Deep learning has proven itself as a successful set of models for learning useful semantic representations of data. These, however, are mostly implicitly learned as part of a classification task. In this paper we propose the triplet network model, which aims to learn useful representations by distance comparisons. A similar model was defined by Wang et al. (2014), tailor made for learning a ranking for image information retrieval. Here we demonstrate using various datasets that our model learns a better representation than that of its immediate competitor, the Siamese network. We also discuss future possible usage as a framework for unsupervised learning.

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We introduce a new low-distortion embedding of d 2 into O(log n) p (p = 1, 2), called the Fast-Johnson-LindenstraussTransform. The FJLT is faster than standard random projections and just as easy to implement. It is based upon the preconditioning of a sparse projection matrix with a randomized Fourier transform. Sparse random projections are unsuitable for low-distortion embeddings. We overcome this handicap by exploiting the "Heisenberg principle" of the Fourier transform, ie, its local-global duality. The FJLT can be used to speed up search algorithms based on lowdistortion embeddings in 1 and 2. We consider the case of approximate nearest neighbors in d 2 . We provide a faster algorithm using classical projections, which we then further speed up by plugging in the FJLT. We also give a faster algorithm for searching over the hypercube.

Abstract. We introduce a new low-distortion embedding of d 2 into O(log n) p (p = 1, 2) called the fast Johnson-Lindenstrauss transform (FJLT). The FJLT is faster than standard random projections and just as easy to implement. It is based upon the preconditioning of a sparse projection matrix with a randomized Fourier transform. Sparse random projections are unsuitable for low-distortion embeddings. We overcome this handicap by exploiting the "Heisenberg principle" of the Fourier transform, i.e., its local-global duality. The FJLT can be used to speed up search algorithms based on low-distortion embeddings in 1 and 2 . We consider the case of approximate nearest neighbors in d 2 . We provide a faster algorithm using classical projections, which we then speed up further by plugging in the FJLT. We also give a faster algorithm for searching over the hypercube.

The Fast Johnson-Lindenstrauss Transform (FJLT) was recently discovered by Ailon and Chazelle as a novel technique for performing fast dimension reduction with small distortion fromachieved by naive multiplication by random dense matrices, an approach followed by several authors as a variant of the seminal result by Johnson and Lindenstrauss (JL) from the mid 80's. In this work we show how to significantly improve the running time, for any arbitrary small fixed δ. This beats the better of FJLT and JL. Our analysis uses a powerful measure concentration bound due to Talagrand applied to Rademacher series in Banach spaces (sums of vectors in Banach spaces with random signs). The set of vectors used is a real embedding of dual BCH code vectors over GF (2). We also discuss the number of random bits used and reduction to 1 space.The connection between geometry and discrete coding theory discussed here is interesting in its own right and may be useful in other algorithmic applications as well.

We address optimization problems in which we are given contradictory pieces of input information and the goal is to find a globally consistent solution that minimizes the extent of disagreement with the respective inputs. Specifically, the problems we address are rank aggregation, the feedback arc set problem on tournaments, and correlation and consensus clustering. We show that for all these problems (and various weighted versions of them), we can obtain improved approximation factors using essentially the same remarkably simple algorithm. Additionally, we almost settle a long-standing conjecture of Bang-Jensen and Thomassen and show that unless NP⊆BPP, there is no polynomial time algorithm for the problem of minimum feedback arc set in tournaments.

The problems of random projections and sparse reconstruction have much in common and individually received much attention. Surprisingly, until now they progressed in parallel and remained mostly separate. Here, we employ new tools from probability in Banach spaces that were successfully used in the context of sparse reconstruction to advance on an open problem in random pojection. In particular, we generalize and use an intricate result by Rudelson and Vershynin for sparse reconstruction which uses Dudley's theorem for bounding Gaussian processes. Our main result states that any set of N = exp(Õ(n)) real vectors in n dimensional space can be linearly mapped to a space of dimension k = O(log N polylog(n)), while (1) preserving the pairwise distances among the vectors to within any constant distortion and (2) being able to apply the transformation in time O(n log n) on each vector. This improves on the best known N = exp(Õ(n 1/2 )) achieved by Ailon and Liberty and N = exp(Õ(n 1/3 )) by Ailon and Chazelle. The dependence in the distortion constant however is believed to be suboptimal and subject to further investigation. For constant distortion, this settles the open question posed by these authors up to a polylog(n) factor while considerably simplifying their constructions.

We study the problem of aggregating partial rankings. This problem is motivated by applications such as meta-searching and information retrieval, search engine spam fighting, e-commerce, learning from experts, analysis of population preference sampling, committee decision making and more. We improve recent constant factor approximation algorithms for aggregation of full rankings and generalize them to partial rankings. Our algorithms improved constant factor approximation with respect to all metrics discussed in Fagin et al's recent important work on comparing partial rankings. We pay special attention to two important types of partial rankings: the well-known top-m lists and the more general p-ratings which we define. We provide first evidence for hardness of aggregating them for constant m, p.