- Titel
- Data-driven Complexity
- Zitierfähige Url:
- https://nbn-resolving.org/urn:nbn:de:bsz:14-qucosa2-1063924
- Erstveröffentlichung
- 2026
- Datum der Einreichung
- 28.10.2025
- Datum der Verteidigung
- 16.06.2026
- Abstract (EN)
- This thesis develops a data-driven complexity measure that quantifies structural non-linearity directly from observed data. The construction rests on a simple comparison: a linear encoding is reduced to an encoding dimension $d_e$, and the data are then reconstructed twice, once with a linear decoder and once with a non-linear decoder. Because both decoders operate from an identical encoding, the residual gap in reconstruction loss isolates precisely those correlations that linear methods cannot exploit. This gap defines the raw complexity $\Delta_\mathcal{L}$ and its bounded counterpart, the normalized complexity $\tilde{\Delta}_\mathcal{L}$. The measure addresses a persistent shortcoming of traditional indicators such as Lyapunov exponents and differential entropy, which increase monotonically with disorder and therefore cannot separate deterministic structure from noise. The framework is developed from the foundations of dimensionality reduction and of statistical measurement, and is then validated on a controlled family of two-dimensional synthetic patterns. These datasets establish the interpretation of the measure: linear relationships yield zero complexity regardless of correlation strength, symmetric non-linear patterns yield maximum complexity despite vanishing correlations, and unstructured data yield near-zero values. Complexity and statistical dependence are shown to be independent properties. The measure is then applied to dynamical systems of increasing richness. In the logistic map, complexity exhibits step-function behaviour at each bifurcation and remains zero whenever the number of periodic points falls below $d_e - 1$, so that $d_e$ acts as a resolution scale for the period-doubling cascade. Within the chaotic regime the measure resolves structural transitions that Lyapunov exponents and differential entropy leave undetected, and it falls to zero once the attractor merges into a single structureless branch. This last observation carries the central conceptual result of the thesis: chaos and structural non-linearity vary independently. Coupled logistic maps extend the analysis to coexisting dynamical regimes, while the simple and double pendula demonstrate scale-dependent, non-monotonic complexity profiles characteristic of continuous systems. The final part transfers the framework to learned representations. Feedforward networks trained on MNIST show progressive complexity refinement with depth, with complexity signatures anticipating overfitting well before conventional metrics respond, while the QWEN-3 transformer exhibits a non-monotonic trajectory of compression followed by selective re-expansion across its blocks. Taken together, the results establish that complexity is relative, to the observation scale, to the encoding dimension, and to the baseline method against which structure is measured, and that this relativity is a feature of the measurement rather than a limitation of it.
- Freie Schlagwörter (EN)
- complexity, autoencoders, dynamical systems, machine learning, language model
- Klassifikation (DDC)
- 530
- Klassifikation (RVK)
- SK 810
- GutachterIn
- Prof. Dr. Jan-Michael Rost
- Prof. Dr. Antonio Politi
- Den akademischen Grad verleihende / prüfende Institution
- Technische Universität Dresden, Dresden
- Sonstige beteiligte Institution
- Max Planck Institute for the Physics of Complex Systems, Dresden
- Version / Begutachtungsstatus
- publizierte Version / Verlagsversion
- URN Qucosa
- urn:nbn:de:bsz:14-qucosa2-1063924
- Veröffentlichungsdatum Qucosa
- 12.08.2026
- Dokumenttyp
- Dissertation
- Sprache des Dokumentes
- Englisch
- Lizenz / Rechtehinweis
CC BY-NC-ND 4.0- Inhaltsverzeichnis
Abstract i Acknowledgements ii I Introduction 1 II Methodology 6 II.1 Dimensionality Reduction 6 II.1.1 PCA 7 II.1.2 Autoencoders 8 II.2 Statistical measurements 11 II.2.1 Entropy 11 II.2.2 Lyapunov exponent 12 II.2.3 Correlation 14 II.3 Complexity Measure 17 II.3.1 Theoretical Foundation 17 II.3.2 Complexity Metrics 19 II.3.3 Implementation Methodology 21 II.3.4 Analysis of Autoencoder Structure 25 IIIResults29 III.1 Interpretation in Low Dimension 29 III.1.1 Design Principles and Expected behaviour 29 III.1.2 Synthetic Data Generation 30 III.1.3 Results and Analysis 33 III.1.4 Interpretation 39 III.1.5 Conclusion 41 III.2 Dynamical systems 43 III.2.1 Logistic Maps 44 III.2.2 Pendula 94 III.3 Machine Learning Complexity 111 III.3.1 Neural Network Architecture and Layer-wise Transformations 111 III.3.2 Classification: MNIST Digit Recognition 112 III.3.3 Large Language Models: QWEN-3 Embedding 117 III.3.4 Conclusion 120 IV Synopsis123 V Outlook128