Low-dimensional Euclidean Space Research Articles

Metric multidimensional scaling for large single-cell datasets using neural networks

Metric multidimensional scaling is one of the classical methods for embedding data into low-dimensional Euclidean space. It creates the low-dimensional embedding by approximately preserving the pairwise distances between the input points. However, current state-of-the-art approaches only scale to a few thousand data points. For larger data sets such as those occurring in single-cell RNA sequencing experiments, the running time becomes prohibitively large and thus alternative methods such as PCA are widely used instead. Here, we propose a simple neural network-based approach for solving the metric multidimensional scaling problem that is orders of magnitude faster than previous state-of-the-art approaches, and hence scales to data sets with up to a few million cells. At the same time, it provides a non-linear mapping between high- and low-dimensional space that can place previously unseen cells in the same embedding.

Towards Metric DBSCAN: Exact, Approximate, and Streaming Algorithms

DBSCAN is a popular density-based clustering algorithm that has many different applications in practice. However, the running time of DBSCAN in high-dimensional space or general metric space (\em e.g., clustering a set of texts by using edit distance) can be as large as quadratic in the input size. Moreover, most of existing accelerating techniques for DBSCAN are only available for low-dimensional Euclidean space. In this paper, we study the DBSCAN problem under the assumption that the inliers (the core points and border points) have a low intrinsic dimension (which is a realistic assumption for many high-dimensional applications), where the outliers can locate anywhere in the space without any assumption. First, we propose a k-center clustering based algorithm that can reduce the time-consuming labeling and merging tasks of DBSCAN to be linear. Further, we propose a linear time approximate DBSCAN algorithm, where the key idea is building a novel small-size summary for the core points. Also, our algorithm can be efficiently implemented for streaming data and the required memory is independent of the input size. Finally, we conduct our experiments and compare our algorithms with several popular DBSCAN algorithms. The experimental results suggest that our proposed approach can significantly reduce the computational complexity in practice.

Using Large Language Models to Understand Suicidality in a Social Media-Based Taxonomy of Mental Health Disorders: Linguistic Analysis of Reddit Posts.

Rates of suicide have increased by over 35% since 1999. Despite concerted efforts, our ability to predict, explain, or treat suicide risk has not significantly improved over the past 50 years. The aim of this study was to use large language models to understand natural language use during public web-based discussions (on Reddit) around topics related to suicidality. We used large language model-based sentence embedding to extract the latent linguistic dimensions of user postings derived from several mental health-related subreddits, with a focus on suicidality. We then applied dimensionality reduction to these sentence embeddings, allowing them to be summarized and visualized in a lower-dimensional Euclidean space for further downstream analyses. We analyzed 2.9 million posts extracted from 30 subreddits, including r/SuicideWatch, between October 1 and December 31, 2022, and the same period in 2010. Our results showed that, in line with existing theories of suicide, posters in the suicidality community (r/SuicideWatch) predominantly wrote about feelings of disconnection, burdensomeness, hopeless, desperation, resignation, and trauma. Further, we identified distinct latent linguistic dimensions (well-being, seeking support, and severity of distress) among all mental health subreddits, and many of the resulting subreddit clusters were in line with a statistically driven diagnostic classification system-namely, the Hierarchical Taxonomy of Psychopathology (HiTOP)-by mapping onto the proposed superspectra. Overall, our findings provide data-driven support for several language-based theories of suicide, as well as dimensional classification systems for mental health disorders. Ultimately, this novel combination of natural language processing techniques can assist researchers in gaining deeper insights about emotions and experiences shared on the web and may aid in the validation and refutation of different mental health theories.

Schematics Retrieval Using Whole-Graph Embedding Similarity

This paper addresses the pressing environmental concern of plastic waste, particularly in the biopharmaceutical production sector, where single-use assemblies (SUAs) significantly contribute to this issue. To address and mitigate this problem, we propose a unique approach centered around the standardization and optimization of SUA drawings through digitization and structured representation. Leveraging the non-Euclidean properties of SUA drawings, we employ a graph-based representation, utilizing graph convolutional networks (GCNs) to capture complex structural relationships. Introducing a novel weakly supervised method for the similarity-based retrieval of SUA graph networks, we optimize graph embeddings in a low-dimensional Euclidean space. Our method demonstrates effectiveness in retrieving similar graphs that share the same functionality, offering a promising solution to reduce plastic waste in pharmaceutical assembly processes.

Metric and fault-tolerant metric dimension for GeSbTe superlattice chemical structure.

The concept of metric dimension has many applications, including optimizing sensor placement in networks and identifying influential persons in social networks, which aids in effective resource allocation and focused interventions; finding the source of a spread in an arrangement; canonically labeling graphs; and inserting typical information in low-dimensional Euclidean spaces. In a graph G, the set S⊆V(G) of minimum vertices from which all other verticescan be uniquely determined by the distances to the vertices in S is called the resolving set. The cardinality of the resolving set is called the metric dimension. The set S is called fault-tolerant resolving set if S\{v} is still a resolving set of G. The minimum cardinality of such a set S is called fault-tolerant metric dimension of G. GeSbTe super lattice is the latest chemical compound to have electronic material that is capable of non-volatile storing phase change memories with minimum energy usage. In this work, we calculate the resolving set (fault tolerant resolving set) to find the metric dimension(fault-tolerant metric dimension) for the molecular structure of the GeSbTe lattice. The results may be useful in comparing network structure and categorizing the structure of the GeSbTe lattice.

Analyzing microbial evolution through gene and genome phylogenies.

Microbiome scientists critically need modern tools to explore and analyze microbial evolution. Often this involves studying the evolution of microbial genomes as a whole. However, different genes in a single genome can be subject to different evolutionary pressures, which can result in distinct gene-level evolutionary histories. To address this challenge, we propose to treat estimated gene-level phylogenies as data objects, and present an interactive method for the analysis of a collection of gene phylogenies. We use a local linear approximation of phylogenetic tree space to visualize estimated gene trees as points in low-dimensional Euclidean space, and address important practical limitations of existing related approaches, allowing an intuitive visualization of complex data objects. We demonstrate the utility of our proposed approach through microbial data analyses, including by identifying outlying gene histories in strains of Prevotella, and by contrasting Streptococcus phylogenies estimated using different gene sets. Our method is available as an open-source R package, and assists with estimating, visualizing, and interacting with a collection of bacterial gene phylogenies.

Fast Vehicle Routing via Knowledge Transfer in a Reproducing Kernel Hilbert Space

Vehicle routing problems (VRPs) are essential in logistics. In the literature, many exact and heuristic optimization algorithms have been proposed to solve the VRPs. These traditional approaches, however, generally start the optimization from scratch and ignore the experiences of solving related VRPs, which may lead to unnecessary computational costs in searching repeated problems and reduce the efficiency of vehicle routing. Recently, transfer optimization (TO) has been presented to speed up vehicle routing by reusing the knowledge learned from similarly solved VRPs. However, existing TO methods build connections across VRPs in a low-dimensional Euclidean space, which has limited modeling ability in the cases of having nonlinear correlations. Keeping this in mind, this article presents a study of TO equipped with the kernel method for fast vehicle routing. In contrast to existing TO methods, in this work, the learning of connections across VRPs for knowledge transfer is conducted in a reproducing kernel Hilbert space (RKHS), which thus has greater modeling capacity in nonlinear customer relationships between VPRs. To evaluate the performance of the proposed method, comprehensive empirical studies have been conducted using well-known VRP benchmarks, against existing state-of-the-art TO methods for vehicle routing. Finally, a well-known real-world VRP application given by a routing company (Jingdong), namely, the package delivery problem (PDP), is investigated to further assess the efficacy of our proposed method.

Greedy Spanners in Euclidean Spaces Admit Sublinear Separators

The greedy spanner in a low dimensional Euclidean space is a fundamental geometric construction that has been extensively studied over three decades as it possesses the two most basic properties of a good spanner: constant maximum degree and constant lightness. Recently, Eppstein and Khodabandeh [28] showed that the greedy spanner in \(\mathbb {R}^2 \) admits a sublinear separator in a strong sense: any subgraph of k vertices of the greedy spanner in \(\mathbb {R}^2 \) has a separator of size \(O(\sqrt {k}) \) . Their technique is inherently planar and is not extensible to higher dimensions. They left showing the existence of a small separator for the greedy spanner in \(\mathbb {R}^d \) for any constant d ≥ 3 as an open problem. In this paper, we resolve the problem of Eppstein and Khodabandeh [28] by showing that any subgraph of k vertices of the greedy spanner in \(\mathbb {R}^d \) has a separator of size O ( k 1 − 1/ d ). We introduce a new technique that gives a simple criterion for any geometric graph to have a sublinear separator that we dub τ -lanky : a geometric graph is τ -lanky if any ball of radius r cuts at most τ edges of length at least r in the graph. We show that any τ -lanky geometric graph of n vertices in \(\mathbb {R}^d \) has a separator of size O ( τn 1 − 1/ d ). We then derive our main result by showing that the greedy spanner is O (1)-lanky. We indeed obtain a more general result that applies to unit ball graphs and point sets of low fractal dimensions in \(\mathbb {R}^d \) . Our technique naturally extends to doubling metrics. We use the τ -lanky criterion to show that there exists a (1 + ϵ)-spanner for doubling metrics of dimension d with a constant maximum degree and a separator of size \(O(n^{1-\frac{1}{d}}) \) ; this result resolves an open problem posed by Abam and Har-Peled [1] a decade ago. We then introduce another simple criterion for a graph in doubling metrics of dimension d to have a sublinear separator. We use the new criterion to show that the greedy spanner of an n -point metric space of doubling dimension d has a separator of size \(O((n^{1-\frac{1}{d}}) + \log \Delta) \) where Δ is the spread of the metric; the factor log ( Δ ) is tightly connected to the fact that, unlike its Euclidean counterpart, the greedy spanner in doubling metrics has unbounded maximum degree . Finally, we discuss algorithmic implications of our results.

A unified approach based on multidimensional scaling for calibration estimation in survey sampling with qualitative auxiliary information.

Survey calibration is a widely used method to estimate the population mean or total score of a target variable, particularly in medical research. In this procedure, auxiliary information related to the variable of interest is used to recalibrate the estimation weights. However, when the auxiliary information includes qualitative variables, traditional calibration techniques may be not feasible or the optimisation procedure may fail. In this article, we propose the use of linear calibration in conjunction with a multidimensional scaling-based set of continuous, uncorrelated auxiliary variables along with a suitable metric in a distance-based regression framework. The calibration weights are estimated using a projection of the auxiliary information on a low-dimensional Euclidean space. The approach becomes one of the linear calibration with quantitative variables avoiding the usual computational problems in the presence of qualitative auxiliary information. The new variables preserve the underlying assumption in linear calibration of a linear relationship between the auxiliary and target variables, and therefore the optimal properties of the linear calibration method remain true. The behaviour of this approach is examined using a Monte Carlo procedure and its value is illustrated by analysing real data sets and by comparing its performance with that of traditional calibration procedures.

Hyperbolic matrix factorization improves prediction of drug-target associations

Past research in computational systems biology has focused more on the development and applications of advanced statistical and numerical optimization techniques and much less on understanding the geometry of the biological space. By representing biological entities as points in a low dimensional Euclidean space, state-of-the-art methods for drug-target interaction (DTI) prediction implicitly assume the flat geometry of the biological space. In contrast, recent theoretical studies suggest that biological systems exhibit tree-like topology with a high degree of clustering. As a consequence, embedding a biological system in a flat space leads to distortion of distances between biological objects. Here, we present a novel matrix factorization methodology for drug-target interaction prediction that uses hyperbolic space as the latent biological space. When benchmarked against classical, Euclidean methods, hyperbolic matrix factorization exhibits superior accuracy while lowering embedding dimension by an order of magnitude. We see this as additional evidence that the hyperbolic geometry underpins large biological networks.

Generative hypergraph models and spectral embedding

Many complex systems involve interactions between more than two agents. Hypergraphs capture these higher-order interactions through hyperedges that may link more than two nodes. We consider the problem of embedding a hypergraph into low-dimensional Euclidean space so that most interactions are short-range. This embedding is relevant to many follow-on tasks, such as node reordering, clustering, and visualization. We focus on two spectral embedding algorithms customized to hypergraphs which recover linear and periodic structures respectively. In the periodic case, nodes are positioned on the unit circle. We show that the two spectral hypergraph embedding algorithms are associated with a new class of generative hypergraph models. These models generate hyperedges according to node positions in the embedded space and encourage short-range connections. They allow us to quantify the relative presence of periodic and linear structures in the data through maximum likelihood. They also improve the interpretability of node embedding and provide a metric for hyperedge prediction. We demonstrate the hypergraph embedding and follow-on tasks—including quantifying relative strength of structures, clustering and hyperedge prediction—on synthetic and real-world hypergraphs. We find that the hypergraph approach can outperform clustering algorithms that use only dyadic edges. We also compare several triadic edge prediction methods on high school and primary school contact hypergraphs where our algorithm improves upon benchmark methods when the amount of training data is limited.

Geometric learning for computational mechanics Part II: Graph embedding for interpretable multiscale plasticity

The history-dependent behaviors of classical plasticity models are often driven by internal variables evolved according to phenomenological laws. The difficulty to interpret how these internal variables represent a history of deformation, the lack of direct measurement of these internal variables for calibration and validation, and the weak physical underpinning of those phenomenological laws have long been criticized as barriers to creating realistic models. In this work, geometric machine learning on graph data (e.g. finite element solutions) is used as a means to establish a connection between nonlinear dimensional reduction techniques and plasticity models. Geometric learning-based encoding on graphs allows the embedding of rich time-history data onto a low-dimensional Euclidean space such that the evolution of plastic deformation can be predicted in the embedded feature space. A corresponding decoder can then convert these low-dimensional internal variables back into a weighted graph such that the dominating topological features of plastic deformation can be observed and analyzed.

Relative Persistent Homology

The alpha complex efficiently computes persistent homology of a point cloud X in Euclidean space when the dimension d is low. Given a subset A of X, relative Čech persistent homology can be computed as the persistent homology of the relative Čech complex {check{mathrm{C}}}(X, A). However, this is not computationally feasible for larger point clouds X. The aim of this note is to present a method for efficient computation of relative Čech persistent homology in low dimensional Euclidean space. We introduce the relative Delaunay–Čech complex {text {Del}check{mathrm{C}}}(X, A) whose homology is the relative Čech persistent homology. It is constructed from the Delaunay complex of an embedding of X in (d+1)-dimensional Euclidean space.

Approximation of discontinuous inverse operators with neural networks

In this work we deal with parametric inverse problems, which consist in recovering a finite number of parameters describing the structure of an unknown object, from indirect measurements. State-of-the-art methods for approximating a regularizing inverse operator by using a dataset of input–output pairs of the forward model rely on deep learning techniques. In these approaches, a neural network (NN) is trained to predict the value of the sought parameters directly from the data. In this paper, we show that these methods provide suboptimal results when a regularizing inverse operator is discontinuous with respect to the Euclidean topology. Hence, we propose a two-step strategy for approximating it by means of a NN, which works under general topological conditions. First, we embed the parameters into a subspace of a low-dimensional Euclidean space; second, we use a NN to approximate a homeomorphism between the subspace and the image of the parameter space through the forward operator. The parameters are then retrieved by applying the inverse of the embedding to the network predictions. The results are shown for the problem of x-ray imaging of solar flares with data from the Spectrometer/Telescope for Imaging X-rays. In this case, the parameter space is homeomorphic to a Moebius strip. Our simulation studies show that the use of a NN for predicting the parameters directly from the data yields systematic errors due to the non-Euclidean topology of the parameter space. The proposed strategy overcomes the discontinuity issues and furnishes stable and accurate reconstructions.

Dimensionality and Coordination in Voting: The Distortion of STV

We study the performance of voting mechanisms from a utilitarian standpoint, under the recently introduced framework of metric-distortion, offering new insights along two main lines. First, if d represents the doubling dimension of the metric space, we show that the distortion of STV is O(d log log m), where m represents the number of candidates. For doubling metrics this implies an exponential improvement over the lower bound for general metrics, and as a special case it effectively answers a question left open by Skowron and Elkind (AAAI '17) regarding the distortion of STV under low-dimensional Euclidean spaces. More broadly, this constitutes the first nexus between the performance of any voting rule and the ``intrinsic dimensionality'' of the underlying metric space. We also establish a nearly-matching lower bound, refining the construction of Skowron and Elkind. Moreover, motivated by the efficiency of STV, we investigate whether natural learning rules can lead to low-distortion outcomes. Specifically, we introduce simple, deterministic and decentralized exploration/exploitation dynamics, and we show that they converge to a candidate with O(1) distortion.

Investigating How Reproducibility and Geometrical Representation in UMAP Dimensionality Reduction Impact the Stratification of Breast Cancer Tumors

Advances in next-generation sequencing have provided high-dimensional RNA-seq datasets, allowing the stratification of some tumor patients based on their transcriptomic profiles. Machine learning methods have been used to reduce and cluster high-dimensional data. Recently, uniform manifold approximation and projection (UMAP) was applied to project genomic datasets in low-dimensional Euclidean latent space. Here, we evaluated how different representations of the UMAP embedding can impact the analysis of breast cancer (BC) stratification. We projected BC RNA-seq data on Euclidean, spherical, and hyperbolic spaces, and stratified BC patients via clustering algorithms. We also proposed a pipeline to yield more reproducible clustering outputs. The results show how the selection of the latent space can affect downstream stratification results and suggest that the exploration of different geometrical representations is recommended to explore data structure and samples’ relationships.

A latent space model for multilayer network data

A Bayesian statistical model to simultaneously characterize two or more social networks defined over a common set of actors is proposed. The key feature of the model is a hierarchical prior distribution that allows the user to represent the entire system jointly, achieving a compromise between dependent and independent networks. Among others things, such a specification provides an easy way to visualize multilayer network data in a low-dimensional Euclidean space, generate a weighted network that reflects the consensus affinity between actors, establish a measure of correlation between networks, assess cognitive judgments that subjects form about the relationships among actors, and perform clustering tasks at different social instances. The model's capabilities are illustrated using real-world and synthetic datasets, taking into account different types of actors, sizes, and relations.

Meticulous process monitoring with multiscale convolutional feature extraction

Due to the interaction of process variables, process data is in essence graph-structured with non-Euclidean nature. Hence, learning the graph representation in a low-dimensional Euclidean space will be helpful for gaining the true insights underlying the industrial process. In this study, a meticulous process monitoring method (PM-MCF) is proposed based on multiscale convolutional feature extraction. For the proposed method, the interactions between different process variables are identified using causality inference. According to the obtained graph structure, convolutional filters are specifically designed for each process variable. In this way, the local correlation within directly related variables and their corresponding dynamic information can be effectively extracted. Besides, with the increasing of convolutional layers, more variables can be involved through the interaction relationship to explore a larger reception field. Based on the obtained feature matrices, sub-models are developed to calculate the monitoring statistics and their corresponding control limits. Finally, the decisions of all the sub-models are integrated to identify the operation status of the process system. It is noted that the proposed PM-MCF can be readily generalized to other existing methods by replacing the selected filter and the developed sub-models. The monitoring performance of the proposed method is illustrated using process data collected from a thermal power plant. Experimental results show that the proposed method can accurately detect the process anomalies using the extracted causal relationship.

An abnormal event detection method based on the Riemannian manifold and LSTM network

In this paper, we propose an abnormal event detection method based on the combination of Riemannian manifold and LSTM network. We divide the video into a series of non overlapping regions, and extract HOG and HOF as original appearance and motion features. Because the existing abnormal event detection methods pay little attention to the changes rate of appearance and motion, we propose a new feature called Riemannian manifold distance feature that can represent the topological relationship of features at different moment in the same position and capture the change rate of the current feature. Once the change rate of the current features changes, the topological relationship of features will change. We use ISOMAP algorithm to embed manifold into low dimensional Euclidean space. In the training phase, the dimension-reduced features of the same position at different historical moment are used as the input of each time step of the LSTM network, and the dimension-reduced feature of the current moment is used as expected output. And in this stage, we only use the normal video, so the LSTM will learn the changing rules of the features contained in normal events. In the testing phase, we use LSTM network to predict current dimension-reduced HOG and HOF features. After that, we can obtain the Riemannian manifold distance feature of actual dimension-reduced feature and predicted dimension-reduced feature. In order to detect abnormal events, we propose an abnormal score, which is used to measure whether the appearance/ movement and the change rate of them are close to the expectation. The abnormal score can detect the abnormal event according to dimension-reduced appearance feature, dimension-reduced motion feature, appearance Riemannian manifold distance feature and motion Riemannian manifold distance feature. Experiments on four benchmark datasets show the competitive performance of our method with the state-of-the-art methods.

A deep learning approach to predict blood-brain barrier permeability.

The blood–brain barrier plays a crucial role in regulating the passage of 98% of the compounds that enter the central nervous system (CNS). Compounds with high permeability must be identified to enable the synthesis of brain medications for the treatment of various brain diseases, such as Parkinson’s, Alzheimer’s, and brain tumors. Throughout the years, several models have been developed to solve this problem and have achieved acceptable accuracy scores in predicting compounds that penetrate the blood–brain barrier. However, predicting compounds with “low” permeability has been a challenging task. In this study, we present a deep learning (DL) classification model to predict blood–brain barrier permeability. The proposed model addresses the fundamental issues presented in former models: high dimensionality, class imbalances, and low specificity scores. We address these issues to enhance the high-dimensional, imbalanced dataset before developing the classification model: the imbalanced dataset is addressed using oversampling techniques and the high dimensionality using a non-linear dimensionality reduction technique known as kernel principal component analysis (KPCA). This technique transforms the high-dimensional dataset into a low-dimensional Euclidean space while retaining invaluable information. For the classification task, we developed an enhanced feed-forward deep learning model and a convolutional neural network model. In terms of specificity scores (i.e., predicting compounds with low permeability), the results obtained by the enhanced feed-forward deep learning model outperformed those obtained by other models in the literature that were developed using the same technique. In addition, the proposed convolutional neural network model surpassed models used in other studies in multiple accuracy measures, including overall accuracy and specificity. The proposed approach solves the problem inevitably faced with obtaining low specificity resulting in high false positive rate.