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Edo Liberty

Edo Liberty appears in the imported research catalog. Authorship, coauthor and topic links are available while profile ownership is still unclaimed.

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Data Structures and AlgorithmsMachine LearningNumerical AnalysisArtificial IntelligenceComputational ComplexityInformation Theorymath.ITmath.NA

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Published work

14 published item(s)

preprint2026arXiv

Nearly Optimal Attention Coresets

We consider the problem of estimating the Attention mechanism in small space, and prove the existence of coresets for it of nearly optimal size. Specifically, we show that for any set of unit-norm keys and values $(K,V)$ in $\mathbb{R}^d$, there exists a subset $(K',V')$ of size at most $O({\sqrt{d} e^{ρ+o(ρ)}/\varepsilon})$ such that \[ \left\| \operatorname{Attn}(q,K,V)- \operatorname{Attn}(q,K',V') \right\| \le \varepsilon \] simultaneously for all queries whose norm is bounded by $ρ$. This outperforms the best known results for this problem. We also offer an improved lower bound showing that $\varepsilon$-coresets must have size $Ω({\sqrt{d} e^ρ/ε})$.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Efficient Frequent Directions Algorithm for Sparse Matrices

This paper describes Sparse Frequent Directions, a variant of Frequent Directions for sketching sparse matrices. It resembles the original algorithm in many ways: both receive the rows of an input matrix $A^{n \times d}$ one by one in the streaming setting and compute a small sketch $B \in R^{\ell \times d}$. Both share the same strong (provably optimal) asymptotic guarantees with respect to the space-accuracy tradeoff in the streaming setting. However, unlike Frequent Directions which runs in $O(nd\ell)$ time regardless of the sparsity of the input matrix $A$, Sparse Frequent Directions runs in $\tilde{O} (nnz(A)\ell + n\ell^2)$ time. Our analysis loosens the dependence on computing the Singular Value Decomposition (SVD) as a black box within the Frequent Directions algorithm. Our bounds require recent results on the properties of fast approximate SVD computations. Finally, we empirically demonstrate that these asymptotic improvements are practical and significant on real and synthetic data.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Optimal Quantile Approximation in Streams

This paper resolves one of the longest standing basic problems in the streaming computational model. Namely, optimal construction of quantile sketches. An $\varepsilon$ approximate quantile sketch receives a stream of items $x_1,\ldots,x_n$ and allows one to approximate the rank of any query up to additive error $\varepsilon n$ with probability at least $1-δ$. The rank of a query $x$ is the number of stream items such that $x_i \le x$. The minimal sketch size required for this task is trivially at least $1/\varepsilon$. Felber and Ostrovsky obtain a $O((1/\varepsilon)\log(1/\varepsilon))$ space sketch for a fixed $δ$. To date, no better upper or lower bounds were known even for randomly permuted streams or for approximating a specific quantile, e.g.,\ the median. This paper obtains an $O((1/\varepsilon)\log \log (1/δ))$ space sketch and a matching lower bound. This resolves the open problem and proves a qualitative gap between randomized and deterministic quantile sketching. One of our contributions is a novel representation and modification of the widely used merge-and-reduce construction. This subtle modification allows for an analysis which is both tight and extremely simple. Similar techniques should be useful for improving other sketching objectives and geometric coreset constructions.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Space Lower Bounds for Itemset Frequency Sketches

Given a database, computing the fraction of rows that contain a query itemset or determining whether this fraction is above some threshold are fundamental operations in data mining. A uniform sample of rows is a good sketch of the database in the sense that all sufficiently frequent itemsets and their approximate frequencies are recoverable from the sample, and the sketch size is independent of the number of rows in the original database. For many seemingly similar problems there are better sketching algorithms than uniform sampling. In this paper we show that for itemset frequency sketching this is not the case. That is, we prove that there exist classes of databases for which uniform sampling is a space optimal sketch for approximate itemset frequency analysis, up to constant or iterated-logarithmic factors.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

An Algorithm for Online K-Means Clustering

This paper shows that one can be competitive with the k-means objective while operating online. In this model, the algorithm receives vectors v_1,...,v_n one by one in an arbitrary order. For each vector the algorithm outputs a cluster identifier before receiving the next one. Our online algorithm generates ~O(k) clusters whose k-means cost is ~O(W*). Here, W* is the optimal k-means cost using k clusters and ~O suppresses poly-logarithmic factors. We also show that, experimentally, it is not much worse than k-means++ while operating in a strictly more constrained computational model.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

Frequent Directions : Simple and Deterministic Matrix Sketching

We describe a new algorithm called Frequent Directions for deterministic matrix sketching in the row-updates model. The algorithm is presented an arbitrary input matrix $A \in R^{n \times d}$ one row at a time. It performed $O(d \times \ell)$ operations per row and maintains a sketch matrix $B \in R^{\ell \times d}$ such that for any $k < \ell$ $\|A^TA - B^TB \|_2 \leq \|A - A_k\|_F^2 / (\ell-k)$ and $\|A - π_{B_k}(A)\|_F^2 \leq \big(1 + \frac{k}{\ell-k}\big) \|A-A_k\|_F^2 $ . Here, $A_k$ stands for the minimizer of $\|A - A_k\|_F$ over all rank $k$ matrices (similarly $B_k$) and $π_{B_k}(A)$ is the rank $k$ matrix resulting from projecting $A$ on the row span of $B_k$. We show both of these bounds are the best possible for the space allowed. The summary is mergeable, and hence trivially parallelizable. Moreover, Frequent Directions outperforms exemplar implementations of existing streaming algorithms in the space-error tradeoff.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

Greedy Minimization of Weakly Supermodular Set Functions

This paper defines weak-$α$-supermodularity for set functions. Many optimization objectives in machine learning and data mining seek to minimize such functions under cardinality constrains. We prove that such problems benefit from a greedy extension phase. Explicitly, let $S^*$ be the optimal set of cardinality $k$ that minimizes $f$ and let $S_0$ be an initial solution such that $f(S_0)/f(S^*) \le ρ$. Then, a greedy extension $S \supset S_0$ of size $|S| \le |S_0| + \lceil αk \ln(ρ/\varepsilon) \rceil$ yields $f(S)/f(S^*) \le 1+\varepsilon$. As example usages of this framework we give new bicriteria results for $k$-means, sparse regression, and columns subset selection.

0 citations0 reviewsNo code linkedOpen access
preprint2014arXiv

Fast Matrix Multiplication with Sketching

We present an approximate algorithm for matrix multiplication based on matrix sketching techniques. First one of the matrix is chosen and sparsified using the online matrix sketching algorithm, and then the matrix product is calculated using the sparsified matrix. We prove when the sample number grows large compared to the sample dimensions the proposed algorithm achieves similar accuracy bound with a smaller computational cost compared to the state-of-the-art algorithms.

0 citations0 reviewsNo code linkedOpen access
preprint2013arXiv

Near-Optimal Entrywise Sampling for Data Matrices

We consider the problem of selecting non-zero entries of a matrix $A$ in order to produce a sparse sketch of it, $B$, that minimizes $\|A-B\|_2$. For large $m \times n$ matrices, such that $n \gg m$ (for example, representing $n$ observations over $m$ attributes) we give sampling distributions that exhibit four important properties. First, they have closed forms computable from minimal information regarding $A$. Second, they allow sketching of matrices whose non-zeros are presented to the algorithm in arbitrary order as a stream, with $O(1)$ computation per non-zero. Third, the resulting sketch matrices are not only sparse, but their non-zero entries are highly compressible. Lastly, and most importantly, under mild assumptions, our distributions are provably competitive with the optimal offline distribution. Note that the probabilities in the optimal offline distribution may be complex functions of all the entries in the matrix. Therefore, regardless of computational complexity, the optimal distribution might be impossible to compute in the streaming model.

0 citations0 reviewsNo code linkedOpen access
preprint2012arXiv

On the Furthest Hyperplane Problem and Maximal Margin Clustering

This paper introduces the Furthest Hyperplane Problem (FHP), which is an unsupervised counterpart of Support Vector Machines. Given a set of n points in Rd, the objective is to produce the hyperplane (passing through the origin) which maximizes the separation margin, that is, the minimal distance between the hyperplane and any input point. To the best of our knowledge, this is the first paper achieving provable results regarding FHP. We provide both lower and upper bounds to this NP-hard problem. First, we give a simple randomized algorithm whose running time is n^O(1/θ^2) where θ is the optimal separation margin. We show that its exponential dependency on 1/θ^2 is tight, up to sub-polynomial factors, assuming SAT cannot be solved in sub-exponential time. Next, we give an efficient approxima- tion algorithm. For any α \in [0, 1], the algorithm produces a hyperplane whose distance from at least 1 - 5α fraction of the points is at least α times the optimal separation margin. Finally, we show that FHP does not admit a PTAS by presenting a gap preserving reduction from a particular version of the PCP theorem.

0 citations0 reviewsNo code linkedOpen access
preprint2012arXiv

Simple and Deterministic Matrix Sketching

We adapt a well known streaming algorithm for approximating item frequencies to the matrix sketching setting. The algorithm receives the rows of a large matrix $A \in \R^{n \times m}$ one after the other in a streaming fashion. It maintains a sketch matrix $B \in \R^ {1/\eps \times m}$ such that for any unit vector $x$ [\|Ax\|^2 \ge \|Bx\|^2 \ge \|Ax\|^2 - \eps \|A\|_{f}^2 \.] Sketch updates per row in $A$ require $O(m/\eps^2)$ operations in the worst case. A slight modification of the algorithm allows for an amortized update time of $O(m/\eps)$ operations per row. The presented algorithm stands out in that it is: deterministic, simple to implement, and elementary to prove. It also experimentally produces more accurate sketches than widely used approaches while still being computationally competitive.

0 citations0 reviewsNo code linkedOpen access
preprint2010arXiv

An Improved Algorithm for Bipartite Correlation Clustering

Bipartite Correlation clustering is the problem of generating a set of disjoint bi-cliques on a set of nodes while minimizing the symmetric difference to a bipartite input graph. The number or size of the output clusters is not constrained in any way. The best known approximation algorithm for this problem gives a factor of 11. This result and all previous ones involve solving large linear or semi-definite programs which become prohibitive even for modestly sized tasks. In this paper we present an improved factor 4 approximation algorithm to this problem using a simple combinatorial algorithm which does not require solving large convex programs. The analysis extends a method developed by Ailon, Charikar and Alantha in 2008, where a randomized pivoting algorithm was analyzed for obtaining a 3-approximation algorithm for Correlation Clustering, which is the same problem on graphs (not bipartite). The analysis for Correlation Clustering there required defining events for structures containing 3 vertices and using the probability of these events to produce a feasible solution to a dual of a certain natural LP bounding the optimal cost. It is tempting here to use sets of 4 vertices, which are the smallest structures for which contradictions arise for Bipartite Correlation Clustering. This simple idea, however, appears to be evasive. We show that, by modifying the LP, we can analyze algorithms which take into consideration subgraph structures of unbounded size. We believe our techniques are interesting in their own right, and may be used for other problems as well.

0 citations0 reviewsNo code linkedOpen access