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28
1Pass RelativeError LpSampling with Applications
"... For any p ∈ [0, 2], we give a 1pass poly(ε −1 log n)space algorithm which, given a data stream of length m with insertions and deletions of an ndimensional vector a, with updates in the range {−M, −M + 1, · · · , M − 1, M}, outputs a sample of [n] = {1, 2, · · · , n} for which for all i th ..."
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Cited by 28 (10 self)
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For any p ∈ [0, 2], we give a 1pass poly(ε −1 log n)space algorithm which, given a data stream of length m with insertions and deletions of an ndimensional vector a, with updates in the range {−M, −M + 1, · · · , M − 1, M}, outputs a sample of [n] = {1, 2, · · · , n} for which for all i the probability that i is returned is (1 ± ɛ) ai  p Fp(a) ± n −C, where ai denotes the (possibly negative) value of coordinate i, Fp(a) = ∑n i=1 aip = a  p p denotes the pth frequency moment (i.e., the pth power of the Lp norm), and C> 0 is an arbitrarily large constant. Here we assume that n, m, and M are polynomially related. Our generic sampling framework improves and unifies algorithms for several communication and streaming problems, including cascaded norms, heavy hitters, and moment estimation. It also gives the first relativeerror forward sampling algorithm in a data stream with deletions, answering an open question of Cormode et al. 1
The data stream space complexity of cascaded norms
 In FOCS
, 2009
"... Abstract — We consider the problem of estimating cascaded aggregates over a matrix presented as a sequence of updates in a data stream. A cascaded aggregate P ◦ Q is defined by evaluating aggregate Q repeatedly over each row of the matrix, and then evaluating aggregate P over the resulting vector of ..."
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Cited by 17 (7 self)
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Abstract — We consider the problem of estimating cascaded aggregates over a matrix presented as a sequence of updates in a data stream. A cascaded aggregate P ◦ Q is defined by evaluating aggregate Q repeatedly over each row of the matrix, and then evaluating aggregate P over the resulting vector of values. This problem was introduced by Cormode and Muthukrishnan, PODS, 2005 [CM]. We analyze the space complexity of estimating cascaded norms on an n × d matrix to within a small relative error. Let Lp denote the pth norm, where p is a nonnegative integer. We abbreviate the cascaded norm L k ◦ Lp by L k,p. (1) For any constant k ≥ p ≥ 2, we obtain a 1pass Õ(n1−2/k d 1−2/p)space algorithm for estimating Lk,p. This is optimal up to polylogarithmic factors and resolves an open question of [CM] regarding the space complexity of L4,2. We also obtain 1pass spaceoptimal algorithms for estimating L∞,k and Lk,∞. (2) We prove a space lower bound of Ω(n1−1/k) on estimating Lk,0 and Lk,1, resolving an open question due to Indyk, IITK Data Streams Workshop (Problem 8), 2006. We also resolve two more questions of [CM] concerning Lk,2 estimation and block heavy hitter problems. Ganguly, Bansal and Dube (FAW, 2008) claimed an Õ(1)space algorithm for estimating Lk,p for any k, p ∈ [0,2]. Our lower bounds show this claim is incorrect. 1.
Sinkhorn distances: Lightspeed computation of optimal transport
 In Advances in Neural Information Processing Systems
, 2013
"... Abstract. Optimal transportation distances are a fundamental family of parameterized distances for histograms. Despite their appealing theoretical properties, excellent performance in retrieval tasks and intuitive formulation, their computation involves the resolution of a linear program whose cos ..."
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Abstract. Optimal transportation distances are a fundamental family of parameterized distances for histograms. Despite their appealing theoretical properties, excellent performance in retrieval tasks and intuitive formulation, their computation involves the resolution of a linear program whose cost is prohibitive whenever the histograms ’ dimension exceeds a few hundreds. We propose in this work a new family of optimal transportation distances that look at transportation problems from a maximumentropy perspective. We smooth the classical optimal transportation problem with an entropic regularization term, and show that the resulting optimum is also a distance which can be computed through SinkhornKnopp’s matrix scaling algorithm at a speed that is several orders of magnitude faster than that of transportation solvers. We also report improved performance over classical optimal transportation distances on the MNIST benchmark problem. 1.
Comparing distributions and shapes using the kernel distance
 In ACM SoCG
, 2011
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Subspace Embeddings for the L1norm with Applications
"... We show there is a distribution over linear mappings R: ℓ n O(d log d) 1 → ℓ1, such that with arbitrarily large constant probability, for any fixed ddimensional subspace L, for all x ∈ L we have ‖x‖1 ≤ ‖Rx‖1 = O(d log d)‖x‖1. This provides the first analogue of the ubiquitous subspace JohnsonLinde ..."
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We show there is a distribution over linear mappings R: ℓ n O(d log d) 1 → ℓ1, such that with arbitrarily large constant probability, for any fixed ddimensional subspace L, for all x ∈ L we have ‖x‖1 ≤ ‖Rx‖1 = O(d log d)‖x‖1. This provides the first analogue of the ubiquitous subspace JohnsonLindenstrauss embedding for the ℓ1norm. Importantly, the target dimension and distortion are independent of the ambient dimension n. We give several applications of this result. First, we give a faster algorithm for computing wellconditioned bases. Our algorithm is simple, avoiding the linear programming machinery required of previous algorithms. We also give faster algorithms for least absolute deviation regression and ℓ1norm best fit hyperplane problems, as well as the first single pass streaming algorithms with low space for these problems. These results are motivated by practical problems in image analysis, spam detection, and statistics, where the ℓ1norm is used in studies where outliers may be safely and effectively ignored. This is because the ℓ1norm is more robust to outliers than the ℓ2norm.
Streaming Algorithms via Precision Sampling
"... ... (STOC 2005) has inspired several recent advances in datastream algorithms. We show that a number of these results follow easily from the application of a single probabilistic method called Precision Sampling. Using this method, we obtain simple datastream algorithms that maintain a randomized s ..."
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... (STOC 2005) has inspired several recent advances in datastream algorithms. We show that a number of these results follow easily from the application of a single probabilistic method called Precision Sampling. Using this method, we obtain simple datastream algorithms that maintain a randomized sketch of an input vector x = (x1,x2,...,xn), which is useful for the following applications: • Estimating the Fkmoment of x, fork>2. • Estimating the ℓpnorm of x, forp ∈ [1, 2], with small update time. • Estimating cascaded norms ℓp(ℓq) for all p, q> 0. • ℓ1 sampling, where the goal is to produce an element i with probability (approximately) xi/‖x‖1. It extends to similarly defined ℓpsampling, for p ∈ [1, 2]. For all these applications the algorithm is essentially the same: scale the vector x entrywise by a wellchosen random vector, and run a heavyhitter estimation algorithm on the resulting vector. Our sketch is a linear function of x, thereby allowing general updates to the vector x. Precision Sampling itself addresses the problem of estimating a sum Pn i=1 ai from weak estimates of each real ai ∈ [0, 1]. More precisely, the estimator first chooses a desired precision ui ∈ (0, 1] for each i ∈ [n], and then it receives an estimate of every ai Pwithin additive ui. Its goal is to provide a good approximation P to ai while keeping a tab on the “approximation cost” i (1/ui). Here we refine previous work (Andoni, Krauthgamer, and Onak, FOCS 2010) which shows that as long as P ai =Ω(1), a good multiplicative approximation can be achieved using total precision of only O(n log n).
Algorithms for the Transportation Problem in Geometric Settings
, 2012
"... For A, B ⊂ R d, A  + B  = n, let a ∈ A have a demand da ∈ Z + and b ∈ B have a supply sb ∈ Z + ∑ a∈A da b∈B sb = U and let d(·, ·) be a distance function. Suppose the diameter of A ∪ B is ∆ under d(·, ·), and ε> 0 is a parameter. We present an algorithm that in O((n √ U log 2 n+U log U)Φ(n) ..."
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For A, B ⊂ R d, A  + B  = n, let a ∈ A have a demand da ∈ Z + and b ∈ B have a supply sb ∈ Z + ∑ a∈A da b∈B sb = U and let d(·, ·) be a distance function. Suppose the diameter of A ∪ B is ∆ under d(·, ·), and ε> 0 is a parameter. We present an algorithm that in O((n √ U log 2 n+U log U)Φ(n) log(∆U/ε)) time computes a solution to the transportation problem on A, B which is within an additive error ε from the optimal solution. Here Φ(n) is the query and update time of a dynamic weighted nearest neighbor data structure under distance function d(·, ·). Note that the (1/ε) appears only in the log term. As among various consequences we obtain, • For A, B ⊂ R d and for the case where d(·, ·) is a metric, an εapproximation algorithm for the transportation problem in O((n √ U log 2 n + U log U)Φ(n) log(U/ε)) time. • For A, B ⊂ [∆] d and the L1 and L ∞ distance, exact algorithm for computing an optimal bipartite matching of A, B that runs in O(n 3/2 log d+O(1) n log ∆) time. • For A, B ⊂ [∆] 2 and RMS distance, exact algorithm for computing an optimal bipartite matching of A, B that runs in O(n 3/2+δ log ∆) time, for an arbitrarily small constant δ> 0. For point sets, A, B ⊂ [∆] d, for the Lp norm and for 0 < α, β < 1, we present a randomized dynamic data structure that maintains a partial solution to the transportation problem under insertions and deletions of points in which at least (1−α)U of the demands are satisfied and whose cost is within (1 + β) of that of the optimal (complete) solution to the transportation problem with high probability. The insertion, deletion and update times are O(poly(log(n∆)/αβ)), provided U = n O(1).
RademacherSketch: A DimensionalityReducing Embedding for SumProduct Norms, with an Application to EarthMover Distance
"... Abstract. Consider a sumproduct normed space, i.e. a space of the form Y = ℓ n 1 ⊗ X, where X is another normed space. Each element in Y consists of a lengthn vector of elements in X, and the norm of an element in Y is the sum of the norms of its coordinates. In this paper we show a constantdisto ..."
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Abstract. Consider a sumproduct normed space, i.e. a space of the form Y = ℓ n 1 ⊗ X, where X is another normed space. Each element in Y consists of a lengthn vector of elements in X, and the norm of an element in Y is the sum of the norms of its coordinates. In this paper we show a constantdistortion embedding from the normed space ℓ n 1 ⊗ X into a lowerdimensional normed space ℓ n′ 1 ⊗ X, where n ′ ≪ n is some value that depends on the properties of the normed space X (namely, on its Rademacher dimension). In particular, composing this embedding with another wellknown embedding of Indyk [18], we get an O(1/ɛ)distortion embedding from the earthmover metric EMD ∆ on the grid [∆] 2 to ℓ ∆O(ɛ) 1 ⊗EEMD∆ɛ (where EEMD is a norm that generalizes earthmover distance). This embedding is stronger (and simpler) than the sketching algorithm of Andoni et al [4], which maps EMD ∆ with O(1/ɛ) approximation into sketches of size ∆ O(ɛ). 1
The Streaming Complexity of Cycle Counting, Sorting By Reversals, and Other Problems
, 2010
"... In this paper we introduce a new technique for proving streaming lower bounds (and oneway communication lower bounds), by reductions from a problem called the Boolean Hidden Hypermatching problem (BHH). BHH is a problem that we introduce and prove the first lower bound for, but it is a generalizati ..."
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Cited by 4 (1 self)
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In this paper we introduce a new technique for proving streaming lower bounds (and oneway communication lower bounds), by reductions from a problem called the Boolean Hidden Hypermatching problem (BHH). BHH is a problem that we introduce and prove the first lower bound for, but it is a generalization of a wellknown problem called the Boolean Hidden Matching, that was used by Gavinsky et al. to prove separations between quantum communication complexity and oneway randomized communication complexity. The hardness of the BHH problem is inherently oneway: it is easy to solve using logarithmic twoway communication, but requires √ n communication if Alice is only allowed to send messages to Bob, and not viceversa. This onewayness allows us to prove lower bounds, via reductions, for streaming problems and related communication problems whose hardness is also inherently oneway. By designing reductions from BHH, we prove lower bounds for the streaming complexity of approximating the sorting by reversal distance, for approximately counting the number of cycles in a 2regular graph, and for other problems. For example, here is one lower bound that we prove, for a cyclecounting problem: Alice gets a perfect matching EA on a set of n nodes, and Bob gets a perfect matching EB on the same set of nodes. The union EA ∪ EB is a collection of cycles, and the goal is to approximate the number of cycles in this collection. We prove that if Alice is allowed to send o ( √ n) bits to Bob (and Bob is not allowed to send anything to Alice), then the number of cycles cannot be approximated to within a factor of 1.999, even using a randomized protocol. We prove that it is not even possible to distinguish the case where all cycles are of length 4, from the case where all cycles are of length 8. This lower bound is “natively ” oneway: With 4 rounds of communication, it is easy to distinguish these two cases. 1