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A central problem in ranking is to design a ranking measure for evaluation of ranking functions. In this paper we study, from a theoretical perspective, the widely used Normalized Discounted Cumulative Gain (NDCG)-type ranking measures. Although there are extensive empirical studies of NDCG, little is known about its theoretical properties. We first show(More)
We study k-SVD that is to obtain the first k singular vectors of a matrix A approximately. Recently, a few breakthroughs have been discovered on k-SVD: Musco and Musco [14] provided the first gap-free theorem for the block Krylov method, Shamir [16] discovered the first variance-reduction stochastic method, and Bhojanapalli et al. [3] provided the fastest(More)
A central problem in ranking is to design a measure for evaluation of ranking functions. In this paper we study, from a theoretical perspective, the Normalized Discounted Cumulative Gain (NDCG) which is a family of ranking measures widely used in practice. Although there are extensive empirical studies of the NDCG family, little is known about its(More)
Word embeddings are ubiquitous in NLP and information retrieval, but it's unclear what they represent when the word is polysemous, i.e., has multiple senses. Here it is shown that multiple word senses reside in linear superposition within the word embedding and can be recovered by simple sparse coding. The success of the method —which applies to several(More)
The papers of Mikolov et al. 2013 as well as subsequent works have led to dramatic progress in solving word analogy tasks using semantic word embeddings. This leverages linear structure that is often found in the word embeddings, which is surprising since the training method is usually nonlinear. There were attempts —notably by Levy and Goldberg and(More)
Semantic word embeddings represent the meaning of a word via a vector, and are created by diverse methods such as Latent Semantic Analysis (LSA), generative text models such as topic models, matrix factorization, neural nets, and energy-based models. Many methods use nonlinear operations —such as Pairwise Mutual Information or PMI— on co-occurrence(More)
We study k-GenEV, the problem of finding the top k generalized eigenvectors, and k-CCA, the problem of finding the top k vectors in canonical-correlation analysis. We propose algorithms LazyEV and LazyCCA to solve the two problems with running times linearly dependent on the input size and on k. Furthermore, our algorithms are doubly-accelerated : our(More)
Semantic word embeddings represent the meaning of a word via a vector, and are created by diverse methods. Many use non-linear operations on co-occurrence statistics, and have hand-tuned hyperparameters and reweighting methods. This paper proposes a new generative model, a dynamic version of the log-linear topic model of Mnih and Hinton (2007). The(More)
Many applications require recovering a ground truth low-rank matrix from noisy observations of the entries. In practice, this is typically formulated as a weighted low-rank approximation problem and solved using non-convex optimization heuristics such as alternating minimization. Such non-convex techniques have few guarantees. Even worse, weighted low-rank(More)