Metric (mathematics)

About: Metric (mathematics) is a research topic. Over the lifetime, 42617 publications have been published within this topic receiving 836571 citations. The topic is also known as: distance function & metric.

Collaborative ranking

Abstract: Typical recommender systems use the root mean squared error (RMSE) between the predicted and actual ratings as the evaluation metric. We argue that RMSE is not an optimal choice for this task, especially when we will only recommend a few (top) items to any user. Instead, we propose using a ranking metric, namely normalized discounted cumulative gain (NDCG), as a better evaluation metric for this task. Borrowing ideas from the learning to rank community for web search, we propose novel models which approximately optimize NDCG for the recommendation task. Our models are essentially variations on matrix factorization models where we also additionally learn the features associated with the users and the items for the ranking task. Experimental results on a number of standard collaborative filtering data sets validate our claims. The results also show the accuracy and efficiency of our models and the benefits of learning features for ranking.

Optimal conditioning of self-scaling variable Metric algorithms

Abstract: Variable Metric Methods are "Newton--Raphson-like" algorithms for unconstrained minimization in which the inverse Hessian is replaced by an approximation, inferred from previous gradients and updated at each iteration. During the past decade various approaches have been used to derive general classes of such algorithms having the common properties of being Conjugate Directions methods and having "quadratic termination". Observed differences in actual performance of such methods motivated recent attempts to identify variable metric algorithms having additional properties that may be significant in practical situations (e.g. nonquadratic functions, inaccurate linesearch, etc.). The SSVM algorithms, introduced by this first author, are such methods that among their other properties, they automatically compensate for poor scaling of the objective function. This paper presents some new theoretical results identifying a subclass of SSVM algorithms that have the additional property of minimizing a sharp bound on the condition number of the inverse Hessian approximation at each iteration. Reducing this condition number is important for decreasing the roundoff error. The theoretical properties of this subclass are explored and two of its special cases are tested numerically in comparison with other SSVM algorithms.

Common Fixed Point Results in Metric-Type Spaces

Abstract: Several fixed point and common fixed point theorems are obtained in the setting of metric-type spaces introduced by M. A. Khamsi in 2010.

Gromov-Wasserstein Alignment of Word Embedding Spaces

Abstract: Cross-lingual or cross-domain correspondences play key roles in tasks ranging from machine translation to transfer learning. Recently, purely unsupervised methods operating on monolingual embeddings have become effective alignment tools. Current state-of-the-art methods, however, involve multiple steps, including heuristic post-hoc refinement strategies. In this paper, we cast the correspondence problem directly as an optimal transport (OT) problem, building on the idea that word embeddings arise from metric recovery algorithms. Indeed, we exploit the Gromov-Wasserstein distance that measures how similarities between pairs of words relate across languages. We show that our OT objective can be estimated efficiently, requires little or no tuning, and results in performance comparable with the state-of-the-art in various unsupervised word translation tasks.

Finite metric spaces: combinatorics, geometry and algorithms

Abstract: In the last several years a number of very interesting results were proved about finite metric spaces. Some of this work is motivated by practical considerations: Large data sets (coming e.g. from computational molecular biology, brain research or data mining) can be viewed as large metric spaces that should be analyzed (e.g. correctly clustered).On the other hand, these investigations connect to some classical areas of geometry - the asymptotic theory of finite-dimensional normed spaces and differential geometry. Finally, the metric theory of finite graphs has proved very useful in the study of graphs per se and the design of approximation algorithms for hard computational problems. In this talk I will try to explain some of the results and review some of the emerging new connections and the many fascinating open problems in this area.