Методы геометрии и топологии для исследования моделей глубокого обучения тема диссертации и автореферата по ВАК РФ 00.00.00, кандидат наук Магай Герман Игоревич

• Магай Герман Игоревич
• кандидат наук
• 2025

Оглавление диссертации кандидат наук Магай Герман Игоревич

Contents

Introduction

1 Preliminaries

1.1 Neural Networks Foundations

1.1.1 Deep neural networks

1.1.2 DNN Generalization and Learning

1.1.3 Manifold learning perspective on Neural Networks

1.2 Deep learning Architectures Paradigms

1.2.1 Convolutional Neural Networks

1.2.2 Transformers and RNN

1.2.3 Foundation models

1.3 Safety AI Tasks and Challenges

1.3.1 Mechanistic interpretability and Explainable AI

1.3.2 Adversarial and spoofing attacks

1.3.3 AI-generated content detection

1.4 Topology in Data Science

1.4.1 Applied algebraic topology

1.4.2 Persistent Homology and Topological Data Analysis

1.5 Geometrical Approaches in Data Science

1.5.1 Manifold learning and Intrinsic dimensions

1.5.2 Persistent Homological Fractal Dimension

1.6 Point cloud similarity measure

1.6.1 Centered Kernel Alignment

1.6.2 Log-Euclidean Signatures

2 Literature Review

2.1 Geometrical and Topological Approaches in Deep Learning: Comprehensive High-Level Survey

2.2 Deep models embedding analysis

2.2.1 Internal representations dynamics across layers

2.2.2 Internal representations similarity on different layers

2.2.3 Geometry and topology inspired embeddings construction

2.3 Deep learning for mathematical reasoning

2.4 Safety AI and Deep models Interpretability

2.4.1 Explainable AI

2.4.2 AI-generated content detection

3 From Layer to Token: A Comprehensive Study of Neural Network Embedding

3.1 Geometry of Embedding and Problem statement

3.1.1 What types of embeddings do we know?

3.1.2 Intrinsic dimension and Topological complexity

3.2 Geometry and Topology of Layers Embeddings

3.2.1 Layer-level data representation flow

3.2.2 Methodology and Experimental setup

3.2.3 Investigation of CNNs Layer Embeddings

3.2.4 Investigation of Text Transformers Layer Embeddings

3.3 Deep models performance analysis via lens of data manifold topology

and geometry

3.3.1 Representation Learning and Feature Compression

3.3.2 Geometry of representations in model training

3.3.3 Topological collapse and Generalization gap estimation

3.4 Token-level embedding manifold in Transformers

3.4.1 Patch-Level Embedding Manifold

3.4.2 Patch level manifold geometry capture visual complexity

3.4.3 Patch-manifold representations geometry

3.5 Domain-specific initial embeddings inductive bias: Topological inspired embedding constructions with GRU

3.5.1 Initial Embedding and Domain-prior knowledge

3.5.2 Background

3.5.3 Tonal Space and topological chords embedding

3.5.4 TonnetzNet: Topological embeddings trick with GRU

3.6 Conclusions

4 Vision Transformers Representations Analysis

4.1 Geometrical and Functional Analysis Approach for the Interpretability

of Vision Transformers

4.1.1 Structure and methodology of experimental framework

4.1.2 Approach for Geometrical and Functional Analysis

4.2 Vision Transformer Representations: Empirical study

4.2.1 ViT in Supervised learning

4.2.2 ViT Foundation models

4.3 Cross-model Representations Analysis

4.3.1 Convergence of Representations Across Architectures

4.3.2 Vision-Text Representation Convergence

4.4 Representation Similarity Analysis: Applications

4.4.1 Transfer learning ability estimation

4.4.2 Data visualization. Domain geometry distance

4.5 Conclusions

5 Evaluating Deep Models for Algebraic Topology Applications

5.1 Algebraic Topology provides challenges for AI

5.2 Homotopy Algebra Background and Motivation

5.2.1 Group theory background

5.2.2 Wu's formula and Homotopy groups of spheres

5.3 Applying AI to Topology: Proof of Concept

5.3.1 Problem statement and Methodology

5.3.2 Non-neural baselines

5.3.3 Normal closure modeling

5.3.4 Intersection of normal closures

5.3.5 LLMs in-context learning

5.4 Conclusions

6 Investigating Geometry in DNNs for AI Safety Applications

6.1 Synthetic images detection via geometry

6.1.1 Experimental setup and methodology

6.1.2 Synthetic images Datasets

6.1.3 Results and discussion

6.1.4 DeepFake and Spoofing attacks

6.1.5 Cross-domain generalization

6.2 Synthetic texts detection via geometry

6.2.1 Recognition of AI-generated texts: robustness evaluation

6.2.2 Synthetic texts boundary detection ROFT

6.3 Adversarial manifold geometry understanding

6.4 Conclusions

Conclusion

References

List of Figures

List of Tables

Введение диссертации (часть автореферата) на тему «Методы геометрии и топологии для исследования моделей глубокого обучения»

Introduction

Topic of the Work

Deep learning has revolutionized a wide range of fields in recent years, including computer vision, natural language processing, robotics, and health informatics, consistently delivering state-of-the-art performance across numerous applications. These breakthroughs stem largely from innovations in developing efficient deep neural network (DNN) architectures and advanced training techniques. The neural network itself is a cornerstone of progress in artificial intelligence, blending fundamental theoretical insights with sophisticated technical solutions and robust computing infrastructure. Deep learning has rapidly evolved by integrating theories, intuitions, and methods from diverse disciplines such as mathematics, computer science, and engineering. These interdisciplinary approaches offer multiple perspectives on deep learning designs and challenges, yielding valuable insights and effective solutions. One unexpected perspective on deep learning models emerges from geometric and topological methods. This opens a wealth of opportunities for exploring and enhancing neural networks, from analyzing DNN models to designing new architectures. This dissertation research contributes to the interdisciplinary field by applying geometric and topological methods to deep learning, providing a systematic study of various aspects of deep learning models from this perspective.

AI Safety motivation. Despite the remarkable progress of modern neural networks, a fundamental challenge in AI endures: the decision-making processes of DNNs often remain uninterpretable, and their internal states poorly understood. DNNs therefore function as black boxes: we can observe their input-output behaviour, yet the mechanisms that govern their information processing and decisions remain obscure. Such opacity can yield unsafe or unintended outcomes in mission-critical domains, medical decision support, autonomous driving, and critical-infrastructure management, where model failures may cause substantial harm. At the same time, rapid advances in generative modelling and the resulting proliferation of realistic AI-generated content pose additional societal risks. These issues lie at the heart of AI safety [153], the discipline devoted to designing AI systems that are verifiably safe. Unravelling the internal logic of DNNs requires a thorough grasp of their hidden data representations. As deep-learning systems become ever more embedded in real-world applications, the imperative to study and interpret these representations—especially their evolution during training—grows increasingly acute. Such understanding is essential for improving system reliability, mitigating ethical hazards, and ensuring the secure deployment of AI technologies, as stressed in the Safety AI Report [31]. These challenges, therefore, motivate the research presented in this dissertation.

Deep models interpretability. The rapid advances in AI, coupled with the concerns raised above about the safety and predictability of AI systems, have given rise to the concept of AI Alignment [200] in academia and industry. This approach in AI philosophy and engineering seeks to develop systems that are beneficial to humans and free from the risk of unintended harm. In this context, the fields of Interpretable and Explainable AI have recently gained momentum. These fields aim to elucidate how DNNs arrive at their predictions, enabling stakeholders—from engineers and re-

searchers to end users—to better understand the reasoning behind model outputs. One particularly prominent area is Mechanistic Interpretability, with methods ranging from local post-hoc techniques, such as saliency maps, gradient-based attribution methods, and counterfactual explanations, to global interpretability approaches that analyze the DNN's internal structures, layer by layer or neuron by neuron [32]. An example of mechanistic interpretability is the analysis of properties of internal representations (probing). A related direction is developmental interpretability, which studies the dynamics of model properties during training. In this thesis, we discuss methods of interpretability and the place of our proposed methods in its taxonomy. Some of our key results constitute both a conceptual development and a geometrical extension of the methodological lens of mechanistic interpretability.

Deep learning model as an object of study. Deep learning models have become pivotal in numerous applications, offering a versatile toolkit for investigating empirical phenomena across diverse scientific fields, including physics, biology, economic analysis, medical support etc. While frequently employed as practical instruments, the intricate nature of these models renders them a compelling object of study in their own right, promising deeper insights into their operational principles and inherent capabilities.

What are the main directions in the study of neural networks?

Here we briefly describe the perspectives within the scientific mainstream of neural network research regarding the methods and stages at which DNNs are studied:

• Post-hoc interpretability research. A significant focus in neural network studies involves analyzing DNNs post-development to enhance their interpretabil-ity. This line of inquiry is crucial for understanding trained models and ensuring they meet regulatory, ethical, or safety standards. Interpretability techniques are typically employed after model training to dissect and explain their decision-making processes, thereby fostering trust and accountability in their applications.

• Research for development. Another key direction centers on leveraging insights from DNN studies to inform the creation of more efficient architectures. This approach integrates research directly into the development process, enabling the design of novel deep learning frameworks or the optimization of existing ones. By studying DNNs during their construction, researchers aim to enhance architectural efficiency and performance tailored to specific tasks.

• Neural network as a mathematical object. Beyond practical applications, neural networks constitute abstract mathematical entities worthy of theoretical exploration. This perspective investigates DNNs through the lens of advanced mathematical disciplines, including functional analysis, algebraic geometry, partial differential equations, tropical geometry, and operator theory, with the potential to contribute to foundational questions in mathematics itself.

We establish the primary object of investigation: deep learning models themselves. The multifaceted character of DNNs means that various research objectives and methodologies scrutinize distinct facets from differing viewpoints. Drawing an analogy from the natural sciences, this research conceptualizes a DNN metaphorically as an organism an entity whose internal mechanisms and characteristics are to be examined

in detail, like an investigation under a microscope. While the broader field of deep learning explores numerous aspects of DNNs such as their weight space, expressive capabilities, architectural configurations, loss function landscapes, decision boundaries, etc., the present study will precisely identify and focus on specific facets. These selected aspects are central to addressing the core research questions posed in our work and will be elaborated upon in the subsequent.

What aspects of the deep models do we study and consider?

Neural networks working pipeline. To clarify the motivation and objectives of this study, we briefly outline the end-to-end operational pipeline of a neural network. We begin by selecting the architectural features of a DNN based on the data and training objectives. The model is then trained by learning internal representations of the data. During training, the model extracts patterns, referred to as knowledge, from the data. Following training, the model's generalization ability and performance are evaluated. Its ultimate capabilities, however, are determined and constrained by its architectural features, the task, and data complexity. Finally, the model is applied to practical applications in real-world scenarios. From this pipeline, we identify the following fundamental components of a DNN's operation: representations, training, knowledge, capabilities, and applications. These components, once formulated more specifically for this study, constitute the primary aspects of our investigation.

Embeddings and Internal Representations. Neural networks are commonly used as feature extractors for various types of data, including sequences, images, audio, and graphs. DNNs are trained to encode these inputs into internal feature representations known as embeddings. These embeddings capture essential semantic and structural properties of the input data as encoded within the DNN. A comprehensive understanding of embeddings is necessary for analyzing and interpreting DNN models [30]. A detailed examination of the properties, applications, and layer-wise evolution of these embeddings constitutes one of the central focuses of this thesis.

Training Dynamics in Deep Models. The study of training dynamics in deep models investigates how internal representations and parameters evolve throughout the training phase, a process intricately linked to optimization methods and the interpretability of training procedures. Such dynamics influence the generalization gap. Investigating these dynamics offers valuable insights into model performance and the specific data features that shape its outcomes. In this thesis, we specifically examine the dynamics of embedding properties during deep models training process.

Domain-specific Knowledge Inductive Bias. This concept refers to the incorporation of domain-specific knowledge into a model's architecture or training process to guide its learning and generalization [51]. Inductive bias is any set of assumptions that a model uses to generalize beyond the observed training data, such as those embodied in a neural network's architecture. Incorporating domain-specific knowledge as such an inductive bias effectively constrains the model's hypothesis space, thereby focusing the learning process on more plausible solutions.

Evaluation of Deep Models' Capabilities. This area of investigation focuses on assessing how effectively deep models learn, generalize, and adapt to diverse tasks and data distributions. Key methodologies for this evaluation include systematic, controlled testing and the comparative performance analysis of various neural architectures using established benchmarks and tailored controlled experiments. In this thesis,

we investigate this aspect specifically within the context of language models applied to specialized symbolic sequences encoding the combinatorial structure of free groups.

Geometry and Topology methods. Viewing neural networks and their multi-faceted aspects as an object of study, geometric and topological methods serve as the conceptual 'lenses' for investigation. These methods encompass a suite of mathematical tools, concepts, and intuitions that determine the point of view or perspective from which we consider various aspects of deep neural networks.

What geometry and topology methods do we use?

Recently, computational topology and geometry approaches have advanced significantly, with methods for calculating invariants from vector representations and graphs under active development. The synergy of geometry, topology, and deep learning has provided powerful, theoretically robust tools to understand, explore, and enhance various aspects of DNNs. These methods enable us to address both fundamental theoretical questions in deep learning and practical engineering challenges. For example, examining the decision boundaries of a ReLU DNN's activation function through polyhedral theory [212]. A distinct field has emerged Topological Deep Learning [92] where topological methods encode complex data relations using structures such as simplicial [68], combinatorial, or cellular complexes, beyond just graphs. Section 2.1 provides a comprehensive survey of recent geometric and topological methods addressing theoretical and practical challenges in modern deep learning.

However, one of the main basic methods of geometry and topology in data analysis is Manifold learning. In this paradigm, it is assumed that vector representations of data lie on a low-dimensional manifold whose curvature and dimension can be calculated [155]. We discuss manifold learning and data intrinsic dimension in Section 1.5.1. Accordingly, information flow inside DNN can be viewed as a sequence of data representation manifolds, induced by a set of nonlinear mappings at each layer. Whether to consider manifold learning as a geometric method is a purely terminological question; following the consensus in the field, we consider it as such.

Topological data analysis (TDA) is a classical method of computational topology that offers a topological view of data by quantifying its topological features such as connected components, loops, and voids. The main tool of TDA is persistent homology, see Section 1.4.2 for details of the topological methods. In this study, TDA methods are used to investigate the topological properties of DNNs' embedding space.

However, topological structure is induced from data even if it is not part of a metric space. Nearly any system of data relationships can be described topologically, capturing hierarchical structures. Therefore, the application of topological methods in our study extends well beyond just computing topological invariants of point clouds. In this study, we primarily focus on, but are not restricted to, the methods mentioned above. We also employ Riemannian geometry, fractal geometry, dimension theory, ho-motopy topology, and other theories. In each subsection presenting the study's results, we provide a detailed description of the mathematical framework for the topological and geometric methods employed.

Motivation and open problems. We are motivated by significant open problems in applying different theories to DNN research, and we formulate the PhD thesis objectives

accordingly. In this work, we examine various aspects of neural networks, analyzing them through the lens of geometry and topology.

Why are we exploring these aspects and what open problems does it solve?

Embeddings topology and geometry dynamics. Given the critical role of embeddings as representational structures and data encoding mechanisms essential for interpreting how DNNs process and understand data, a significant gap exists in the study of their layer-wise and cross-layer dynamic properties. Specifically, we examine how DNN architecture (e.g., CNNs, Transformers, MLP-Mixers), activation functions, and training modes (supervised, self-unsupervised, contrastive learning) influence embedding dynamics. By analyzing various DNN embeddings through their geometric and topological properties, we address this identified gap.

Training geometry evolution and generalization gap estimation. Investigating the dynamics of embeddings during training is crucial for understanding how a model learns effective data representations, revealing the interaction between optimization processes and the layer-specific characteristics of the feature space. This is particularly valuable for detecting overfitting. However, developing methods to automatically estimate generalization ability remains an open challenge, one that is particularly significant in the context of limited data, common in practical applications. We address this challenge by proposing a geometric analysis of embeddings during training and developing an approach to estimate the generalization gap.

Topological embeddings with domain-specific inductive bias. Integrating domain-specific knowledge into model training pipelines has long been recognized as effective for improving performance. This technique is common for graph and text data but has been largely overlooked in music signal processing. Moreover, there has been insufficient focus on developing approaches to solving music processing tasks using domain-specific knowledge from the fundamental and theoretically justified field of topological music theory. Our research addresses this gap by exploring the integration of topological data properties into DNN training for performance improvement.

Evaluation of deep models capabilities in homotopy topology task Given the success of deep learning methods in mathematical reasoning, it is crucial to explore their application to increasingly complex challenges in pure mathematics. Specifically, we evaluate the potential of deep language models to learn complex structural rules in data, a capability related to compositional generalization. Inspired by the beauty of pure mathematics and the success of deep language models, we evaluate their capability to model sequences representing normal closures of free groups. This is a first step towards solving the fundamental problem of automatically finding non-trivial elements of homotopy groups of spheres according to the combinatorial Wu's formula.

Geometry of embeddings in Safety AI application. With the rapid development and widespread adoption of generative models, the realism and prevalence of AI-generated content raise significant social concerns. These issues fall within the broader field of AI Safety, which seeks to mitigate AI-related risks (see Section 1.3 for details). Developing robust methods for AI-generated content detection is one of the fundamental challenges in AI Safety. However, the trustworthiness of these methods depends on their interpretability. Motivated by the problem's societal importance, we develop approaches to detect AI-generated content using geometric information.

Summarizing the above-mentioned key problems associated with various aspects of deep learning models, we investigate the application of topological and geometric

methods to address them.

Goals and tasks This study aims to examine various aspects of DNNs through the lens of geometry and topology. Ultimately, we aim to develop a unified methodological and conceptual framework for a comprehensive understanding and interpretation of deep models in terms of geometry and topology. We tackle the following tasks:

1. Development of a method for analyzing and interpreting various types of deep neural network embeddings from the point of view of geometry and topology;

2. Development of an approach to evaluate the generalization ability and knowledge transfer learning ability of deep learning models based on the geometric properties of internal representations;

3. Development of a method for integrating domain-specific knowledge derived from topological properties of data into a neural network training pipeline in the task of musical signals processing;

4. Assessment of the applicability of geometric properties of Transformer embed-dings to detecting AI-generated content, deepfakes, and spoofing attacks;

5. Investigation of the capability of neural networks in tasks of generation and modeling symbolic sequences which represent the structure of normal closures of free groups in problems inspired by homotopy topology.

Main results and conclusions

From Layer to Token: Geometrical and Topological Study of Neural Network Embedding. This research proposes a method for studying and interpreting the dynamics of data representation flow across layers of diverse DNN embeddings in text and image processing tasks. We establish a relationship between architectural features (e.g. CNNs, Transformers, and MLP-Mixers), activation functions, datasets, DNN training modes (supervised, self-supervised, contrastive learning), and the geometric and topological properties of deep neural network models' internal representations. Particular emphasis is placed on Foundation Models. An approach to testing the hypothesis of platonic representation in DNNs in an experimental setup of cross-model convergence and cross-modality convergence of geometric properties of embed-dings in foundation models is also proposed. Furthermore, we discover and propose an interpretation for a two-phase layer-wise dynamic of intrinsic dimensions in the hidden representations of Transformers and CNNs. The study also proposes a method for investigating the geometry of cross-layer and layer-wise evolution for various ViT embedding types, including layer-level, attention-head, and patch-token embeddings. Detailed results are presented in Section 3.2 for CNNs and Section 4.2 for Transformers.

Analysis of Deep Model Performance through the Lens of Data Manifold Topology and Geometry. In Section 3.3 drawing upon our study of embedding geometry, we develop methods for estimating DNN performance in supervised learning mode based on topological and geometrical properties of DNN final layer data representation. We also proposed an approach for assessing the transfer learning capabilities

of deep neural network models. Furthermore, we propose an interpretation of the dynamics of embedding geometry during DNN training. For the experimental validation of these proposed methods, we collected a dataset of pre-trained CNN and ViT models.

Topologically Inspired Domain-Specific Initial Embeddings for Neural Networks. Our research introduces TonnetZNet, a novel recurrent deep neural network architecture for music signal processing tasks. Its key feature is the integration of domain-specific knowledge prior, based on topological data properties, into the model training pipeline via topological embeddings. In Section 3.5, we investigate the effect of inductive bias, introduced through these topological initial embeddings, on the model's accuracy and convergence. The experiments focus on the chord prediction task using the JS Bach Chorales classical polyphonic music dataset and confirm the performance improvements achieved by the proposed approach. This result demonstrates the potential of topological methods in developing new neural network architectures.

Evaluating Deep Models' capabilities for Algebraic Topology Problems. We

evaluate the ability of neural networks to model symbolic sequences representing the group structure motivated by algebraic topology. In Section 6, we develop an approach to systematically compare various DNNs architectures on the problems of classification and generation of elements in specific subgroups (normal closures) of finitely generated groups related to problems inspired by the study of homotopy groups of spheres.

Application of Geometry to Safety AI. We investigate the applicability of geometric properties of Transformer embeddings to detecting shifts in data sources. We develop a method based on the intrinsic dimension of embeddings to detect AI-generated images, deepfakes, and spoofing display attacks. The generalization robustness of the proposed method is evaluated in a cross-domain and leave-one-domain-out experimental setup. We also evaluate the generalization robustness of the approach for detecting AI-generated text in a cross-model setup with modern LLMs. The line of research presented in Section 7 is related to the fields of AI Security and AI Safety.

Theses for Defense

1. An approach utilizing geometric and topological methods is developed to study and interpret the layer-wise flow of data representation in diverse neural network architectures, improving the interpretability of hidden representations;

2. We propose an approach to interpret the training dynamics of deep neural networks through the lens of geometric and topological properties of embedding spaces. This analysis of embedding dynamics forms the basis for a new method to evaluate neural network model performance;

3. A novel deep neural network architecture, TonnetzNet, is proposed, leveraging topological embeddings. Experiments demonstrate enhanced accuracy and convergence in training for chord prediction tasks;

4. An approach is developed to evaluate capabilities of neural networks for modeling symbolic sequences that encode the structure of elements in free groups, specifically in contexts related to Wu's formula in homotopy topology. This includes an investigation into the capacity of various DNN architectures for this task.

5. This thesis investigates and demonstrates the applicability of geometric methods for effectively robust detecting data source shifts and for recognizing AI-generated and malicious content;

Additional results. Supplementary additional results, not submitted for defense but supporting and complementing the main findings, are also highlighted. Section 3.1.2 presents a comparison of the accuracy of various methods for estimating intrinsic dimensionality. In Section 3.4, we introduce a method for the automatic assessment of image visual complexity, based on the geometry of token-patch embeddings in Transformers. Section 6.1.2 describes the curation of a dataset featuring AI-generated content from modern diffusion models. Furthermore, we propose a method for the visual representation of image dataset structures, based on Transformer embeddings, in Section 4.4.2. Our investigation into the geometric properties of adversarial attack embeddings is discussed in Section 6.3. Finally, Section 2.1 presents the development of a taxonomy, along with a systematization, review, and analysis of existing approaches for studying DNNs via geometric and topological methods.

Novelty Each section of this study presents novel results. To the best of our knowledge, this work presents the first systematic investigation of the topological properties of embeddings in CNNs and the geometry of the embedding space in Vision Transformers and foundation models for computer vision tasks. Additionally, this work introduces the study of the geometry of the patch-token manifold embedding and proposes practical applications based on these findings. Furthermore, our research has first introduced an approach to evaluating the ability of language models, in particular modern LLMs, for compositional generalization and symbolic reasoning in tasks involving sequences that encode the structure of elements in finitely generated groups.

Personal Contribution The author's personal contribution to this work lies in conducting the main theoretical and practical presented research. All results were obtained and all theses proposed for defense were developed by the author. This research was conducted under the expert supervision of Anton Ayzenberg.

The main results of this dissertation are published in [149, 150, 148], where the author is the sole or principal co-author. Some aspects of the research presented in the text, and necessary for a complete and coherent narrative, were developed in collaboration with co-authors. These joint efforts enriched the work and ensured its completeness. In the paper [39] on evaluating DNNs for modeling elements in free groups, the author participated in a team under the insightful scientific supervision of Fedor Pavutnitsky, who formulated the problem of generating elements at the intersection of normal closures as follows from Wu's theorem. Within the framework of this direction, the author participated in the development of the methodology and implementation of approaches based on SeqGAN, BERT, GPT-Ignore and modern LLMs for evaluating DNNs in modeling symbolic sequences inspired by homotopy topology. Kirill Brilliantov is the author of the GPT-based methods of controlled generation and symbolic method for dataset collection. Additionally, in addressing the detection of AI-generated content, the author collaborated with a team led by Irina Piontkovskaya and Sergey Barannikov. The author advanced the existing geometric analysis for detecting fake texts and conducted a study of the cross-domain robustness of the existing method for detecting AI-generated text using new data generated by modern LLMs.

These findings were incorporated into paper [130]. In the research on topological em-beddings, published as [18], the author studied the topology of the tonal space and integrated domain-specific embeddings into the DNN training pipeline for the chord prediction.

Theoretical and Practical Significance. The theoretical and methodological significance of the obtained results lies in developing a conceptual and methodological framework that enables the description and analysis of DNNs' internal data representation properties from a geometric and topological perspective during learning and layer-wise evolution. In particular, this work compares various DNN architectures (ViT, CNN, and MLP), representing different design paradigms, based on the geometric and topological properties of their embedding spaces, considering both layer-wise information flow and training dynamics. This is a contribution to the field of Explainable AI and extends the methodological lens of mechanistic interpretability with geometrical tools. When developing the TonnetZNet architecture, it was revealed that the specific music data under consideration lie on a topological space homeomorphic to a thickened torus T2 x I, which formed the basis for introducing a metric on it and developing topological embeddings. The study of how integrating a topological prior into DNN embeddings influences the modeling of musical sequences also holds theoretical significance.

The practical significance stems from applying the theoretical results and insights from empirical observations to address real-world problems. This involves developing applications based on these foundations. This includes the development of a new deep learning method for predicting chords, which is important for applications in music generation. A method for automatic evaluation of neural network performance, including its software implementation, was developed. This is particularly valuable for practical applications involving the training and assessment of neural network generalization ability, especially under data insufficiency conditions. The study of modeling symbolic sequences representing finitely generated groups holds clear practical significance as an initial key step towards creating a software tool. This tool aims to assist mathematicians and specialists in algebraic topology and homological algebra in solving specific computational problems within their domains. Of particular practical significance is the development and software implementation of a method for automatic detection of deepfakes and images generated by various generative models. Given the prevalence of AI-generated content on the internet, this method holds practical significance for its potential to combat disinformation.

Publications and Presentations of the Work

The results of this thesis are published in several publications.

1. «Applying language models to algebraic topology: generating simplicial cycles using multi-labeling in Wu's formula», Proceedings of the 41st International Conference on Machine Learning, 235:4542-4560, (ICML 2024). (A* CORE) [39]

2. Robust AI-Generated Text Detection by Restricted Embeddings, The 2024 Conference on Empirical Methods in Natural Language Processing, pages 17036-17055, Association for Computational Linguistics. (EMNLP 2024) (A* CORE) [130]

3. «Chordal Embeddings Based on Topology of the Tonal Space». In Artificial Intelligence in Music, Sound, Art and Design. EvoMUSART 2023. Lecture Notes in Computer Science, vol 13988. Springer, Cham. (Scopus/WoS) [18]

4. «Deep Neural Networks Architectures from the Perspective of Manifold Learn-ing», 2023 IEEE 6th International Conference on Pattern Recognition and Artificial Intelligence, Haikou, China, 2023, pp. 1021-1031 (Scopus/WoS) [149]

5. «Estimating the Transfer Learning Ability of a Deep Neural Networks by Means of Representations». In: Advances in Neural Computation, Machine Learning, and Cognitive Research VII. NEUROINFORMATICS 2023. Studies in Computational Intelligence, vol 1120. Springer, Cham. (Scopus/WoS) [150]

6. «Study of Foundation Models Knowledge Representations: Geometry Perspective» In: Advances in Neural Computation, Machine Learning, and Cognitive Research VIII. NEUROINFORMATICS 2024. Studies in Computational Intelligence, Springer, vol 1179. Springer, Cham. (Scopus/WoS) [148]

Various parts of the work were presented for report and discussion at the following conferences and seminars.

1. «Assessing the generalizing ability of neural networks», Young Topologists Meeting Conference, University of Copenhagen, Denmark, 2022

2. «Topological and geometry properties of embedding manifold in internal representation of CNNs and Vision Transformers», Conference «Algebraic Topology: Methods, Computation, and Science» 10, Mathematical Institute Oxford University, United Kingdom, 2022

3. «Analysis of representations of deep learning models», Workshop on Topological Data Analysis: Mathematics, Physics and beyond, KIAS (Korea Institute For Advanced Study), Seoul, South Korea, 2023

4. «GPT or Human? Robust Approach to AI-generated Content Detection: Topo-logical perspective, and not only», FCS HSE seminar, 2024

5. «Deep neural networks from the perspective of manifold learning», FCS HSE seminar, 2023

Volume and Structure of the Work The thesis contains an Introduction chapter, which provides context for the work, motivation, and results. The Preliminaries chapter introduces the background and methods used. The Literature Review chapter includes systematization, review, and analysis of existing approaches related to our study. There are four content chapters with the main results and additional context, as well as a Conclusion chapter, which summarizes the results and concludes the study. The full volume of the thesis is 149 pages, 45 figures, and 279 citations.

Заключение диссертации по теме «Другие cпециальности», Магай Герман Игоревич

Заключение

Данная диссертация представляет собой вклад в разработку систематического концептуального и методологического фреймворка описания моделей глубинного обучения, основанного на теории и методах геометрии и топологии. Этот фреймворк обеспечивает исследование различных аспектов области глубокого обучения, в особенности интерпретации моделей глубинного обучения.

Все поставленные цели и задачи исследования выполнены и получены обоснованные теоретически и валидируемые практические результаты. Настоящее междисциплинарное исследование продемонстрировало, что анализ и изучение разнообразных аспектов DNN через призму геометрических и топологических методов позволяют понять внутреннюю логику, выявить ограничения и оценить возможности глубоких моделей, одновременно открывая перспективы для создания новых практических приложений. Далее выделяются следующие группы ключевых теоретических, методологических и практических результатов, составляющих вклад исследования:

• Теоретический и методологический результат заключается в разработке метода геометрического и топологического анализа эмбеддингов нейронных сетей. А его применение выявило ряд наблюдений и закономерностей в динамике представлений данных внутри глубоких моделей (CNN, ViT, Foundation models, MLP). В частности обнаружена устойчивая тенденция к послойному упрощению топологической сложности и уменьшению внутренней размерности (ID). Был разработан методологический фреймворк для интерпретации ViT через геометрический и функциональный анализ представлений ViT на уровнях слоя, головы и токена-патча. Исследование выявило двухфазную динамику ID и блочную структуру в моделях на основе ViT и Foundation models. А также обнаружено влияние способа обучения модели (обучение с учителем, самообучение и контрастное обучение) на послойную динамику геометрических и функциональных свойств представлений. В заключении, была предложена интерпретация полученных наблюдений связывающая особенности послойной динамики с различными этапами извлечения знаний.

• В настоящем исследовании также разработаны новые подходы к анализу производительности DNN. Был предложен метод оценки способности моделей к обобщению на основе геометрических и топологических свойств финальных и промежуточных эмбеддингов, а также разработан метод оценки способности моделей к переносу знаний между доменами на основе структуры схожести представлений на разных слоях и их динамики при дообучении.

• В развитие теоретических достижений диссертация предложила новую математическую модель для топологического описания пространства музыкальных аккордов и геометрии этого пространства через индуцирование на нём метрики. Этот результат непосредственно лёг в основу разработки архитектуры TonnetZNet, интегрирующей топологическое априорное доменно-специфичное знание посредством специально разработанных топологических начальных эмбеддингов. Предложенная архитектура экспериментально проверена на наборе данных полифонической музыки, превосходя по точности бейзлайн.

• Практическая значимость исследования охватывает ряд ключевых областей. В сфере безопасности и защиты ИИ продемонстрирована эффективность использования геометрических свойств (внутренних размерностей) вложений Трансформеров для надежного выявления ИИ-сгенерированного контента, включая синтетические изображения и дипфейки. Разработанные методы показали высокую устойчивость к обобщению в сложных сценариях кросс-доменного анализа и исключения одного домена, решая важную задачу противодействия цифровой дезинформации. Хотя обнаружение ИИ-сгенерированного текста на основе аналогичных принципов оказалось более сложным при работе с продвинутыми ЬЬМ, исследование предоставило ценные выводы о текущих ограничениях и сложностях этой динамично развивающейся области.

• Еще один значимый практический вклад заключается в новаторском применении глубокого обучения к сложным вычислительным задачам вдохновленным алгебраической топологии. Были исследованы возможности различных архитектур ВКК, включая современные ЬЬМ, для моделирования и генерации символьных последовательностей, представляющих элементы в свободных группах (в частности, в контексте формулы Ву и гомотопических групп). Это направление тесно связано с областями символьного ризонинга, композиционного обобщения и релевантно области приложения ИИ в математических рассуждениях.

В целом, диссертация подтвердила перспективность и важность геометрической и топологической перспективы не только как инструмента анализа, но и как основы для разработки подходов к интерпретации, оценке, применению и совершенствованию моделей глубокого обучения. Разнообразные направления исследования, от изучения динамики представлений и разработки новых архитектур до создания практических методов оценки моделей, объединены в целостную концептуальную основу, охватывающую важные аспекты глубокого обучения. Этот результат подчеркивает необходимость более глубокого, математически обоснованного понимания ВКК для их дальнейшего развития и применения.

Сочетая теоретические основы с практическими приложениями, исследование акцентирует значимость топологических и геометрических методов и подходов для анализа и разработки систем, основанных на глубоких нейронных сетях.

Перспективные направления дальнейших исследований включают расширение топологического анализа внутренних представлений на современные ЬЬМ, углубленное изучение геометрических свойств траекторий весов обучения для генеративных моделей с целью оценки их производительности, а также исследование интеграции геометрического априорного знания предметной области в продвинутые архитектуры на основе Трансформеров. Продолжение развития таких фундаментальных подходов будет критически важным для создания более прозрачных, надежных и практически полезных систем ИИ.