Search results for “number-theory”

64th International Mathematical Olympiad. Chiba, Japan, 2023.

Eigenvectors and eigenvalues, the characteristic equation, eigenspaces, and diagonalization of matrices.

The basic properties of determinants with brief proofs, and their use in simplifying computations.

A complete local ring with a finitely generated maximal ideal is Noetherian. Drop completeness and it fails: an ultraproduct construction gives a non-Noetherian local ring whose maximal ideal is principal.

This article is devoted to the problem of calculating an exact upper bound for the functionals of the generalization ability of a family of one-dimensional threshold decision rules. An algorithm is investigated that solves the stated problem and is polynomial in the total number of samples used for training and validation and in the number of training samples. A theorem is proved for calculating an estimate for the functional of expected overfitting and an estimate for the error rate of the method for minimizing empirical risk on a validation set. The exact bounds calculated using the theorem are compared with the previously known quick-to-compute upper bounds so as to estimate the orders of overestimation of the bounds and to identify the bounds that could be used in real problems.

Linear dependence and independence of the rows of a matrix, the criterion via a linear combination, and the connection to rank.

Nonsingular matrices, the invertibility criterion, the adjugate-matrix formula, and the Gauss–Jordan method.

Minors and the rank of a matrix, invariance of rank under elementary transformations, and a practical method for computing it.

Linear dependence of vectors, basis and coordinates, dimension, subspaces, and linear span.

The determinant of a square matrix — its meaning, minors and cofactors, Laplace expansion, and computation methods.

This paper introduces a new approach to estimating the text document complexity.Common readability indices are based on average length of sentences and words.In contrast to these methods, we propose to count the number of rare words occurring abnormally often in the document.We use the reference corpus of texts and the quantile approach in order to determine what words are rare, and what frequencies are abnormal.We construct a general text complexity model, which can be adjusted for the specific task, and introduce two special models.The experimental design is based on a set of thematically similar pairs of Wikipedia articles, labeled using crowdsourcing.The experiments demonstrate the competitiveness of the proposed approach.

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.

Rapid growth of scientific publications and intensive emergence of new directions and approaches poses a challenge to the scientific community to identify trends in a timely and automatic manner. We denote trend as a semantically homogeneous theme that is characterized by a lexical kernel steadily evolving in time and a sharp, often exponential, increase in the number of publications. In this paper, we investigate recent topic modeling approaches to accurately extract trending topics at an early stage. In particular, we customize the standard ARTM-based approach and propose a novel incremental training technique which helps the model to operate on data in real-time. We further create the Artificial Intelligence Trends Dataset (AITD) that contains a collection of early-stage articles and a set of key collocations for each trend. The conducted experiments demonstrate that the suggested ARTM-based approach outperforms the classic PLSA, LDA models and a neural approach based on BERT representations. Our models and dataset are open for research purposes.

Modern machine translation systems are trained on large volumes of parallel data obtained using heuristic methods of bypassing the Internet. The poor quality of the data leads to systematic translation errors, which can be quite noticeable to humans. To fix such errors, human-based models for reranking hypotheses is introduced in this study. In this paper the use of human markup is shown not only to increase the overall quality of the translation but also to significantly reduce the number of systematic translation errors. In addition, the relative simplicity of human markup and its integration in the model training process opens up new opportunities in the field of domain adaptation of translation models for new domains like online retail.

Modern machine translation systems are trained on large volumes of parallel data obtained using heuristic methods of the Internet bypassing. The poor quality of the data leads to systematic translation errors, which can be quite noticeable from the human point of view. To fix such errors a human based models hypotheses re-ranking is introduced in this work. In this paper the use of human markup is shown not only to increase the overall quality of translation, but also to significantly reduce the number of systematic translation errors. In addition, the relative simplicity of human markup and its integration in the model training process opens up new opportunities in the field of domain adaptation of translation models for new domains like online retail.

In the exploratory search paradigm of information retrieval, the user has a complicated search demand that can not be formulated in a short query. The user collects thematically relevant information iteratively in a “query-browse-refine” process being motivated by learning, understanding, and knowledge acquisition purposes. We consider an elementary step of this scenario in which the search intent can be expressed by a long text query. For this case, we develop an exploratory search engine based on probabilistic topic modeling. Topic model gives a low-dimensional sparse interpretable vector representation (topical embedding) of a text. The search engine uses these embeddings for ranking documents by their similarity to the query. We show that performing only one query, the topic-based search engine achieves better precision and recall that human assessors do spending up to one hour in a conventional browse-refine loop. We use additive regularization for topic modeling (ARTM) to make the model simultaneously sparse, decorrelated, n-gram, multimodal and hierarchical. We show experimentally that each of these features of the model is important to achieve precision and recall higher than 90