🍩 Database of Original & Non-Theoretical Uses of Topology

(found 12 matches in 0.001841s)

  1. Towards a Philological Metric Through a Topological Data Analysis Approach (2020)

    Eduardo Paluzo-Hidalgo, Rocio Gonzalez-Diaz, Miguel A. Gutiérrez-Naranjo
    Abstract The canon of the baroque Spanish literature has been thoroughly studied with philological techniques. The major representatives of the poetry of this epoch are Francisco de Quevedo and Luis de Góngora y Argote. They are commonly classified by the literary experts in two different streams: Quevedo belongs to the Conceptismo and G\ńgora to the Culteranismo. Besides, traditionally, even if Quevedo is considered the most representative of the Conceptismo, Lope de Vega is also considered to be, at least, closely related to this literary trend. In this paper, we use Topological Data Analysis techniques to provide a first approach to a metric distance between the literary style of these poets. As a consequence, we reach results that are under the literary experts' criteria, locating the literary style of Lope de Vega, closer to the one of Quevedo than to the one of G\'ǵora.

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  2. Using Persistent Homology to Reveal Hidden Information in Neural Data (2015)

    Gard Spreemann, Benjamin Dunn, Magnus Bakke Botnan, Nils A. Baas
    Abstract We propose a method, based on persistent homology, to uncover topological properties of a priori unknown covariates of neuron activity. Our input data consist of spike train measurements of a set of neurons of interest, a candidate list of the known stimuli that govern neuron activity, and the corresponding state of the animal throughout the experiment performed. Using a generalized linear model for neuron activity and simple assumptions on the effects of the external stimuli, we infer away any contribution to the observed spike trains by the candidate stimuli. Persistent homology then reveals useful information about any further, unknown, covariates.

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  3. Unsupervised Topological Learning for Identification of Atomic Structures (2022)

    Sébastien Becker, Emilie Devijver, Rémi Molinier, Noël Jakse
    Abstract We propose an unsupervised learning methodology with descriptors based on topological data analysis (TDA) concepts to describe the local structural properties of materials at the atomic scale. Based only on atomic positions and without a priori knowledge, our method allows for an autonomous identification of clusters of atomic structures through a Gaussian mixture model. We apply successfully this approach to the analysis of elemental Zr in the crystalline and liquid states as well as homogeneous nucleation events under deep undercooling conditions. This opens the way to deeper and autonomous study of complex phenomena in materials at the atomic scale.

  4. The Shape of Word Embeddings: Quantifying Non-Isometry With Topological Data Analysis (2024)

    Ondřej Draganov, Steven Skiena
    Abstract Word embeddings represent language vocabularies as clouds of d-dimensional points. We investigate how information is conveyed by the general shape of these clouds, instead of representing the semantic meaning of each token. Specifically, we use the notion of persistent homology from topological data analysis (TDA) to measure the distances between language pairs from the shape of their unlabeled embeddings. These distances quantify the degree of non-isometry of the embeddings. To distinguish whether these differences are random training errors or capture real information about the languages, we use the computed distance matrices to construct language phylogenetic trees over 81 Indo-European languages. Careful evaluation shows that our reconstructed trees exhibit strong and statistically-significant similarities to the reference.

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  5. Using Zigzag Persistent Homology to Detect Hopf Bifurcations in Dynamical Systems (2020)

    Sarah Tymochko, Elizabeth Munch, Firas A. Khasawneh
    Abstract Bifurcations in dynamical systems characterize qualitative changes in the system behavior. Therefore, their detection is important because they can signal the transition from normal system operation to imminent failure. While standard persistent homology has been used in this setting, it usually requires analyzing a collection of persistence diagrams, which in turn drives up the computational cost considerably. Using zigzag persistence, we can capture topological changes in the state space of the dynamical system in only one persistence diagram. Here we present Bifurcations using ZigZag (BuZZ), a one-step method to study and detect bifurcations using zigzag persistence. The BuZZ method is successfully able to detect this type of behavior in two synthetic examples as well as an example dynamical system.

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  6. Exploring the Geometry and Topology of Neural Network Loss Landscapes (2022)

    Stefan Horoi, Jessie Huang, Bastian Rieck, Guillaume Lajoie, Guy Wolf, Smita Krishnaswamy
    Abstract Recent work has established clear links between the generalization performance of trained neural networks and the geometry of their loss landscape near the local minima to which they converge. This suggests that qualitative and quantitative examination of the loss landscape geometry could yield insights about neural network generalization performance during training. To this end, researchers have proposed visualizing the loss landscape through the use of simple dimensionality reduction techniques. However, such visualization methods have been limited by their linear nature and only capture features in one or two dimensions, thus restricting sampling of the loss landscape to lines or planes. Here, we expand and improve upon these in three ways. First, we present a novel “jump and retrain” procedure for sampling relevant portions of the loss landscape. We show that the resulting sampled data holds more meaningful information about the network’s ability to generalize. Next, we show that non-linear dimensionality reduction of the jump and retrain trajectories via PHATE, a trajectory and manifold-preserving method, allows us to visualize differences between networks that are generalizing well vs poorly. Finally, we combine PHATE trajectories with a computational homology characterization to quantify trajectory differences.

  7. Using Persistent Homology and Dynamical Distances to Analyze Protein Binding (2016)

    Violeta Kovacev-Nikolic, Peter Bubenik, Dragan Nikolić, Giseon Heo
    Abstract Persistent homology captures the evolution of topological features of a model as a parameter changes. The most commonly used summary statistics of persistent homology are the barcode and the persistence diagram. Another summary statistic, the persistence landscape, was recently introduced by Bubenik. It is a functional summary, so it is easy to calculate sample means and variances, and it is straightforward to construct various test statistics. Implementing a permutation test we detect conformational changes between closed and open forms of the maltose-binding protein, a large biomolecule consisting of 370 amino acid residues. Furthermore, persistence landscapes can be applied to machine learning methods. A hyperplane from a support vector machine shows the clear separation between the closed and open proteins conformations. Moreover, because our approach captures dynamical properties of the protein our results may help in identifying residues susceptible to ligand binding; we show that the majority of active site residues and allosteric pathway residues are located in the vicinity of the most persistent loop in the corresponding filtered Vietoris-Rips complex. This finding was not observed in the classical anisotropic network model.

  8. Detecting Bifurcations in Dynamical Systems With CROCKER Plots (2022)

    İsmail Güzel, Elizabeth Munch, Firas A. Khasawneh
    Abstract Existing tools for bifurcation detection from signals of dynamical systems typically are either limited to a special class of systems or they require carefully chosen input parameters and a significant expertise to interpret the results. Therefore, we describe an alternative method based on persistent homology—a tool from topological data analysis—that utilizes Betti numbers and CROCKER plots. Betti numbers are topological invariants of topological spaces, while the CROCKER plot is a coarsened but easy to visualize data representation of a one-parameter varying family of persistence barcodes. The specific bifurcations we investigate are transitions from periodic to chaotic behavior or vice versa in a one-parameter collection of differential equations. We validate our methods using numerical experiments on ten dynamical systems and contrast the results with existing tools that use the maximum Lyapunov exponent. We further prove the relationship between the Wasserstein distance to the empty diagram and the norm of the Betti vector, which shows that an even more simplified version of the information has the potential to provide insight into the bifurcation parameter. The results show that our approach reveals more information about the shape of the periodic attractor than standard tools, and it has more favorable computational time in comparison with the Rösenstein algorithm for computing the maximum Lyapunov exponent.

  9. Using Persistent Homology as Preprocessing of Early Warning Signals for Critical Transition in Flood (2021)

    Syed Mohamad Sadiq Syed Musa, Mohd Salmi Md Noorani, Fatimah Abdul Razak, Munira Ismail, Mohd Almie Alias, Saiful Izzuan Hussain
    Abstract Flood early warning systems (FLEWSs) contribute remarkably to reducing economic and life losses during a flood. The theory of critical slowing down (CSD) has been successfully used as a generic indicator of early warning signals in various fields. A new tool called persistent homology (PH) was recently introduced for data analysis. PH employs a qualitative approach to assess a data set and provide new information on the topological features of the data set. In the present paper, we propose the use of PH as a preprocessing step to achieve a FLEWS through CSD. We test our proposal on water level data of the Kelantan River, which tends to flood nearly every year. The results suggest that the new information obtained by PH exhibits CSD and, therefore, can be used as a signal for a FLEWS. Further analysis of the signal, we manage to establish an early warning signal for ten of the twelve flood events recorded in the river; the two other events are detected on the first day of the flood. Finally, we compare our results with those of a FLEWS constructed directly from water level data and find that FLEWS via PH creates fewer false alarms than the conventional technique.

  10. Quantifying Genetic Innovation: Mathematical Foundations for the Topological Study of Reticulate Evolution (2020)

    Michael Lesnick, Raúl Rabadán, Daniel I. S. Rosenbloom
    Abstract A topological approach to the study of genetic recombination, based on persistent homology, was introduced by Chan, Carlsson, and Rabadán in 2013. This associates a sequence of signatures called barcodes to genomic data sampled from an evolutionary history. In this paper, we develop theoretical foundations for this approach. First, we present a novel formulation of the underlying inference problem. Specifically, we introduce and study the novelty profile, a simple, stable statistic of an evolutionary history which not only counts recombination events but also quantifies how recombination creates genetic diversity. We propose that the (hitherto implicit) goal of the topological approach to recombination is the estimation of novelty profiles. We then study the problem of obtaining a lower bound on the novelty profile using barcodes. We focus on a low-recombination regime, where the evolutionary history can be described by a directed acyclic graph called a galled tree, which differs from a tree only by isolated topological defects. We show that in this regime, under a complete sampling assumption, the \$1\textasciicircum\mathrm\st\\$ barcode yields a lower bound on the novelty profile, and hence on the number of recombination events. For \$i\textgreater1\$, the \$i\textasciicircum\\mathrm\th\\\$ barcode is empty. In addition, we use a stability principle to strengthen these results to ones which hold for any subsample of an arbitrary evolutionary history. To establish these results, we describe the topology of the Vietoris--Rips filtrations arising from evolutionary histories indexed by galled trees. As a step towards a probabilistic theory, we also show that for a random history indexed by a fixed galled tree and satisfying biologically reasonable conditions, the intervals of the \$1\textasciicircum\\mathrm\st\\\$ barcode are independent random variables. Using simulations, we explore the sensitivity of these intervals to recombination.

  11. Weighted Persistent Homology for Biomolecular Data Analysis (2020)

    Zhenyu Meng, D. Vijay Anand, Yunpeng Lu, Jie Wu, Kelin Xia
    Abstract In this paper, we systematically review weighted persistent homology (WPH) models and their applications in biomolecular data analysis. Essentially, the weight value, which reflects physical, chemical and biological properties, can be assigned to vertices (atom centers), edges (bonds), or higher order simplexes (cluster of atoms), depending on the biomolecular structure, function, and dynamics properties. Further, we propose the first localized weighted persistent homology (LWPH). Inspired by the great success of element specific persistent homology (ESPH), we do not treat biomolecules as an inseparable system like all previous weighted models, instead we decompose them into a series of local domains, which may be overlapped with each other. The general persistent homology or weighted persistent homology analysis is then applied on each of these local domains. In this way, functional properties, that are embedded in local structures, can be revealed. Our model has been applied to systematically study DNA structures. It has been found that our LWPH based features can be used to successfully discriminate the A-, B-, and Z-types of DNA. More importantly, our LWPH based principal component analysis (PCA) model can identify two configurational states of DNA structures in ion liquid environment, which can be revealed only by the complicated helical coordinate system. The great consistence with the helical-coordinate model demonstrates that our model captures local structure variations so well that it is comparable with geometric models. Moreover, geometric measurements are usually defined in local regions. For instance, the helical-coordinate system is limited to one or two basepairs. However, our LWPH can quantitatively characterize structure information in regions or domains with arbitrary sizes and shapes, where traditional geometrical measurements fail.

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  12. Weighted Persistent Homology for Osmolyte Molecular Aggregation and Hydrogen-Bonding Network Analysis (2020)

    D. Vijay Anand, Zhenyu Meng, Kelin Xia, Yuguang Mu
    Abstract It has long been observed that trimethylamine N-oxide (TMAO) and urea demonstrate dramatically different properties in a protein folding process. Even with the enormous theoretical and experimental research work on these two osmolytes, various aspects of their underlying mechanisms still remain largely elusive. In this paper, we propose to use the weighted persistent homology to systematically study the osmolytes molecular aggregation and their hydrogen-bonding network from a local topological perspective. We consider two weighted models, i.e., localized persistent homology (LPH) and interactive persistent homology (IPH). Boltzmann persistent entropy (BPE) is proposed to quantitatively characterize the topological features from LPH and IPH, together with persistent Betti number (PBN). More specifically, from the localized persistent homology models, we have found that TMAO and urea have very different local topology. TMAO is found to exhibit a local network structure. With the concentration increase, the circle elements in these networks show a clear increase in their total numbers and a decrease in their relative sizes. In contrast, urea shows two types of local topological patterns, i.e., local clusters around 6 Å and a few global circle elements at around 12 Å. From the interactive persistent homology models, it has been found that our persistent radial distribution function (PRDF) from the global-scale IPH has same physical properties as the traditional radial distribution function. Moreover, PRDFs from the local-scale IPH can also be generated and used to characterize the local interaction information. Other than the clear difference of the first peak value of PRDFs at filtration size 4 Å, TMAO and urea also shows very different behaviors at the second peak region from filtration size 5 Å to 10 Å. These differences are also reflected in the PBNs and BPEs of the local-scale IPH. These localized topological information has never been revealed before. Since graphs can be transferred into simplicial complexes by the clique complex, our weighted persistent homology models can be used in the analysis of various networks and graphs from any molecular structures and aggregation systems.

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