When waves propagate through weakly scattering but correlated, disordered environments they are randomly focused into pronounced branchlike structures, a phenomenon referred to as branched flow, which has been studied in a wide range of isotropic random media. In many natural environments, however, the fluctuations of the random medium typically show pronounced anisotropies. A prominent example is the focusing of tsunami waves by the anisotropic structure of the ocean floor topography. We study the influence of anisotropy on such natural focusing events and find a strong and nonintuitive dependence on the propagation angle which we explain by semiclassical theory.

Extreme Classification (XC) refers to supervised learning where each training/test instance is labeled with small subset of relevant labels that are chosen from a large set of possible target labels. The framework of XC has been widely employed in web applications such as automatic labeling of web-encyclopedia, prediction of related searches, and recommendation systems.While most state-of-the-art models in XC achieve high overall accuracy by performing well on the frequently occurring labels, they perform poorly on a large number of infrequent (tail) labels. This arises from two statistical challenges, (i) missing labels, as it is virtually impossible to manually assign every relevant label to an instance, and (ii) highly imbalanced data distribution where a large fraction of labels are tail labels. In this work, we consider common loss functions that decompose over labels, and calculate unbiased estimates that compensate missing labels according to Natarajan et al. [26]. This turns out to be disadvantageous from an optimization perspective, as important properties such as convexity and lower-boundedness are lost. To circumvent this problem, we use the fact that typical loss functions in XC are convex surrogates of the 0-1 loss, and thus propose to switch to convex surrogates of its unbiased version. These surrogates are further adapted to the label imbalance by combining with label-frequency-based rebalancing.We show that the proposed loss functions can be easily incorporated into various different frameworks for extreme classification. This includes (i) linear classifiers, such as DiSMEC, on sparse input data representation, (ii) attention-based deep architecture, Atten-tionXML, learnt on dense Glove embeddings, and (iii) XLNet-based transformer model for extreme classification, APLC-XLNet. Our results demonstrate consistent improvements over the respective vanilla baseline models, on the propensity-scored metrics for precision and nDCG. This paper is published under the Creative Commons Attribution 4.0 International (CC-BY 4.0) license. Authors reserve their rights to disseminate the work on their personal and corporate Web sites with the appropriate attribution.

In this paper, we show that a simple, data dependent way of setting the initial vector can be used to substantially speed up the training of linear one-versus-all classifiers in extreme multi-label classification (XMC). We discuss the problem of choosing the initial weights from the perspective of three goals. We want to start in a region of weight space (a) with low loss value, (b) that is favourable for second-order optimization, and (c) where the conjugate-gradient (CG) calculations can be performed quickly. For margin losses, such an initialization is achieved by selecting the initial vector such that it separates the mean of all positive (relevant for a label) instances from the mean of all negatives – two quantities that can be calculated quickly for the highly imbalanced binary problems occurring in XMC. We demonstrate a training speedup of up to $$5\times$$ 5 × on Amazon-670K dataset with 670,000 labels. This comes in part from the reduced number of iterations that need to be performed due to starting closer to the solution, and in part from an implicit negative-mining effect that allows to ignore easy negatives in the CG step. Because of the convex nature of the optimization problem, the speedup is achieved without any degradation in classification accuracy. The implementation can be found at https://github.com/xmc-aalto/dismecpp.

The propensity model introduced by Jain et al. [18] has become a standard approach for dealing with missing and long-tail labels in extreme multi-label classification (XMLC). In this paper, we critically revise this approach showing that despite its theoretical soundness, its application in contemporary XMLC works is debatable. We exhaustively discuss the flaws of the propensity-based approach, and present several recipes, some of them related to solutions used in search engines and recommender systems, that we believe constitute promising alternatives to be followed in XMLC. CCS CONCEPTS• Computing methodologies → Supervised learning by classification.