1,172 research outputs found

    A Lower Bound for Sampling Disjoint Sets

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    Suppose Alice and Bob each start with private randomness and no other input, and they wish to engage in a protocol in which Alice ends up with a set x subseteq[n] and Bob ends up with a set y subseteq[n], such that (x,y) is uniformly distributed over all pairs of disjoint sets. We prove that for some constant beta0 of the uniform distribution over all pairs of disjoint sets of size sqrt{n}

    The quantum communication complexity of sampling

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    Sampling is an important primitive in probabilistic and quantum algorithms. In the spirit of communication complexity, given a function f : X × Y → {0, 1} and a probability distribution D over X × Y , we define the sampling complexity of (f,D) as the minimum number of bits that Alice and Bob must communicate for Alice to pick x ∈ X and Bob to pick y ∈ Y as well as a value z such that the resulting distribution of (x, y, z) is close to the distribution (D, f(D)). In this paper we initiate the study of sampling complexity, in both the classical and quantum models. We give several variants of a definition. We completely characterize some of these variants and give upper and lower bounds on others. In particular, this allows us to establish an exponential gap between quantum and classical sampling complexity for the set-disjointness function

    Simple Heuristics Yield Provable Algorithms for Masked Low-Rank Approximation

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    In masked lowrank approximationmasked\ low-rank\ approximation, one is given ARn×nA \in \mathbb{R}^{n \times n} and binary mask matrix W{0,1}n×nW \in \{0,1\}^{n \times n}. The goal is to find a rank-kk matrix LL for which: cost(L)=i=1nj=1nWi,j(Ai,jLi,j)2OPT+ϵAF2,cost(L) = \sum_{i=1}^{n} \sum_{j = 1}^{n} W_{i,j} \cdot (A_{i,j} - L_{i,j} )^2 \leq OPT + \epsilon \|A\|_F^2 , where OPT=minrankk L^cost(L^)OPT = \min_{rank-k\ \hat{L}} cost(\hat L) and ϵ\epsilon is a given error parameter. Depending on the choice of WW, this problem captures factor analysis, low-rank plus diagonal decomposition, robust PCA, low-rank matrix completion, low-rank plus block matrix approximation, and many problems. Many of these problems are NP-hard, and while some algorithms with provable guarantees are known, they either 1) run in time nΩ(k2/ϵ)n^{\Omega(k^2/\epsilon)} or 2) make strong assumptions, e.g., that AA is incoherent or that WW is random. In this work, we show that a common polynomial time heuristic, which simply sets AA to 00 where WW is 00, and then finds a standard low-rank approximation, yields bicriteria approximation guarantees for this problem. In particular, for rank k>kk' > k depending on the $public\ coin\ partition\ numberof of W,theheuristicoutputsrank, the heuristic outputs rank-k' Lwithcost with cost(L) \leq OPT + \epsilon \|A\|_F^2.Thispartitionnumberisinturnboundedbythe. This partition number is in turn bounded by the randomized\ communication\ complexityof of W,wheninterpretedasatwoplayercommunicationmatrix.Formanyimportantexamplesofmaskedlowrankapproximation,includingallthoselistedabove,thisresultyieldsbicriteriaapproximationguaranteeswith, when interpreted as a two-player communication matrix. For many important examples of masked low-rank approximation, including all those listed above, this result yields bicriteria approximation guarantees with k' = k \cdot poly(\log n/\epsilon)$. Further, we show that different models of communication yield algorithms for natural variants of masked low-rank approximation. For example, multi-player number-in-hand communication complexity connects to masked tensor decomposition and non-deterministic communication complexity to masked Boolean low-rank factorization.Comment: ITCS 202

    Gaussian distribution of short sums of trace functions over finite fields

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    We show that under certain general conditions, short sums of \ell-adic trace functions over finite fields follow a normal distribution asymptotically when the origin varies, generalizing results of Erd\H{o}s-Davenport, Mak-Zaharescu and Lamzouri. In particular, this applies to exponential sums arising from Fourier transforms such as Kloosterman sums or Birch sums, as we can deduce from the works of Katz. By approximating the moments of traces of random matrices in monodromy groups, a quantitative version can be given as in Lamzouri's article, exhibiting a different phenomenon than the averaging from the central limit theorem.Comment: 42 page

    Quantum energy inequalities and local covariance II: Categorical formulation

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    We formulate Quantum Energy Inequalities (QEIs) in the framework of locally covariant quantum field theory developed by Brunetti, Fredenhagen and Verch, which is based on notions taken from category theory. This leads to a new viewpoint on the QEIs, and also to the identification of a new structural property of locally covariant quantum field theory, which we call Local Physical Equivalence. Covariant formulations of the numerical range and spectrum of locally covariant fields are given and investigated, and a new algebra of fields is identified, in which fields are treated independently of their realisation on particular spacetimes and manifestly covariant versions of the functional calculus may be formulated.Comment: 27 pages, LaTeX. Further discussion added. Version to appear in General Relativity and Gravitatio

    Robust Bell Inequalities from Communication Complexity

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    The question of how large Bell inequality violations can be, for quantum distributions, has been the object of much work in the past several years. We say a Bell inequality is normalized if its absolute value does not exceed 1 for any classical (i.e. local) distribution. Upper and (almost) tight lower bounds have been given in terms of number of outputs of the distribution, number of inputs, and the dimension of the shared quantum states. In this work, we revisit normalized Bell inequalities together with another family: inefficiency-resistant Bell inequalities. To be inefficiency-resistant, the Bell value must not exceed 1 for any local distribution, including those that can abort. Both these families of Bell inequalities are closely related to communication complexity lower bounds. We show how to derive large violations from any gap between classical and quantum communication complexity, provided the lower bound on classical communication is proven using these lower bounds. This leads to inefficiency-resistant violations that can be exponential in the size of the inputs. Finally, we study resistance to noise and inefficiency for these Bell inequalities
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