Search results for “functional-equation”

Reference table of the derivatives of the elementary functions: power, exponential, logarithmic, trigonometric and their inverses.

Reference table of indefinite integrals of the exponential function and its products with powers of x.

Topic modeling is an area of text analysis actively developing over the past 20 years. A probabilistic topic model (PTM) finds a set of hidden topics from a collection of text documents. It defines each topic as a probability distribution over words and describes each document as a probability mixture of topic distributions. Learning algorithms for topic models are usually based on Bayesian inference or log-likelihood maximization. In both cases, EM-like algorithms are used. In this paper, we propose to replace the logarithm in the log-likelihood by an arbitrary smooth loss function. We prove that such a modification preserves both the structure of the algorithm and compatibility with any regularizers in terms of additive regularization of topic models (ARTM). Moreover, in the case of a linear loss, the Estep becomes much faster due to the omission of a normalization. We study combinations of the fast and usual E-steps and compare them to regularization using different number of topics in both offline and online versions of EM-algorithm. For an empirical comparison of the algorithms, we estimate perplexity, coherence, and learning time. We use an efficient parallel implementation of the EM-algorithm from the BigARTM open-source library. We show that in most cases the two-stage strategy wins, which uses fast E-steps at the beginning of iterations, then proceeds with usual E-steps.

This paper introduces an approach to measuring the cognitive complexity of texts on various language levels. While standard readability indices are based on the linear combination of primary statistics, our general approach allows us to estimate complexity on morphological, lexical, syntactic, and discursive levels. Each model is defined by the tokens for the specific language level and the complexity function of a single token. We then use the reference collection of moderately complex texts and the quantile-based approach to spot the abnormally rare tokens. The proposed supervised ensemble, based on the ElasticNet model, incorporates models from all language levels. Having collected a labeled dataset through crowdsourcing, consisting of pairs of articles from the Russian Wikipedia, we consider several models and ensembles and compare them to common baselines. Suggested models are flexible due to the freedom in choosing the reference collection. The described experiments confirm the competitiveness of the proposed approach, as the ensembles demonstrate the best target metric value.