Too Many or Too Few? Sampling Bounds for Topological Descriptors

Network Intrusion Detection Systems (NIDSs) are an increasingly important tool for the prevention and mitigation of cyber attacks. Over the past years, a lot of research efforts have aimed at leveraging the increasingly powerful models of Machine Learning (ML) for this purpose. A number of labelled synthetic datasets have been generated and made publicly available by researchers, and they have become the benchmarks via which new ML-based NIDS classifiers are being evaluated. Recently published results show excellent classification performance with these datasets, increasingly approaching 100 percent performance across key evaluation metrics such as Accuracy, F1 score, AUC, etc. Unfortunately, we have not yet seen these excellent academic research results translated into practical NIDS systems with such near-perfect performance. This motivated our research presented in this paper, where we analyse the statistical properties of the benign traffic in three of the more recent and relevant NIDS datasets, (CIC_IDS, UNSW_NB15, TON_IOT), by converting them into a common flow format. As a comparison, we consider two datasets obtained from real-world production networks, one from a university network and one from a medium size Internet Service Provider (ISP). Our results show that the two real-world datasets are quite similar among themselves in regards to most of the considered statistical features. Equally, the three synthetic datasets are also relatively similar within their group. However, and most importantly, our results show a distinct difference of most of the considered statistical features between the three synthetic datasets and the two real-world datasets. Since ML relies on the basic assumption of training and test datasets being sampled from the same distribution, this raises the question of how well the performance results of ML-classifiers trained on the considered synthetic datasets can translate and generalise to real-world networks. We believe this is an interesting and relevant question which provides motivation for further research in this space.

The microstructure of porous electrodes of solid oxide cells (SOCs) significantly impacts their electrochemical performance. Therefore, various attempts have been made to analyze their complex porous microstructure in three dimensions. The obtained microstructures have been quantitatively evaluated using structural metrics, such as volume fraction, surface area density, and tortuosity factor of the constituent phases, as well as double- and triple-phase boundaries. These structural characteristics are useful to correlate the electrode performance with their microstructure, thereby determining optimal electrode design. However, these intuitive metrics often fails to explain the electrode performance. In fact, structural changes during long term operation of SOCs are subtle so that the performance degradation of the electrodes cannot be fully detected by these intuitive characteristics. Therefore, it is necessary to find a way to extract hidden structural metrics that characterize the electrode microstructure and to clarify their correlation with electrode performance.One of the promising candidates for such structural metrics is topological information. Topology is a structural property that is invariant to successive deformation operations, such as connected structures and the number of holes in the structures. Topology-based structural analysis has many applications in material science, where macroscopic properties of materials are correlated with topological information about their internal structures (e.g., crystal structures). For example, Wang et al. [1] performed a topological analysis on the crystal structure of double-phase steel to characterize its structure and to predict its macroscopic properties, such as stress-strain curves. Application of topological data analysis is also found in the analysis of SOFC electrodes; Pawlowski et al. [2] attempted to capture the structural changes in SOFC anodes during long-term operation from a topological perspective.In addition, topological analysis is regarded as a tool for dimensional reduction of complex structural information. Therefore, the information extracted from the topological analysis can be used to evaluate conventional metrics of porous microstructures, such as surface area and triple-phase boundary. If these metrics can be accurately evaluated from a limited number of information from the topological analysis, the numerical cost is expected to be significantly reduced.Therefore, this study investigates the applicability of the topological analysis to the electrode microstructure analysis. Fig. 1 shows the schematic diagram of the structural analysis in this study. First, persistent homology analysis [3] is employed to extract the topological information of the electrodes, where the birth and death of topological features are detected during the filtration process within the structure datasets to produce the persistent diagram (PD). The persistent diagram is then discretized and concatenated to obtain the persistent image (PI). Subsequently, principal component analysis (PCA) is conducted to further reduce the dimensionality of the structural information extracted in the persistent image. The obtained values of the principal components are correlated with the conventional structural metrics of the porous electrodes, such as volume fraction, surface area density, and triple-phase boundary density.Moreover, neural networks (NNs) consisting of fully-connected layers are constructed to quantify the structural metrics from the principal components. The constructed neural networks are trained using the real electrode microstructure datasets obtained using the focused ion beam and scanning electron microstructure [4]. To validate the developed neural networks, they are compared with the convolutional neural network (CNN), which directly quantifies the microstructural metrics from the three-dimensional structures, in terms of quantification accuracy, the number of training datasets, and required training time.The values obtained from the topological analysis followed by the principal component analysis are found to contain essential structural information in the porous electrodes. This is implied by the fact that the principal component values obtained from the real structure datasets are significantly different from those from the artificial sphere-packing structures, even though the conventional structural metrics, i.e., volume fraction and surface density, are identical between the real and artificial structures. In addition, the principal component values have sensitivity to the conventional statistical metrics, because the structures with different solid compositions form clusters in different locations in a principal component space. These suggest that the topological information will be useful not only to detect subtle structural changes in the porous electrodes undetectable in conventional structural metrics, but also to reduce the dimensionality of the complex porous electrodes without losing the information quality.Since the information extracted using the persistent homology and principal component analysis possess essential information about the electrode structures, conventional microstructural metrics are accurately quantified by the constructed neural network. It should be emphasized that the number of parameters in the neural network that need to be adjusted during the training process is significantly reduced compared with the convolutional neural network. As a result, the number of required datasets for the training of the network is significantly reduced. These clearly indicate the effectiveness of the dimensional reduction in the quantification of the electrode microstructures.

Scalar field comparison is a fundamental task in scientific visualization. In topological data analysis, we compare topological descriptors of scalar fields-such as persistence diagrams and merge trees-because they provide succinct and robust abstract representations. Several similarity measures for topological descriptors seem to be both asymptotically and practically efficient with polynomial time algorithms, but they do not scale well when handling large-scale, time-varying scientific data and ensembles. In this paper, we propose a new framework to facilitate the comparative analysis of merge trees, inspired by tools from locality sensitive hashing (LSH). LSH hashes similar objects into the same hash buckets with high probability. We propose two new similarity measures for merge trees that can be computed via LSH, using new extensions to Recursive MinHash and subpath signature, respectively. Our similarity measures are extremely efficient to compute and closely resemble the results of existing measures such as merge tree edit distance or geometric interleaving distance. Our experiments demonstrate the utility of our LSH framework in applications such as shape matching, clustering, key event detection, and ensemble summarization.

The Vietoris–Rips filtration is a versatile tool in topological data analysis. It is a sequence of simplicial complexes built on a metric space to add topological structure to an otherwise disconnected set of points. It is widely used because it encodes useful information about the topology of the underlying metric space. This information is of-ten extracted from its so-called persistence diagram. Unfortunately, this filtration is often too large to construct in full. We show how to construct an O(n)-size filtered simplicial complex on an n-point metric space such that its persistence diagram is a good approximation to that of the Vietoris–Rips filtration. This new filtration can be constructed in O(n log n) time. The constant factors in both the size and the run-ning time depend only on the doubling dimension of the metric space and the desired tightness of the approximation. For the first time, this makes it computationally tractable to approximate the persistence di-agram of the Vietoris–Rips filtration across all scales for large data sets. We describe two different sparse filtrations. The first is a zigzag filtration that removes points as the scale increases. The second is a (non-zigzag) filtration that yields the same persistence diagram. Both methods are based on a hierarchical net-tree and yield the same guar-antees.

In this paper we develop an alternative topological data analysis (TDA) approach for studying graph representations of time series of dynamical systems. Specifically, we show how persistent homology, a tool from TDA, can be used to yield a compressed, multi-scale representation of the graph that can distinguish between dynamic states such as periodic and chaotic behavior. We show the approach for two graph constructions obtained from the time series. In the first approach the time series is embedded into a point cloud which is then used to construct an undirected k-nearest-neighbor graph. The second construct relies on the recently developed ordinal partition framework. In either case, a pairwise distance matrix is then calculated using the shortest path between the graph's nodes, and this matrix is utilized to define a filtration of a simplicial complex that enables tracking the changes in homology classes over the course of the filtration. These changes are summarized in a persistence diagram's two-dimensional summary of changes in the topological features. We then extract existing as well as new geometric and entropy point summaries from the persistence diagram and compare to other commonly used network characteristics. Our results show that persistence-based point summaries yield a clearer distinction of the dynamic behavior and are more robust to noise than existing graph-based scores, especially when combined with ordinal graphs.

Abstract. The storage of spatial data that consists of spatial and non-spatial properties requires a database management system that possesses spatial functions that can cater to the spatial characteristics of data. These characteristics include the geometrical shape, topological and positional information. Parallel to how geometries describe the shape of an object, topological information is also an important spatial property which describes how the geometries in a space are related to each other. This information describes the connectivity, containment and adjacencies of spatial objects which are the foundation for more complex analysis such as navigation, data reconstruction, spatial queries and others. However, the topological support provided by spatial databases varies. This paper provided an overview on the current implementations of topological support in spatial databases such as ArcGIS, QGIS, PostgreSQL and others. The native topology in most spatial databases was found to be 2D topology maintained by 2D topology rules with limited representation of 3D topological relationships. Consequently, 3D objects represented by 2D topology had to be decomposed into objects of lower dimensions. Approaches to implement additional topological support for spatial databases included the use of topological data models, data structures, operators, and rules. 3D applications such as 3D cadastre required more detailed representations of topological information which required a more comprehensive 3D topological data model. Nonetheless, comprehensive preservation of topological information also mandates voluminous storage and higher computational efficiency. Thus, the appropriate 3D topological support should be provided in spatial databases to accurately represent 3D objects and meet 3D analysis requirements.

Persistent homology (PH) is a robust method to compute multi-dimensional geometric and topological features of a dataset. Because these features are often stable under certain perturbations of the underlying data, are often discriminating, and can be used for visualization of structure in high-dimensional data and in statistical and machine learning modeling, PH has attracted the interest of researchers across scientific disciplines and in many industry applications. However, computational costs may present challenges to effectively using PH in certain data contexts, and theoretical stability results may not hold in practice. In this paper, we define, implement, and investigate a simplicial complex construction for computing persistent homology of Euclidean point cloud data, which we call the Delaunay-Rips complex (DR). By only considering simplices that appear in the Delaunay triangulation of the point cloud and assigning the Vietoris-Rips weights to simplices, DR avoids potentially costly computations in the persistence calculations. We document and compare a Python implementation of DR with other simplicial complex constructions for generating persistence diagrams. By imposing sufficient conditions on point cloud data, we are able to theoretically justify the stability of the persistence diagrams produced using DR. When the Delaunay triangulation of the point cloud changes under perturbations of the points, we prove that DR-produced persistence diagrams exhibit instability. Since we cannot guarantee that real-world data will satisfy our stability conditions, we demonstrate the practical robustness of DR for persistent homology in comparison with other simplicial complexes in machine learning applications. We find in our experiments that using DR in an ML-TDA pipeline performs comparatively well as using other simplicial complex constructions.

Label noise and class imbalance are common challenges encountered in real-world datasets. Existing approaches for robust learning often focus on addressing either label noise or class imbalance individually, resulting in suboptimal performance when both biases are present. To bridge this gap, this work introduces a novel meta-learning-based dynamic loss that adapts the objective functions during the training process to effectively learn a classifier from long-tailed noisy data. Specifically, our dynamic loss consists of two components: a label corrector and a margin generator. The label corrector is responsible for correcting noisy labels, while the margin generator generates per-class classification margins by capturing the underlying data distribution and the learning state of the classifier. In addition, we employ a hierarchical sampling strategy that enriches a small amount of unbiased metadata with diverse and challenging samples. This enables the joint optimization of the two components in the dynamic loss through meta-learning, allowing the classifier to effectively adapt to clean and balanced test data. Extensive experiments conducted on multiple real-world and synthetic datasets with various types of data biases, including CIFAR-10/100, Animal-10N, ImageNet-LT, and Webvision, demonstrate that our method achieves state-of-the-art accuracy. The code for our approach will soon be made publicly available.

The manifold surface reconstruction in multi-view stereo often fails in retaining thin structures due to incomplete and noisy reconstructed point clouds. In this paper, we address this problem by leveraging spatial curves. The curve representation in nature is advantageous in modeling thin and elongated structures, implying topology and connectivity information of the underlying geometry, which exactly compensates the weakness of scattered point clouds. We present a novel surface reconstruction method using both curves and point clouds. First, we propose a 3D curve reconstruction algorithm based on the initialize-optimize-extend strategy. Then, tetrahedra are constructed from points and curves, where the volumes of thin structures are robustly preserved by the Curve-conformed Delaunay Refinement. Finally, the mesh surface is extracted from tetrahedra by a graph optimization. The method has been intensively evaluated on both synthetic and real-world datasets, showing significant improvements over state-of-the-art methods.

Summarizing topological information from datasets and maps defined on them is a central theme in topological data analysis. Mapper, a tool for such summarization, takes as input both a possibly high dimensional dataset and a map defined on the data, and produces a summary of the data by using a cover of the codomain of the map. This cover, via a pullback operation to the domain, produces a simplicial complex connecting the data points.The resulting view of the data through a cover of the codomain offers flexibility in analyzing the data. However, it offers only a view at a fixed scale at which the cover is constructed. Inspired by the concept, we explore a notion of a tower of covers which induces a tower of simplicial complexes connected by simplicial maps, which we call multiscale mapper. We study the resulting structure, and design practical algorithms to compute its persistence diagrams efficiently. Specifically, when the domain is a simplicial complex and the map is a real-valued piecewise-linear function, the algorithm can compute the exact persistence diagram only from the 1-skeleton of the input complex. For general maps, we present a combinatorial version of the algorithm that acts only on vertex sets connected by the 1-skeleton graph, and this algorithm approximates the exact persistence diagram thanks to a stability result that we show to hold.

Phishing remains a persistent cybersecurity threat, often bypassing traditional detection methods due to evolving attack techniques. This study presents a Reinforcement Learning (RL)-based phishing detection framework, leveraging a Deep Q-Network (DQN) to enhance detection accuracy, reduce false positives, and improve classification performance. The model was trained and evaluated using a real-world dataset comprising 5000 emails (2500 phishing and 2500 benign) and externally validated against a synthetic phishing dataset of 1000 samples simulating unseen attacks. It achieved a 95% accuracy, 96% precision, 94% recall, and a 2% false positive rate on the real-world dataset and a 93% accuracy, 94% precision, and a 4% false positive rate on the synthetic dataset. Area Under the Curve (AUC) analysis yielded a score of 0.92, confirming excellent classification separability and alignment with the model’s high accuracy and low false positive rate. This work contributes to scalable, real-world phishing defense by addressing the limitations of static detection systems and improving detection reliability.

Given a large amount of data and appropriate hyperparameters, deep learning techniques can deliver impressive performance if several challenging issues with training, such as vanishing gradients, can be overcome. Often, deep learning training techniques produce suboptimal results because the parameter search space is large and populated with many less-than-ideal solutions. Automatic hyperparameter tuning algorithms, known as autotuners, offer an attractive alternative for automating the training process, though they can be computationally expensive. Additionally, autotuners democratize state-of-the-art machine learning approaches and increase the accessibility of deep learning technology to different scientific communities and novice users. In this paper, we investigate the efficacy of autotuning using Keras Tuner on both synthetic and real-world datasets. We show that autotuning performed well on synthetic datasets but was inadequate on real data. As we increase model complexity, autotuning produces errors that are tedious to resolve for those with limited experience in machine learning. Avoiding overfitting, for example, requires extensive knowledge of an algorithm's unique characteristics (e.g., adding dropout layers). Autotuning tools are excellent for creating baseline models on new datasets, but they need more attention to formulate optimal solutions for end-users with less background in deep learning. Because of this, manual tuning based on domain knowledge and experience is still preferred in machine learning because it produces better performance, even though it requires extensive machine learning expertise.

Problem statement: The processing of the XML queries needs an efficient indexing method. The index generation includes the labeling of the nodes. But for the dynamic data, when the changes are made, the re-computation of the labeling is needed. Approach: Based on the structural indexing used the performance of the query processing system will be affected. The dynamic XML document allows inserting, deleting and updating operations. Results: Suppose for the frequently changed documents, the indexes will also be changed. i.e., it requires the re-computing the labels frequently. This leads to an inconvenient system. To avoid this problem, in the scheme New Labeling Scheme for XML (NLSX), only the small, capital letters and digits are used to generate persistent labels for the nodes in the document. In the proposed system New Labeling Scheme for XML using Unicode Characters, characters from Unicode Characters (NLSXU) are used. Thus it can provide more combinations of the characters for persistent labeling the nodes in the document so that it will very much reduce the space needed to store the labels. Using the proposed scheme NLSXU, the index size of the real world data sets will be greatly reduced by 81% of the existing scheme NLSX. The results shows that the proposed scheme NLSXU will reduce the size of the indexes of the synthetic data sets up to 26, 34, 71 and 95% than the NLSX, LSDX, GRP and SP schemes respectively. Also when compared to LSDX scheme, the NLSXU will reduce the time taken for generating the labels by 96 and 80% for the real world datasets and the synthetic data sets respectively. Conclusion: Finally when compared to NLSX scheme, the NLSXU will reduce the time taken for generating the labels by 66 and 15% for the real world datasets and the synthetic data sets respectively. Thus it will improve the performance of the query system.

Autonomous mobile robots like self-flying drones and industrial robots heavily depend on depth images to perform tasks such as 3D reconstruction and visual SLAM. However, the presence of inaccuracies in these depth images can greatly hinder the effectiveness of these applications, resulting in sub-optimal results. Depth images produced by commercially available cameras frequently exhibit noise, which manifests as flickering pixels and erroneous patches. Machine Learning (ML)-based methods to rectify these images are unsuitable for edge devices that have very limited computational resources. Non-ML methods are much faster but have limited accuracy, especially for correcting errors that are a result of occlusion and camera movement. We propose a scheme called VoxDepth that is fast, accurate, and runs very well on edge devices such as the NVIDIA Jetson Nano board. It relies on a host of novel techniques: 3D point cloud construction and fusion, and using it to create a 2D template to fix erroneous depth images. VoxDepth shows superior results on both synthetic and real-world datasets. We specifically demonstrate a 31% improvement in quality as compared with state-of-the-art methods on real-world depth datasets, while maintaining a competitive frame rate of 27 FPS (frames per second).

Point cloud segmentation is a fundamental task in 3D. Despite recent progress on point cloud segmentation with the power of deep networks, current learning methods based on the clean label assumptions may fail with noisy labels. Yet, class labels are often mislabeled at both instance-level and boundary-level in real-world datasets. In this work, we take the lead in solving the instance-level label noise by proposing a Point Noise-Adaptive Learning (PNAL) framework. Compared to noise-robust methods on image tasks, our framework is noise-rate blind, to cope with the spatially variant noise rate specific to point clouds. Specifically, we propose a point-wise confidence selection to obtain reliable labels from the historical predictions of each point. A cluster-wise label correction is proposed with a voting strategy to generate the best possible label by considering the neighbor correlations. To handle boundary-level label noise, we also propose a variant "PNAL-boundary" with a progressive boundary label cleaning strategy. Extensive experiments demonstrate its effectiveness on both synthetic and real-world noisy datasets. Even with 60% symmetric noise and high-level boundary noise, our framework significantly outperforms its baselines, and is comparable to the upper bound trained on completely clean data. Moreover, we cleaned the popular real-world dataset ScanNetV2 for rigorous experiment. Our code and data is available at https://github.com/pleaseconnectwifi/PNAL.