Cross-entropy loss is crucial in training many deep neural networks. In this context, we show a number of novel and strong correlations among various related divergence functions. In particular, we demonstrate that, in some circumstances, (a) cross-entropy is almost perfectly correlated with the little-known triangular divergence, and (b) cross-entropy is strongly correlated with the Euclidean distance over the logits from which the softmax is derived. The consequences of these observations are as follows. First, triangular divergence may be used as a cheaper alternative to cross-entropy. Second, logits can be used as features in a Euclidean space which is strongly synergistic with the classification process. This justifies the use of Euclidean distance over logits as a measure of similarity, in cases where the network is trained using softmax and cross-entropy. We establish these correlations via empirical observation, supported by a mathematical explanation encompassing a number of strongly related divergence functions.

Most research into similarity search in metric spaces relies on the triangle inequality property. This property allows the space to be arranged according to relative distances to avoid searching some subspaces. We show that many common metric spaces, notably including those using Euclidean and Jensen-Shannon distances, also have a stronger property, sometimes called the four-point property: In essence, these spaces allow an isometric embedding of any four points in three-dimensional Euclidean space, as well as any three points in two-dimensional Euclidean space. In fact, we show that any space that is isometrically embeddable in Hilbert space has the stronger property. This property gives stronger geometric guarantees, and one in particular, which we name the Hilbert Exclusion property, allows any indexing mechanism which uses hyperplane partitioning to perform better. One outcome of this observation is that a number of state-of-the-art indexing mechanisms over high-dimensional spaces can be easily refined to give a significant increase in performance; furthermore, the improvement given is greater in higher dimensions. This therefore leads to a significant improvement in the cost of metric search in these spaces.

Many valuable earth science data are not available in a digital format. Manual entry of such information into databases is time consuming, unrewarding, and prone to introducing errors. Taxonomic descriptions of fossils are a good example of valuable data that are overwhelming and available only in printed volumes and journals, some of which are increasingly rare and inaccessible. The highly structured nature of taxonomic procedures and nomenclature means that many previously published data remain equally valid to the present day, and contain information that is currently not available on the World Wide Web; these data would be of great use to a wide variety of scientists and other end users in government, industry, academia and the general public. This paper describes an XML (extensible markup language) parsing technique that allows taxonomic descriptions to be fully digitized much more rapidly than would be possible by manual entry of the data into a database. The technique exploits the high degree of structure in taxonomic descriptions, which are written in a standardized format, to automate the processing of tagging separate sections of the text. Once tagged using XML, the data can be subjected to complex searches using queries written in any of the XML query standards. The XML-tagged data can potentially be imported into existing databases, in effect removing the necessity to manually enter the information, and hence overcoming the main bottleneck in generating digital data from printed material. Individual parsers can be tailored precisely to the nature of the text being analyzed, and once the underlying concepts and procedures are understood, those interested in acquiring and using digital data will be able to generate XML parsers dedicated to text with different styles of standardized formatting