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Machine Learning 2. Lecture 12. Interpretability and Explainability

2026LectureКонстантин Вячеславович Воронцов

Interpretability and explainability of machine learning models: goals, tasks and basic notions, who needs explanations and why; interpretable models; feature importance in linear regression: the t-statistic, net effects, contributions and situational importance; partial dependence plots (PDP) and individual conditional expectation plots (ICE); permutation feature importance; the Shapley value from cooperative game theory, its axioms and Monte Carlo estimation, its use in linear regression; surrogate modelling in the neighbourhood of an object, the LIME, Anchors, SHAP, Shapley Kernel and SAGE methods; the Shapley value for training objects, Data Shapley and Gradient Shapley; counterfactual explanations.
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230

IMO 1973

2026Problem sheet

International mathematics competition for pre-university students, held annually since 1959. Six problems over two days.
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240

IMO 2023

2026Problem sheet

64th International Mathematical Olympiad. Chiba, Japan, 2023.
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430

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

Sharpness Estimation of Combinatorial Generalization Ability Bounds for Threshold Decision Rules

2021Journal articleSh. Kh. Ishkina, K. V. Vorontsov

Automation and Remote Control

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.
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6

RuSciBench: Open Benchmark for Russian and English Scientific Document Representations

2024Journal articleA. Vatolin, N. Gerasimenko, A. Ianina, K. Vorontsov

Doklady Mathematics

Sharing scientific knowledge in the community is an important endeavor. However, most papers are written in English, which makes dissemination of knowledge in countries where English is not spoken by the majority of people harder. Nowadays, machine translation and language models may help to solve this problem, but it is still complicated to train and evaluate models in languages other than English with no or little data in the required language. To address this, we propose the first benchmark for evaluating models on scientific texts in Russian. It consists of papers from Russian electronic library of scientific publications. We also present a set of tasks which can be used to fine-tune various models on our data and provide a detailed comparison between state-of-the-art models on our benchmark.
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4

Incremental Learning of Topic Models for Finding Trend Topics in Scientific Publications

2022Journal articleN. A. Gerasimenko, A. S. Chernyavsky, M. A. Nikiforova, M. D. Nikitin +1

Doklady Mathematics

With a soaring number of scientific publications and rapid emergence of new directions and approaches, the scientific community faces the task of timely identification of trends. By a trend, we mean a semantically homogeneous topic characterized by a steady lexical kernel and a sharp, often exponential increase in the number of publications [1]. Examples of trends in machine learning are “LSTM,” “deep learning,” “word2vec,” “BERT,” and “fake news detection.” For real-time detection of trend topics from a stream of scientific publications, we use incremental methods of probabilistic topic modeling. An ARTM-based approach to early trend detection has been shown to outperform popular classical and neural network approaches to this task. A dataset of 91 trends for performance evaluation has been manually collected and made available for public use.

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7

Incremental Topic Modeling for Scientific Trend Topics Extraction

2023Conference paperNikolai Gerasimenko, Alexander Chernyavskiy, Maria Nikiforova, Anastasia Ianina +1

Computational Linguistics and Intellectual Technologies

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.
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4

Combinatorial probability and the tightness of generalization bounds

2008Journal articleK. V. Vorontsov

Pattern Recognition and Image Analysis

Accurate prediction of the generalization ability of a learning algorithm is an important problem in computational learning theory. The classical Vapnik-Chervonenkis (VC) generalization bounds are too general and therefore overestimate the expected error. Recently obtained data-dependent bounds are still overestimated. To find out why the bounds are loose, we reject the uniform convergence principle and apply a purely combinatorial approach that is free of any probabilistic assumptions, makes no approximations, and provides an empirical control of looseness. We introduce new data-dependent complexity measures: a local shatter coefficient and a nonscalar local shatter profile , which can give much tighter bounds than the classical VC shatter coefficient . An experiment on real datasets shows that the effective local measures may take very small values; thus, the effective local VC dimension takes values in [0, 1] and therefore is not related to the dimension of the space.

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7

The torsion component of the Picard scheme

2026OtherBogdan Zavyalov

Every finite flat commutative group scheme over a noetherian local ring is the torsion component of the Picard scheme of a smooth projective scheme with 3-dimensional fibers, built as a quotient of a complete intersection. An application: Hodge numbers that jump in a smooth projective family.
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500

A curiosity: “supersmooth” varieties

2026TheorySean Cotner

A rigidity condition on schemes, strictly stronger than smoothness, introduced with examples and a look at where it fails to be geometric — no known application, just a curiosity.
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120

Properties of Determinants

2026TheorySergey

The basic properties of determinants with brief proofs, and their use in simplifying computations.
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260

Basic Operations on Matrices

2026TheorySergey

Addition, scalar multiplication, transposition and multiplication of matrices, along with subtraction, powers, and trace.
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150

Properties of Matrix Operations

2026TheorySergey

The basic properties of addition, scalar multiplication, transposition, and multiplication of matrices, with justifications.
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180

Determinant via Permutations

2026TheorySergey

The direct definition of a determinant via permutations: inversions, parity, and a sum of n! terms.

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170

Matrix Rank

2026TheorySergey

Minors and the rank of a matrix, invariance of rank under elementary transformations, and a practical method for computing it.
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190

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

IMO 2021

2026Problem sheet

62nd International Mathematical Olympiad, 2021.
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330

IMO 2022

2026Problem sheet

63rd International Mathematical Olympiad. Oslo, 2022.
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410

IMO 2024

2026Problem sheet

65th International Mathematical Olympiad, 2024.
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560