Search results for “linear algebra”

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

Linear systems in matrix form, the Kronecker–Capelli theorem, homogeneous systems, and the fundamental system of solutions.

The axioms of a linear (vector) space, examples and counterexamples.

A linear operator and its matrix, the action x' = Ax, similar matrices, and the formula B = C⁻¹AC.

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

Addition, scalar multiplication, transposition and multiplication of matrices, along with subtraction, powers, and trace.

The basic properties of addition, scalar multiplication, transposition, and multiplication of matrices, with justifications.

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

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

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.

Two methods for solving a SLAE — Cramer's rule via determinants and the universal Gaussian elimination method, with examples.

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

The change-of-basis matrix, recomputing a vector's coordinates and an operator's matrix, and the orthonormal case.

Quadratic forms and their matrix, definiteness, Sylvester's criterion, and canonical form.

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.

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.

The problem of probabilistic topic modeling is as follows. Given a collection of text documents, find the conditional distribution over topics for each document and the conditional distribution over words (or terms) for each topic. Log-likelihood maximization is used to solve this problem. The problem generally has an infinite set of solutions and is ill-posed according to Hadamard. In the framework of Additive Regularization of Topic Models (ARTM), a weighted sum of regularization criteria is added to the main log-likelihood criterion. The numerical method for solving this optimization problem is a kind of an iterative EM-algorithm written in a general form for an arbitrary smooth regularizer as well as for a linear combination of smooth regularizers. This paper studies the problem of convergence of the EM iterative process. Sufficient conditions are obtained for the convergence to a stationary point of the regularized log-likelihood. The constraints imposed on the regularizer are not too restrictive. We give their interpretations from the point of view of the practical implementation of the algorithm. A modification of the algorithm is proposed that improves the convergence without additional time and memory costs. Experiments on a news text collection have shown that our modification both accelerates the convergence and improves the value of the criterion to be optimized.