arXiv subjects

Paweł Dłotko

Publications and source records attributed to Paweł Dłotko.

At least 19 recordsLinked to original sources

arXiv2026

Frame-invariant topological representations of trabecular bone microarchitecture for strength prediction

Directional topological representations of trabecular bone should retain interpretable structural information without depending on an arbitrary transverse coordinate frame. We develop a frame-invariant directional filtration and compare its strength prediction with signed distance persistent homology and conventional morphometry. Twenty-four human trabecular cores were analyzed using persistence images, directional Betti tensors, and nested ridge regression over 13 validation groups. The directional construction combines cone occupancy, directional covariance, and principal axis degeneracy. Equal bone removal experiments and a pair closely matched in morphometry were used to examine whether topological differences tracked changes in simulated elastic stiffness. The original combined persistence image model had a root mean squared error (RMSE) of 1.952 MPa, compared with 2.001 MPa for morphometry; the paired difference was $-0.049$ MPa with a 95% bootstrap interval of $[-0.491,0.393]$ MPa. Exploratory signed distance persistent homology in dimension zero gave an RMSE of 1.639 MPa. The post hoc frame-invariant directional dimension zero model gave 1.610 MPa, compared with 1.877 MPa for dimension one and 1.630 MPa for a harmonized signed distance dimension zero model. The paired RMSE difference between the invariant and signed distance models was $-0.021$ MPa with a 95% interval of $[-0.234,0.210]$ MPa. Localized removal reduced stiffness more than diffuse removal in 19 of 24 cores despite producing smaller $H_0$ persistence image changes. Frame invariance removes the dependence of directional topology on an arbitrary transverse coordinate frame. Connected component representations warrant external evaluation for strength prediction, while the mechanical experiments limit their interpretation as scalar stiffness surrogates.

q-bio.QM↗

arXiv2026

Topological Characteristics for the Analysis of Vector Fields. New stable characteristics for discrete and continuous dynamical systems

The qualitative analysis of nonlinear dynamical systems relies on identifying geometric and topological structures that govern phase-space organization. We develop a computational framework based on novel topological and geometric descriptors of vector fields that enables robust comparison of dynamical regimes from both analytical models and sampled data. The framework comprises the Begin-End Point Embedding (BEPE), Density of Directions (DOD) and Euler characteristic-based summaries (Euler Characteristic Curve (ECC) and Euler Characteristic Profile (ECP)). BEPE captures local variations in vector field, while DoD characterizes the global organization of vector directions. ECCs and ECPs provide filtration-based topological summaries of vector-field structure, with ECPs extending these summaries to multiparameter settings. Together, these descriptors encode complementary geometric and topological information about flow organization. We formulate both continuous and sampled versions of the descriptors and establish their robustness with respect to perturbations, finite sampling and selected transformations. The effectiveness of the proposed approach is demonstrated on a collection of benchmark systems (the Hopf bifurcation, the FitzHugh-Nagumo model, the Lorenz system). Experiments on the discrete Hénon map demonstrate the applicability of the framework to displacement fields generated by discrete dynamical systems, while experiments on high-dimensional turbulent flows assess its scalability beyond low-dimensional phase spaces. Comparisons with classical approaches, including Conley-index-based methods and $L_p$-type metrics, show that the proposed descriptors provide complementary structural information while offering favorable robustness, interpretability and computational efficiency, establishing a practical connection between dynamical systems theory and topological data analysis.

math.DS↗

arXiv2026

Topological cell-openness index for porous materials

We propose a method of estimating and parametrising the proportion of open and closed cells in a porous material based on measuring Betti numbers on the structures. We define a cell-openness index τ which can be used to complement the proportion of open-celled volume reported by gas pycnometry, which is the current gold standard for pore type characterization. We discuss in what types of structures mismatches between the two measures can occur and how such mismatches convey additional information about the structure. We demonstrate examples of significant correlations between τ and measurable physical quantities in both numerical and experimental structures. We also discuss how Betti curves can be used to estimate characteristic feature sizes in porous structures.

cond-mat.soft↗

arXiv2026

Direction-aware topological descriptors for elastic stiffness tensor prediction in porous materials

Classical topological descriptors used in topological data analysis (TDA) are invariant under permutations of spatial axes and therefore cannot represent the loading direction, which is essential for modeling anisotropic mechanical response. Here, this limitation is addressed by introducing a direction-aware TDA framework in which the loading axis is explicitly embedded into filtration functions used to compute both persistent homology and Euler characteristic profile descriptors. We apply this framework to predict the full elastic stiffness tensor of porous microstructures using non-directional as well as direction-aware descriptors of the structures as well as convolutional neural networks trained directly on the voxelized structure. We show that the performance of all those are comparable on the diagonal uniaxial, Poisson, and shear components. However for the twelve off-diagonal, normal shear coupling components - which govern elastic anisotropy and are the hardest to predict - only direction-aware topology retains meaningful predictive power, with all baselines, non-directional descriptors and the CNN collapsing to near-chance accuracy. When used as inputs to gradient-boosted tree models, the proposed descriptors match or exceed the accuracy of the convolutional neural network specifically on these hardest-to-predict coupling terms, despite relying on a compact, physically interpretable representation that is orders of magnitude smaller than the raw voxel grid. Overall, the results establish direction-aware TDA as a practical route for linking porous microstructure to the full anisotropic elastic response, capturing coupling terms that conventional descriptors and end-to-end deep learning models fail to resolve.

physics.comp-ph↗

arXiv2026

Transferable 3D Convolutional Neural Networks for Elastic Constants Prediction in Nanoporous Metals

The topology of nanoporous metals is crucial for determining their mechanical response. In this work, we generated 6,000 gold and 422 silver nanoporous structures and calculated three components of elastic modulus with Molecular Dynamics simulations, resulting in 19,263 data points. This study compared two distinct approaches of predicting elastic modulus: a Fully-Connected neural network trained on precomputed topological descriptors, and several 3D Convolutional neural network architectures adapted from computer vision. The 3D CNNs outperformed the descriptor-based baseline model ($R^2 = 0.704$), with to-performing DenseNet-201 architecture achieving $R^2 = 0.955$. Additionally, the effects of training grid resolution, dataset size, and descriptor integration into a model were investigated. We further demonstrated model robustness through Transfer learning: a pretrained model was fine-tuned on a much smaller dataset of denser gold structures and the dataset of denser silver structures. Finally, the trained model was employed to evaluate the mechanical properties of 100,000 stochastic nanoporous gold structures and identify the Pareto optimal designs.

cond-mat.mtrl-sci↗

arXiv2026

Physics-informed convolutional neural networks for fluid flow through porous media

Accurate simulation of fluid flow in porous media is challenging due to complex pore-space geometries and the computational cost of solving the Navier-Stokes equations. This difficulty is particularly important when repeated simulations are required, as standard numerical solvers may converge slowly in intricate porous domains. We present a neural-network-based framework for predicting pore-scale velocity fields directly from sample geometry. The method uses a convolutional encoder-decoder architecture with skip connections to preserve spatial detail while extracting multi-scale features. Physical consistency is encouraged through a custom loss function combining velocity reconstruction with incompressibility, no-flow conditions inside solids, periodicity constraints, and agreement with the global tortuosity index. We analyze the influence of the corresponding loss weights and quantify the contribution of individual loss components to prediction accuracy. Several CNN backbones are evaluated to identify architectures providing accurate and robust predictions. The generalization ability of the trained model is tested on samples outside the training distribution, including changes in obstacle geometry, boundary conditions, porosity, and realistic porous structures. Finally, we demonstrate a practical use of the predicted velocity fields as initial conditions for Lattice-Boltzmann simulations. This warm-start strategy accelerates solver convergence, reducing the number of iterations in over 90% of tested cases.

cs.LG↗

arXiv2025

A Survey of Dimension Estimation Methods

It is a standard assumption that datasets in high dimension have an internal structure which means that they in fact lie on, or near, subsets of a lower dimension. In many instances it is important to understand the real dimension of the data, hence the complexity of the dataset at hand. A great variety of dimension estimators have been developed to find the intrinsic dimension of the data but there is little guidance on how to reliably use these estimators. This survey reviews a wide range of dimension estimation methods, categorising them by the geometric information they exploit: tangential estimators which detect a local affine structure; parametric estimators which rely on dimension-dependent probability distributions; and estimators which use topological or metric invariants. The paper evaluates the performance of these methods, as well as investigating varying responses to curvature and noise. Key issues addressed include robustness to hyperparameter selection, sample size requirements, accuracy in high dimensions, precision, and performance on non-linear geometries. In identifying the best hyperparameters for benchmark datasets, overfitting is frequent, indicating that many estimators may not generalise well beyond the datasets on which they have been tested.

stat.ML↗

arXiv2025

Consonance in music -- the Pythagorean approach revisited

The Pythagorean school attributed consonance in music to simplicity of frequency ratios between musical tones. In the last two centuries, the consonance curves developed by Helmholtz, Plompt and Levelt shifted focus to psycho-acoustic considerations in perceiving consonances. The appearance of peaks of these curves at the ratios considered by the Pythagorean school, and which were a consequence of an attempt to understand the world by nice mathematical proportions, remained a curiosity. This paper addresses this curiosity, by describing a mathematical model of musical sound, along with a mathematical definition of consonance. First, we define pure, complex and mixed tones as mathematical models of musical sound. By a sequence of numerical experiments and analytic calculations, we show that continuous cosine similarity, abbreviated as cosim, applied to these models quantifies the elusive concept of consonance as a frequency ratio which gives a local maximum of the cosim function. We prove that these maxima occur at the ratios considered as consonant in classical music theory. Moreover, we provide a simple explanation why the number of musical intervals considered as consonant by musicians is finite, but has been increasing over the centuries. Specifically, our formulas show that the number of consonant intervals changes with the depth of the tone (the number of harmonics present).

math.HO↗

arXiv2024

ClusterGraph: a new tool for visualization and compression of multidimensional data

Understanding the global organization of complicated and high dimensional data is of primary interest for many branches of applied sciences. It is typically achieved by applying dimensionality reduction techniques mapping the considered data into lower dimensional space. This family of methods, while preserving local structures and features, often misses the global structure of the dataset. Clustering techniques are another class of methods operating on the data in the ambient space. They group together points that are similar according to a fixed similarity criteria, however unlike dimensionality reduction techniques, they do not provide information about the global organization of the data. Leveraging ideas from Topological Data Analysis, in this paper we provide an additional layer on the output of any clustering algorithm. Such data structure, ClusterGraph, provides information about the global layout of clusters, obtained from the considered clustering algorithm. Appropriate measures are provided to assess the quality and usefulness of the obtained representation. Subsequently the ClusterGraph, possibly with an appropriate structure--preserving simplification, can be visualized and used in synergy with state of the art exploratory data analysis techniques.

cs.CG↗

arXiv2024

Testing topological conjugacy of time series

This paper considers a problem of testing, from a finite sample, a topological conjugacy of two dynamical systems $(X,f)$ and $(Y,g)$. More precisely, given $x_1,\ldots, x_n \subset X$ and $y_1,\ldots,y_n \subset Y$ such that $x_{i+1} = f(x_i)$ and $y_{i+1} = g(y_i)$ as well as $h: X \rightarrow Y$, we deliver a number of tests to check if $f$ and $g$ are topologically conjugated via $h$. The values of the tests are close to zero for conjugated systems and large for systems that are not conjugated. Convergence of the test values, in case when sample size goes to infinity, is established. A number of numerical examples indicating scalability and robustness of the methods are given. In addition, we show how the presented method specialize to a test of sufficient embedding dimension in Takens' embedding theorem. Our methods also apply to the situation when we are given two observables of deterministic processes, of a form of one or higher dimensional time-series. In this case, their similarity can be accessed by comparing the dynamics of their Takens' reconstructions.

math.DS↗

arXiv2024

Combinatorial Topological Models for Phylogenetic Networks and the Mergegram Invariant

Mutations of genetic sequences are often accompanied by their recombinations, known as phylogenetic networks. These networks are typically reconstructed from coalescent processes that may arise from optimal merging or fitting together a given set of phylogenetic trees. Nakhleh formulated the phylogenetic network reconstruction problem (PNRP): Given a family of phylogenetic trees over a common set of taxa, is there a unique minimal phylogenetic network whose set of spanning trees contains the family? Inspired by ideas from topological data analysis (TDA), we devise lattice-diagram models for phylogenetic networks and filtrations, the cliquegram and the facegram, both generalizing the dendrogram (filtered partition) model of phylogenetic trees. Both models allow us to solve the PNRP rigorously. The solutions are obtained by taking the join of the dendrograms on the lattice of cliquegrams or facegrams. Furthermore, computing the join-facegram is polynomial in the size and number of the input trees. Cliquegrams and facegrams can be challenging to work with when the number of taxa is large. We propose a topological invariant of facegrams and filtrations, called the mergegram, by extending a construction by Elkin and Kurlin defined on dendrograms. We show that the mergegram is invariant of weak equivalences of filtrations which, in turn, implies that it is a 1-Lipschitz stable invariant with respect to Mémoli's tripod distance. The mergegram, can be used as a computable proxy for phylogenetic networks and filtrations of datasets. We illustrate the utility of these new TDA-concepts to phylogenetics, by performing experiments with artificial and biological data.

math.AT↗

arXiv2023

Topology-Driven Goodness-of-Fit Tests in Arbitrary Dimensions

This paper adopts a tool from computational topology, the Euler characteristic curve (ECC) of a sample, to perform one- and two-sample goodness of fit tests. We call our procedure TopoTests. The presented tests work for samples of arbitrary dimension, having comparable power to the state-of-the-art tests in the one-dimensional case. It is demonstrated that the type I error of TopoTests can be controlled and their type II error vanishes exponentially with increasing sample size. Extensive numerical simulations of TopoTests are conducted to demonstrate their power for samples of various sizes.

stat.ME↗

arXiv2023

Euler Characteristic Curves and Profiles: a stable shape invariant for big data problems

Tools of Topological Data Analysis provide stable summaries encapsulating the shape of the considered data. Persistent homology, the most standard and well studied data summary, suffers a number of limitations; its computations are hard to distribute, it is hard to generalize to multifiltrations and is computationally prohibitive for big data-sets. In this paper we study the concept of Euler Characteristics Curves, for one parameter filtrations and Euler Characteristic Profiles, for multi-parameter filtrations. While being a weaker invariant in one dimension, we show that Euler Characteristic based approaches do not possess some handicaps of persistent homology; we show efficient algorithms to compute them in a distributed way, their generalization to multifiltrations and practical applicability for big data problems. In addition we show that the Euler Curves and Profiles enjoys certain type of stability which makes them robust tool in data analysis. Lastly, to show their practical applicability, multiple use-cases are considered.

math.AT↗

arXiv2023

Mapper-type algorithms for complex data and relations

Mapper and Ball Mapper are Topological Data Analysis tools used for exploring high dimensional point clouds and visualizing scalar-valued functions on those point clouds. Inspired by open questions in knot theory, new features are added to Ball Mapper that enable encoding of the structure, internal relations and symmetries of the point cloud. Moreover, the strengths of Mapper and Ball Mapper constructions are combined to create a tool for comparing high dimensional data descriptors of a single dataset. This new hybrid algorithm, Mapper on Ball Mapper, is applicable to high dimensional lens functions. As a proof of concept we include applications to knot and game theory, as well as material science and cancer research.

math.AT↗

arXiv2022

Bottleneck Profiles and Discrete Prokhorov Metrics for Persistence Diagrams

In topological data analysis (TDA), persistence diagrams have been a succesful tool. To compare them, Wasserstein and Bottleneck distances are commonly used. We address the shortcomings of these metrics and show a way to investigate them in a systematic way by introducing bottleneck profiles. This leads to a notion of discrete Prokhorov metrics for persistence diagrams as a generalization of the Bottleneck distance. They satisfy a stability result and bounds with respect to Wasserstein metrics. We provide algorithms to compute the newly introduced quantities and end with an discussion about experiments.

cs.CG↗

arXiv2020

Hotspot identification for Mapper graphs

Mapper algorithm can be used to build graph-based representations of high-dimensional data capturing structurally interesting features such as loops, flares or clusters. The graph can be further annotated with additional colouring of vertices allowing location of regions of special interest. For instance, in many applications, such as precision medicine, Mapper graph has been used to identify unknown compactly localized subareas within the dataset demonstrating unique or unusual behaviours. This task, performed so far by a researcher, can be automatized using hotspot analysis. In this work we propose a new algorithm for detecting hotspots in Mapper graphs. It allows automatizing of the hotspot detection process. We demonstrate the performance of the algorithm on a number of artificial and real world datasets. We further demonstrate how our algorithm can be used for the automatic selection of the Mapper lens functions.

math.NA↗

arXiv2019

Persistence Bag-of-Words for Topological Data Analysis

Persistent homology (PH) is a rigorous mathematical theory that provides a robust descriptor of data in the form of persistence diagrams (PDs). PDs exhibit, however, complex structure and are difficult to integrate in today's machine learning workflows. This paper introduces persistence bag-of-words: a novel and stable vectorized representation of PDs that enables the seamless integration with machine learning. Comprehensive experiments show that the new representation achieves state-of-the-art performance and beyond in much less time than alternative approaches.

stat.ML↗

arXiv2019

Cyclicality, Periodicity and the Topology of Time Series

Periodic and semi periodic patterns are very common in nature. In this paper we introduce a topological toolbox aiming in detecting and quantifying periodicity. The presented technique is of a general nature and may be employed wherever there is suspected cyclic behaviour in a time series with no trend. The approach is tested on a number of real-world examples enabling us to consistently demonstrate an ability to recognise periodic behaviour where conventional techniques fail to do so. Quicker to react to changes in time series behaviour, and with a high robustness to noise, the toolbox offers a powerful way to deeper understanding of time series dynamics.

math.AT↗