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Machine Learning 2. Lecture 3. Vector Representations of Texts and Graphs

2026LectureКонстантин Вячеславович Воронцов

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.
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250

Topology 2. Lecture 2

2026LectureФёдор Вылегжанин

Where simplicial homology shows up in real life — topological data analysis. The union of balls around a point cloud, persistence modules and their decomposition into a sum of interval modules over a field (the structure theorem and the algorithm), the barcode; the Vietoris–Rips and Čech complexes; the stability theorem for barcodes, the Hausdorff, Gromov–Hausdorff and Wasserstein distances; open covers and their nerves, Leray covers and the nerve theorem, the category of covers, paracompactness, Lebesgue covering dimension, Čech homology and cohomology.
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520

Machine Learning 2. Lecture 5. Topic Modelling

2026LectureКонстантин Вячеславович Воронцов

Probabilistic topic modelling of text document collections: the bag-of-words and conditional independence hypotheses, the maximum likelihood criterion, three interpretations of the problem (soft clustering, stochastic matrix factorisation, autoencoder); the lemma on maximising a function on unit simplices; the ill-posedness of the problem and the additive regularisation of topic models ARTM, the EM algorithm and its interpretation; PLSA and LDA, the Dirichlet distribution, the maximum a posteriori probability and giving up Bayesian inference; smoothing, sparsing, background topics, decorrelation; multimodal and multilingual models, unigrams and bigrams; regularisers for classification and regression on texts, for word co-occurrence in short texts, for links between documents and for topic hierarchies.
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130

Machine Learning 2. Lecture 4. Attention Models and Transformers

2026LectureКонстантин Вячеславович Воронцов

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.
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450

Machine Learning 2. Lecture 2. Neural Networks for Unsupervised Learning

2026LectureКонстантин Вячеславович Воронцов

Kohonen networks: clustering as a two-layer neural network, the hard (WTA) and soft (WTM) competition rules, self-organising Kohonen maps and their interpretation; autoencoders: the formulation of the problem, the linear autoencoder and the principal component method, sparse, denoising, contractive, relational and variational autoencoders, reparametrisation, autoencoders for supervised learning, multilayer autoencoders; joint training of several models: pre-training and transfer learning, multi-task learning and few-shot learning, self-supervised learning, distillation and surrogate modelling, learning using privileged information (LUPI), generative adversarial networks (GAN).
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210

Machine Learning. Additional Lecture 1. The Theory of Overfitting

2026LectureКонстантин Вячеславович Воронцов

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.
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220

Algebra 1. Lecture 1

2026LectureАртём Авилов

What algebra studies — algebraic structures. Groups (definition, examples, additive notation, uniqueness of the identity and of inverses), rings (variants of the axioms, rings with and without identity), fields and division rings, vector spaces and modules, algebras over a field. The symmetric group SnS_n — permutations, non-commutativity, two-row notation, cycles and transpositions, decomposition into transpositions, disjoint permutations commute, decomposition into disjoint cycles and its uniqueness, powers, inversions, parity and the sign of a permutation, multiplying by a transposition changes the sign.

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310

Topology 2. Lecture 1

2026LectureФёдор Вылегжанин

Why homology is needed — the fundamental group as a functor, homotopy groups, bordism and its combinatorial approximation; chain complexes and their homology; abstract simplicial complexes, geometric realisation and triangulations; the complex of simplicial chains, the proof of d2=0d^2 = 0, the simplicial homology of a segment, of two points and of the boundary of a triangle; reduced homology.

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470

Линейные пространства

2026LectureTaras Panov

Линейные пространства над произвольным полем, подпространства, линейная зависимость, базис и размерность, сумма и пересечение подпространств, замена базиса.
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530

Аффинные пространства

2026LectureTaras Panov

Аффинные пространства и подпространства, аффинные системы координат и формулы их замены, прямые и плоскости в аффинных пространствах.
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460

Machine Learning 1. Lecture 6. Linear Regression Models

2026LectureКонстантин Вячеславович Воронцов

Multiple linear regression and least squares, the pseudoinverse and the geometric meaning of least squares; the solution through the singular value decomposition; multicollinearity and the condition number; ridge regression, choosing the regularization parameter and the effective dimension; LASSO and non-smooth regularizers for feature selection; principal component analysis as a low-rank matrix decomposition; the spectral method of least squares.
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130

Machine Learning 2. Adaptive Forecasting Methods

2026LectureКонстантин Вячеславович Воронцов

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.
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140

Machine Learning 1. Lecture 5. The Support Vector Machine

2026LectureКонстантин Вячеславович Воронцов

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.
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190

Topology 2. Lecture 3

2026LectureФёдор Вылегжанин

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.
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550

Machine Learning 1. Lecture 3. Neural Networks: Gradient Optimization Methods

2026LectureКонстантин Вячеславович Воронцов

The neuron as a linear model; implementing Boolean functions and the exclusive-OR problem; two-layer networks as universal approximators, Cybenko's theorem; the multilayer network and the backpropagation method; the adaptive gradient methods RMSProp, AdaDelta, Adam, Nadam; gradient explosion and gradient clipping; Dropout, ReLU activation functions, batch normalization, initialization and network pruning (Optimal Brain Damage).
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130