Search results for “граф расслоения–связности”

The combinatorial theory of overfitting: the probability of overfitting and why the Vapnik–Chervonenkis bounds are loose; experiments with overfitting — a monotone chain of algorithms, the effect of splitting and connectivity of the family on the probability of overfitting and on the complete cross-validation estimate, overfitting of chains, the exact formula for the probability of overfitting when choosing between two algorithms; the splitting–connectivity graph of a family of algorithms, examples for linear classifiers, a monotone chain and a two-dimensional net; generating and forbidden sets, upper connectivity and inferiority of an algorithm; the splitting–connectivity upper bound, the idea of its proof and its properties.

The continuity and compactness hypotheses; the Minkowski metric and feature-free distances; the generalized metric classifier, the k nearest neighbours method, the Parzen window and the method of potential functions as a linear classifier; complete cross-validation, the compactness profile and the exact CCV formula for the nearest neighbour method; prototype selection; Nadaraya–Watson kernel smoothing, the choice of kernel and bandwidth.

The general scheme of supervised learning for regression, binary and multiclass classification; the margin and continuous approximations of the threshold loss; the linear classifier as a model of a neuron; the stochastic gradient method, momentum, Nesterov's accelerated gradient and heuristics; multicollinearity and L2 regularization; the maximum likelihood principle, its link to empirical risk minimization and the probabilistic meaning of regularization.

The interpretability of models; decision trees and the greedy ID3 algorithm; the axioms of impurity, the entropy and Gini criteria; handling missing values; the merits and drawbacks of the greedy strategy; tree pruning (post-pruning); CART regression trees and Minimal Cost-Complexity Pruning; rule induction: logical regularities, families of rules (conjunctions, syndromes, half-planes, balls), search metaheuristics, a tree as a covering set of conjunctions; the Pareto front in (p,n) space, logical and statistical regularities, Fisher's exact test, informativeness criteria; building a classifier from regularities.

Collaborative filtering problems: transaction data, the clients–items matrix, filling in missing values and ranking recommendations, the Netflix Prize competition; user-based and item-based correlation methods, the Pearson, cosine and Jaccard similarity measures, Fisher's exact test; the sparse linear model SLIM; latent models and matrix factorisations: SVD, the alternating least squares method ALS, the latent factor model LFM and stochastic gradient, nonnegative matrix factorisations NNMF, PLSA; taking implicit preferences into account (implicit feedback); matrix factorisation as a linear autoencoder, neural network and denoising autoencoders; taking features of clients and items into account, factorisation machines libFM, a hypergraph topic model; the quality of recommendations: RMSE, precision@k, recall@k, diversity, novelty, coverage, serendipity, bias, offline and online evaluation.

Vector representations (embeddings) of texts and graphs: the distributional semantics hypothesis, syntagmatic and paradigmatic similarity of words, the CBOW and Skip-gram models of the word2vec program, hierarchical SoftMax on a Huffman tree, negative sampling (SGNS) and the connection of word2vec with the matrix factorisation of the PMI matrix, semantic analogies of words, the FastText model on character n-grams; texts as graphs; multidimensional scaling and the stress criterion, stochastic neighbour embedding SNE, perplexity, t-SNE; matrix factorisations of graphs, the random-walk models DeepWalk and node2vec; graph embedding methods as autoencoders, the generalised GraphEDM model, graph neural networks and message passing.

Sequence-to-sequence problems: the recurrent network, the encoder–decoder architecture and its limitations; the recurrent network with an attention model, interpreting the attention matrix in machine translation and image captioning; similarity functions, the query–key–value attention model, the attention formula, self-attention; multi-head attention and its regularisation, hierarchical attention for document classification, the graph attention model GAT; the transformer for machine translation: the encoder, positional encoding, layer normalisation and skip connections, the decoder with masking, the training criterion, BLEU and WER; the transformer encoder BERT, the MLM and NSP criteria, fine-tuning, BPE and WordPiece tokenisation, model sizes.

Singular homology — standard simplices and the inclusions of facets, singular simplices and chains, the differential; zeroth homology and path components, the homology of a point, reduced homology; chain maps and maps of homology, functoriality, the chain map induced by a continuous map; chain homotopies and the prism; the homotopy invariance theorem with a full proof via the prism operator; subcomplexes, quotient complexes and the homology of a pair.

The optimal separating hyperplane and the width of the separating strip; the hinge loss and L2 regularization; the Karush–Kuhn–Tucker conditions, the dual problem and the types of objects; kernels, rectifying spaces and the constructive building of kernels; SVM as a two-layer neural network; choosing the constant C; SVM regression; L1 regularization, Elastic Net SVM, SFM and RVM.

Евклидовы и эрмитовы пространства, скалярное произведение, неравенство Коши–Буняковского и неравенство треугольника, ортогональные системы векторов.

The learning-to-rank problem: recovering a partial order relation, the pointwise, pairwise and listwise approaches, the linear ranking model, applications; ordinal regression and the ordinal SVM (SVOR); the pairwise criterion, Ranking SVM, RankNet, the transition to the listwise method LambdaRank; ranking search results: query–document objects, assessor ratings, text, link and click features, TF-IDF and its probabilistic meaning, Okapi BM25, PageRank; quality criteria: precision and recall, MAP, the share of defective pairs and its link with AUC, DCG and NDCG, the pFound user behaviour model; the DSSM neural model: a siamese network, word hashing by letter trigrams, training on clicks with negative sampling.

Time series forecasting problems: the time series model, the forecasting horizon, the least squares method; econometrics as the main source of problems — trends, seasonalities, change points; examples of forecasting sales volumes and electricity prices; the linear autoregression model; a brief survey of forecasting methods; adaptive short-term forecasting methods: the exponential moving average and the smoothing parameter, the models of Holt, Theil–Wage, Winters with linear and exponential trend, adaptive autoregression; the tracking signal and a statistical test of model adequacy, the Trigg–Leach model; adaptive selection and adaptive composition of models, the LAVR method — local adaptation of weights with regularisation, the monotonicity constraint as a regulariser; experiments on the sales forecasting problem.

Naive set theory and Russell's paradox, Kuratowski's ordered pair and the Cartesian product, a function as a subset of a product, composition, injections, surjections and bijections, the group of bijections and the embedding of any group into it, equinumerosity and indicator functions, the Cantor–Bernstein and Cantor theorems.