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.

On Sampled Metrics for Item Recommendation

Abstract: The task of item recommendation requires ranking a large catalogue of items given a context. Item recommendation algorithms are evaluated using ranking metrics that depend on the positions of relevant items. To speed up the computation of metrics, recent work often uses sampled metrics where only a smaller set of random items and the relevant items are ranked. This paper investigates sampled metrics in more detail and shows that they are inconsistent with their exact version, in the sense that they do not persist relative statements, e.g., recommender A is better than B, not even in expectation. Moreover, the smaller the sampling size, the less difference there is between metrics, and for very small sampling size, all metrics collapse to the AUC metric. We show that it is possible to improve the quality of the sampled metrics by applying a correction, obtained by minimizing different criteria such as bias or mean squared error. We conclude with an empirical evaluation of the naive sampled metrics and their corrected variants. To summarize, our work suggests that sampling should be avoided for metric calculation, however if an experimental study needs to sample, the proposed corrections can improve the quality of the estimate.

Riemannian Ricci curvature lower bounds in metric measure spaces with -finite measure

Abstract: In prior work (4) of the first two authors with Savare, a new Riemannian notion of lower bound for Ricci curvature in the class of metric measure spaces (X,d,m) was introduced, and the corresponding class of spaces denoted by RCD(K,∞). This notion relates the CD(K,N) theory of Sturm and Lott-Villani, in the case N = ∞, to the Bakry-Emery approach. In (4) the RCD(K,∞) property is defined in three equivalent ways and several properties of RCD(K,∞) spaces, including the regularization properties of the heat flow, the connections with the theory of Dirichlet forms and the stability under tensor products, are provided. In (4) only finite reference measures m have been considered. The goal of this paper is twofold: on one side we extend these results to general σ-finite spaces, on the other we remove a technical assumption appeared in (4) concerning a strengthening of the CD(K,∞) condition. This more general class of spaces includes Euclidean spaces endowed with Lebesgue measure, complete noncompact Riemannian manifolds with bounded geometry and the pointed metric measure limits of manifolds with lower Ricci curvature bounds.

Haar wavelets for efficient similarity search of time-series: with and without time warping

Abstract: We address the handling of time series search based on two important distance definitions: Euclidean distance and time warping distance. The conventional method reduces the dimensionality by means of a discrete Fourier transform. We apply the Haar wavelet transform technique and propose the use of a proper normalization so that the method can guarantee no false dismissal for Euclidean distance. We found that this method has competitive performance from our experiments. Euclidean distance measurement cannot handle the time shifts of patterns. It fails to match the same rise and fall patterns of sequences with different scales. A distance measure that handles this problem is the time warping distance. However, the complexity of computing the time warping distance function is high. Also, as time warping distance is not a metric, most indexing techniques would not guarantee any false dismissal. We propose efficient strategies to mitigate the problems of time warping. We suggest a Haar wavelet-based approximation function for time warping distance, called Low Resolution Time Warping, which results in less computation by trading off a small amount of accuracy. We apply our approximation function to similarity search in time series databases, and show by experiment that it is highly effective in suppressing the number of false alarms in similarity search.

Variable Metric Method for Minimization

Abstract: A method is presented for numerically determining local minima of differentiable functions of several variables. In the proeess of locating each minimum, a matrix is determined which characterizes the behavior of the function about the minimum. For a region in which thc function depends quadratically on the variables, no more than N iterations are required, where N is the number of variables. By suitable choice of starting values and without modification of the procedure, linear constraints can be imposed upon the variables. (auth)