebraic branching programs) is closed under factoring. We show that given a polynomial f of degree nO(1) and formula (resp. ABP) size nO(logn) we can find a similar size formula (resp. ABP) factor in randomized poly(nlogn)-time. Consequently, if determinant requires nΩ(logn) size formula, then the same can be said about any of its nonzero multiples. As part of our proofs, we identify a new property of multivariate polynomial factorization. We show that under a random linear transformation τ, f(τx) completely factors via power series roots. Moreover, the factorization adapts well to circuit complexity analysis. This with allRootsNI are the techniques that help us make progress towards the old open problems; supplementing the large body of classical results and concepts in algebraic circuit factorization (eg. Zassenhaus, J.NT 1969; Kaltofen, STOC 1985-7 & B'urgisser, FOCS 2001).
{"title":"Discovering the roots: uniform closure results for algebraic classes under factoring","authors":"P. Dutta, Nitin Saxena, Amit Sinhababu","doi":"10.1145/3188745.3188760","DOIUrl":"https://doi.org/10.1145/3188745.3188760","url":null,"abstract":"Newton iteration (NI) is an almost 350 years old recursive formula that approximates a simple root of a polynomial quite rapidly. We generalize it to a matrix recurrence (allRootsNI) that approximates all the roots simultaneously. In this form, the process yields a better circuit complexity in the case when the number of roots r is small but the multiplicities are exponentially large. Our method sets up a linear system in r unknowns and iteratively builds the roots as formal power series. For an algebraic circuit f(x1,…,xn) of size s we prove that each factor has size at most a polynomial in: s and the degree of the squarefree part of f. Consequently, if f1 is a 2Ω(n)-hard polynomial then any nonzero multiple ∏i fiei is equally hard for arbitrary positive ei’s, assuming that ∑ideg(fi) is at most 2O(n). It is an old open question whether the class of poly(n)-sized formulas (resp. algebraic branching programs) is closed under factoring. We show that given a polynomial f of degree nO(1) and formula (resp. ABP) size nO(logn) we can find a similar size formula (resp. ABP) factor in randomized poly(nlogn)-time. Consequently, if determinant requires nΩ(logn) size formula, then the same can be said about any of its nonzero multiples. As part of our proofs, we identify a new property of multivariate polynomial factorization. We show that under a random linear transformation τ, f(τx) completely factors via power series roots. Moreover, the factorization adapts well to circuit complexity analysis. This with allRootsNI are the techniques that help us make progress towards the old open problems; supplementing the large body of classical results and concepts in algebraic circuit factorization (eg. Zassenhaus, J.NT 1969; Kaltofen, STOC 1985-7 & B'urgisser, FOCS 2001).","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90530989","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Paulsen problem is a basic open problem in operator theory: Given vectors u1, …, un ∈ ℝd that are є-nearly satisfying the Parseval’s condition and the equal norm condition, is it close to a set of vectors v1, …, vn ∈ ℝd that exactly satisfy the Parseval’s condition and the equal norm condition? Given u1, …, un, the squared distance (to the set of exact solutions) is defined as infv ∑i=1n || ui − vi ||22 where the infimum is over the set of exact solutions. Previous results show that the squared distance of any є-nearly solution is at most O(poly(d,n,є)) and there are є-nearly solutions with squared distance at least Ω(d є). The fundamental open question is whether the squared distance can be independent of the number of vectors n. We answer this question affirmatively by proving that the squared distance of any є-nearly solution is O(d13/2 є). Our approach is based on a continuous version of the operator scaling algorithm and consists of two parts. First, we define a dynamical system based on operator scaling and use it to prove that the squared distance of any є-nearly solution is O(d2 n є). Then, we show that by randomly perturbing the input vectors, the dynamical system will converge faster and the squared distance of an є-nearly solution is O(d5/2 є) when n is large enough and є is small enough. To analyze the convergence of the dynamical system, we develop some new techniques in lower bounding the operator capacity, a concept introduced by Gurvits to analyzing the operator scaling algorithm.
Paulsen问题是算子理论中的一个基本开放问题:给定向量u1，…，un∈，∈，∈，满足Parseval条件和等范数条件є-nearly，是否接近于恰好满足Parseval条件和等范数条件的向量v1，…，vn∈，∈，∈，∈，满足Parseval条件的集合?给定u1，…，un，到精确解集合的距离平方定义为inv∑i=1n || ui−vi ||22，其中极小值在精确解集合上。先前的结果表明，任何є-nearly解的平方距离不超过O(poly(d,n, n))，并且存在平方距离至少为Ω(d,n)的є-nearly解。基本的开放问题是距离的平方是否可以独立于向量n的数量。我们通过证明任何є-nearly解的平方距离为O(d13/2 n)来肯定地回答这个问题。我们的方法是基于连续版本的算子缩放算法，由两部分组成。首先，我们定义了一个基于算子尺度的动力系统，并用它证明了任意є-nearly解的平方距离为O(d2 n n)。然后，我们证明了通过随机扰动输入向量，动力系统将收敛得更快，并且当n足够大且k足够小时，є-nearly解的平方距离为O(d5/2)。为了分析动力系统的收敛性，我们发展了一些新的算子容量下限技术，这是Gurvits引入的一个概念，用于分析算子缩放算法。
{"title":"The Paulsen problem, continuous operator scaling, and smoothed analysis","authors":"T. C. Kwok, L. Lau, Y. Lee, Akshay Ramachandran","doi":"10.1145/3188745.3188794","DOIUrl":"https://doi.org/10.1145/3188745.3188794","url":null,"abstract":"The Paulsen problem is a basic open problem in operator theory: Given vectors u1, …, un ∈ ℝd that are є-nearly satisfying the Parseval’s condition and the equal norm condition, is it close to a set of vectors v1, …, vn ∈ ℝd that exactly satisfy the Parseval’s condition and the equal norm condition? Given u1, …, un, the squared distance (to the set of exact solutions) is defined as infv ∑i=1n || ui − vi ||22 where the infimum is over the set of exact solutions. Previous results show that the squared distance of any є-nearly solution is at most O(poly(d,n,є)) and there are є-nearly solutions with squared distance at least Ω(d є). The fundamental open question is whether the squared distance can be independent of the number of vectors n. We answer this question affirmatively by proving that the squared distance of any є-nearly solution is O(d13/2 є). Our approach is based on a continuous version of the operator scaling algorithm and consists of two parts. First, we define a dynamical system based on operator scaling and use it to prove that the squared distance of any є-nearly solution is O(d2 n є). Then, we show that by randomly perturbing the input vectors, the dynamical system will converge faster and the squared distance of an є-nearly solution is O(d5/2 є) when n is large enough and є is small enough. To analyze the convergence of the dynamical system, we develop some new techniques in lower bounding the operator capacity, a concept introduced by Gurvits to analyzing the operator scaling algorithm.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75268488","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The asymptotic restriction problem for tensors s and t is to find the smallest β ≥ 0 such that the nth tensor power of t can be obtained from the (β n+o(n))th tensor power of s by applying linear maps to the tensor legs — this is called restriction — when n goes to infinity. Applications include computing the arithmetic complexity of matrix multiplication in algebraic complexity theory, deciding the feasibility of an asymptotic transformation between pure quantum states via stochastic local operations and classical communication in quantum information theory, bounding the query complexity of certain properties in algebraic property testing, and bounding the size of combinatorial structures like tri-colored sum-free sets in additive combinatorics. Naturally, the asymptotic restriction problem asks for obstructions (think of lower bounds in computational complexity) and constructions (think of fast matrix multiplication algorithms). Strassen showed that for obstructions it is sufficient to consider maps from k-tensors to nonnegative reals, that are monotone under restriction, normalised on diagonal tensors, additive under direct sum and multiplicative under tensor product, named spectral points (SFCS 1986 and J. Reine Angew. Math. 1988). Strassen introduced the support functionals, which are spectral points for oblique tensors, a strict subfamily of all tensors (J. Reine Angew. Math. 1991). On the construction side, an important work is the Coppersmith-Winograd method for tight tensors and tight sets. We present the first nontrivial spectral points for the family of all complex tensors, named quantum functionals. Finding such universal spectral points has been an open problem for thirty years. We use techniques from quantum information theory, invariant theory and moment polytopes. We present comparisons among the support functionals and our quantum functionals, and compute generic values. We relate the functionals to instability from geometric invariant theory, in the spirit of Blasiak et al. (Discrete Anal. 2017). We prove that the quantum functionals are asymptotic upper bounds on slice-rank and multi-slice rank, extending a result of Tao and Sawin. Furthermore, we make progress on the construction side of the combinatorial version of the asymptotic restriction problem by extending the Coppersmith–Winograd method via combinatorial degeneration. The regular method constructs large free diagonals in powers of any tight set. Our extended version works for any set that has a combinatorial degeneration to a tight set. This generalizes a result of Kleinberg, Sawin and Speyer. As an application we reprove in hindsight recent results on tri-colored sum-free sets by reducing this problem to a result of Strassen on reduced polynomial multiplication. Proofs are in the full version of this paper, available at https://arxiv.org/abs/1709.07851.
{"title":"Universal points in the asymptotic spectrum of tensors","authors":"M. Christandl, Péter Vrana, Jeroen Zuiddam","doi":"10.1145/3188745.3188766","DOIUrl":"https://doi.org/10.1145/3188745.3188766","url":null,"abstract":"The asymptotic restriction problem for tensors s and t is to find the smallest β ≥ 0 such that the nth tensor power of t can be obtained from the (β n+o(n))th tensor power of s by applying linear maps to the tensor legs — this is called restriction — when n goes to infinity. Applications include computing the arithmetic complexity of matrix multiplication in algebraic complexity theory, deciding the feasibility of an asymptotic transformation between pure quantum states via stochastic local operations and classical communication in quantum information theory, bounding the query complexity of certain properties in algebraic property testing, and bounding the size of combinatorial structures like tri-colored sum-free sets in additive combinatorics. Naturally, the asymptotic restriction problem asks for obstructions (think of lower bounds in computational complexity) and constructions (think of fast matrix multiplication algorithms). Strassen showed that for obstructions it is sufficient to consider maps from k-tensors to nonnegative reals, that are monotone under restriction, normalised on diagonal tensors, additive under direct sum and multiplicative under tensor product, named spectral points (SFCS 1986 and J. Reine Angew. Math. 1988). Strassen introduced the support functionals, which are spectral points for oblique tensors, a strict subfamily of all tensors (J. Reine Angew. Math. 1991). On the construction side, an important work is the Coppersmith-Winograd method for tight tensors and tight sets. We present the first nontrivial spectral points for the family of all complex tensors, named quantum functionals. Finding such universal spectral points has been an open problem for thirty years. We use techniques from quantum information theory, invariant theory and moment polytopes. We present comparisons among the support functionals and our quantum functionals, and compute generic values. We relate the functionals to instability from geometric invariant theory, in the spirit of Blasiak et al. (Discrete Anal. 2017). We prove that the quantum functionals are asymptotic upper bounds on slice-rank and multi-slice rank, extending a result of Tao and Sawin. Furthermore, we make progress on the construction side of the combinatorial version of the asymptotic restriction problem by extending the Coppersmith–Winograd method via combinatorial degeneration. The regular method constructs large free diagonals in powers of any tight set. Our extended version works for any set that has a combinatorial degeneration to a tight set. This generalizes a result of Kleinberg, Sawin and Speyer. As an application we reprove in hindsight recent results on tri-colored sum-free sets by reducing this problem to a result of Strassen on reduced polynomial multiplication. Proofs are in the full version of this paper, available at https://arxiv.org/abs/1709.07851.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75478437","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
One of the prominent current challenges in complexity theory is the attempt to prove lower bounds for TC0, the class of constant-depth, polynomial-size circuits with majority gates. Relying on the results of Williams (2013), an appealing approach to prove such lower bounds is to construct a non-trivial derandomization algorithm for TC0. In this work we take a first step towards the latter goal, by proving the first positive results regarding the derandomization of TC0 circuits of depth d>2. Our first main result is a quantified derandomization algorithm for TC0 circuits with a super-linear number of wires. Specifically, we construct an algorithm that gets as input a TC0 circuit C over n input bits with depth d and n1+exp(−d) wires, runs in almost-polynomial-time, and distinguishes between the case that C rejects at most 2n1−1/5d inputs and the case that C accepts at most 2n1−1/5d inputs. In fact, our algorithm works even when the circuit C is a linear threshold circuit, rather than just a TC0 circuit (i.e., C is a circuit with linear threshold gates, which are stronger than majority gates). Our second main result is that even a modest improvement of our quantified derandomization algorithm would yield a non-trivial algorithm for standard derandomization of all of TC0, and would consequently imply that NEXP⊈TC0. Specifically, if there exists a quantified derandomization algorithm that gets as input a TC0 circuit with depth d and n1+O(1/d) wires (rather than n1+exp(−d) wires), runs in time at most 2nexp(−d), and distinguishes between the case that C rejects at most 2n1−1/5d inputs and the case that C accepts at most 2n1−1/5d inputs, then there exists an algorithm with running time 2n1−Ω(1) for standard derandomization of TC0.
{"title":"Quantified derandomization of linear threshold circuits","authors":"R. Tell","doi":"10.1145/3188745.3188822","DOIUrl":"https://doi.org/10.1145/3188745.3188822","url":null,"abstract":"One of the prominent current challenges in complexity theory is the attempt to prove lower bounds for TC0, the class of constant-depth, polynomial-size circuits with majority gates. Relying on the results of Williams (2013), an appealing approach to prove such lower bounds is to construct a non-trivial derandomization algorithm for TC0. In this work we take a first step towards the latter goal, by proving the first positive results regarding the derandomization of TC0 circuits of depth d>2. Our first main result is a quantified derandomization algorithm for TC0 circuits with a super-linear number of wires. Specifically, we construct an algorithm that gets as input a TC0 circuit C over n input bits with depth d and n1+exp(−d) wires, runs in almost-polynomial-time, and distinguishes between the case that C rejects at most 2n1−1/5d inputs and the case that C accepts at most 2n1−1/5d inputs. In fact, our algorithm works even when the circuit C is a linear threshold circuit, rather than just a TC0 circuit (i.e., C is a circuit with linear threshold gates, which are stronger than majority gates). Our second main result is that even a modest improvement of our quantified derandomization algorithm would yield a non-trivial algorithm for standard derandomization of all of TC0, and would consequently imply that NEXP⊈TC0. Specifically, if there exists a quantified derandomization algorithm that gets as input a TC0 circuit with depth d and n1+O(1/d) wires (rather than n1+exp(−d) wires), runs in time at most 2nexp(−d), and distinguishes between the case that C rejects at most 2n1−1/5d inputs and the case that C accepts at most 2n1−1/5d inputs, then there exists an algorithm with running time 2n1−Ω(1) for standard derandomization of TC0.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":"37 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73167262","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give a constant-factor approximation algorithm for the asymmetric traveling salesman problem. Our approximation guarantee is analyzed with respect to the standard LP relaxation, and thus our result confirms the conjectured constant integrality gap of that relaxation. Our techniques build upon the constant-factor approximation algorithm for the special case of node-weighted metrics. Specifically, we give a generic reduction to structured instances that resemble but are more general than those arising from node-weighted metrics. For those instances, we then solve Local-Connectivity ATSP, a problem known to be equivalent (in terms of constant-factor approximation) to the asymmetric traveling salesman problem.
{"title":"A constant-factor approximation algorithm for the asymmetric traveling salesman problem","authors":"O. Svensson, Jakub Tarnawski, László A. Végh","doi":"10.1145/3188745.3188824","DOIUrl":"https://doi.org/10.1145/3188745.3188824","url":null,"abstract":"We give a constant-factor approximation algorithm for the asymmetric traveling salesman problem. Our approximation guarantee is analyzed with respect to the standard LP relaxation, and thus our result confirms the conjectured constant integrality gap of that relaxation. Our techniques build upon the constant-factor approximation algorithm for the special case of node-weighted metrics. Specifically, we give a generic reduction to structured instances that resemble but are more general than those arising from node-weighted metrics. For those instances, we then solve Local-Connectivity ATSP, a problem known to be equivalent (in terms of constant-factor approximation) to the asymmetric traveling salesman problem.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72574671","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ittai Abraham, Arnold Filtser, Anupam Gupta, Ofer Neiman
We study the problem of embedding weighted graphs of pathwidth k into ℓp spaces. Our main result is an O(kmin{1p,12})-distortion embedding. For p=1, this is a super-exponential improvement over the best previous bound of Lee and Sidiropoulos. Our distortion bound is asymptotically tight for any fixed p >1. Our result is obtained via a novel embedding technique that is based on low depth decompositions of a graph via shortest paths. The core new idea is that given a geodesic shortest path P, we can probabilistically embed all points into 2 dimensions with respect to P. For p>2 our embedding also implies improved distortion on bounded treewidth graphs (O((klogn)1p)). For asymptotically large p, our results also implies improved distortion on graphs excluding a minor.
{"title":"Metric embedding via shortest path decompositions","authors":"Ittai Abraham, Arnold Filtser, Anupam Gupta, Ofer Neiman","doi":"10.1145/3188745.3188808","DOIUrl":"https://doi.org/10.1145/3188745.3188808","url":null,"abstract":"We study the problem of embedding weighted graphs of pathwidth k into ℓp spaces. Our main result is an O(kmin{1p,12})-distortion embedding. For p=1, this is a super-exponential improvement over the best previous bound of Lee and Sidiropoulos. Our distortion bound is asymptotically tight for any fixed p >1. Our result is obtained via a novel embedding technique that is based on low depth decompositions of a graph via shortest paths. The core new idea is that given a geodesic shortest path P, we can probabilistically embed all points into 2 dimensions with respect to P. For p>2 our embedding also implies improved distortion on bounded treewidth graphs (O((klogn)1p)). For asymptotically large p, our results also implies improved distortion on graphs excluding a minor.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81349006","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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