Vector Space Research Articles

Toward Robust Car-Following Dynamics Modeling via Blackbox Models: Methodology, Analysis, and Recommendations

The selection of the target variable is critical for the performance of classical car-following (CF) models. There is a vast body of literature on which target variable is optimal for classical CF models, but there is no study that evaluates the selection of optimal target variables for black-box models prevalent in machine learning, which are increasingly being used to model CF behavior. This paper evaluates different target variables, for example acceleration, velocity, and headway, for three models: Gaussian processes (GP), kernel ridge regression, and long short-term memory (LSTM). These models have different objective functions and work in different vector spaces, for example, GP work in function space, and LSTM in parameter space. Furthermore, this paper evaluates the efficacy of black box and classical CF models across datasets, including diverse traffic scenarios and human versus automated control. Finally, this paper evaluates the coupling of design choices across selection of model type, target variable during model calibration, and dataset characteristics. The experiments show that the optimal target variable recommendations for black-box models differ from classical CF models, depending on the objective function and the vector space. Statistical analysis indicates that selection of models couples greatly with target variables, while recommended target variables do not depend on the datasets used during model calibration. These results provide a path toward more robust CF models, potentially resulting in more accurate and generalizable capacity and/or traffic flow estimates for roadways, even as driving behaviors change with the introduction of new types of automation.

Some Lie Algebra Structures on Symmetric Powers

Let k be a field of any characteristic, V a finite-dimensional vector space over k, and S d ( V * ) be the d-th symmetric power of the dual space V * . Given a linear map φ on V and an eigenvector w of φ , we prove that the pair ( φ , w ) can be used to construct a new Lie algebra structure on S d ( V * ) . We prove that this Lie algebra structure is solvable, and in particular, it is nilpotent if φ is a nilpotent map. We also classify the Lie algebras for all possible pairs ( φ , w ) , when k = C and V is two-dimensional.

LEAF: A Lifestyle Approximation Framework Based on Analysis of Mobile Network Data in Smart Cities

The increasing use of mobile networks is an opportunity to collect and model users’ movement data for extracting knowledge about life and health while considering privacy leakage risk. This study aims to approximate the lifestyles of urban residents, employing statistical information derived from their movements among various Points of Interest (PoI). Our investigations comprehend a multidimensional analysis of key urban factors to provide insights into the population’s daily routines, preferences, and characteristics. To this end, we developed a framework called LEAF that models lifestyles by interpreting anonymized cell phone mobility data and integrating it with information from other sources, such as geographical layers of land use and sets of PoI. LEAF presents the information in a vector space model capable of responding to spatial queries about lifestyle. We also developed a consolidated lifestyle pattern framework to systematically identify and analyze the dominant activity patterns in different urban areas. To evaluate the effectiveness of the proposed framework, we tested it on movement data from individuals in a medium-sized city and compared the results with information collected through surveys. The RMSE of 5.167 between the proposed framework’s results and survey-based data indicates that the framework provides a reliable estimation of lifestyle patterns across diverse urban areas. Additionally, summarized patterns of criteria ordering were created, offering a concise and intuitive representation of lifestyles. The analysis revealed high consistency between the two methods in the derived patterns, underscoring the framework’s robustness and accuracy in modeling urban lifestyle dynamics.

Stiefel manifold interpolation for non-intrusive model reduction of parameterized fluid flow problems

Many engineering problems are parameterized. In order to minimize the computational cost necessary to evaluate a new operating point, the interpolation of singular matrices representing the data seems natural. Unfortunately, interpolating such data by conventional methods usually leads to unphysical solutions, as the data live on manifolds and not vector spaces. An alternative is to perform the interpolation in the tangent space to the Grassmann manifold to obtain interpolated spatial modes. Temporal modes are afterwards determined via the Galerkin projection of the high-fidelity model onto the interpolated spatial basis. This method, which is known for some fifteen years, is intrusive. Recently, Oulghelou and Allery (JCP, 2021) have proposed a non-intrusive approach (equation-free), but requiring the resolution of two low-dimensional optimization problems after interpolation. In this paper, a non-intrusive alternative based on Interpolation on the Tangent Space of the Stiefel Manifold (ITSSM) is presented. This approach has the advantage of not requiring a calibration phase after interpolation. To assess the method, we compare our results with those obtained using global POD on the one hand, and two methods based on Grassmann interpolation on the other. These comparisons are performed for two classical configurations encountered in fluid dynamics. The first corresponds to the one-dimensional non-linear Burgers' equation. The second example is the two-dimensional cylinder wake flow. We show that the proposed strategy can accurately reconstruct the physical quantities associated with a new operating point. Moreover, the estimation is fast enough to allow real-time computation.

Vector space model, term frequency-inverse document frequency with linear search, and object-relational mapping Django on hadith data search

For Muslims, the Hadith ranks as the secondary legal authority following the Quran. This research leverages hadith data to streamline the search process within the nine imams’ compendium using the vector space model (VSM) approach. The primary objective of this research is to enhance the efficiency and effectiveness of the search process within Hadith collections by implementing pre-filtering techniques. This study aims to demonstrate the potential of linear search and Django object-relational mapping (ORM) filters in reducing search times and improving retrieval performance, thereby facilitating quicker and more accurate access to relevant Hadiths. Prior studies have indicated that VSM is efficient for large data sets because it assigns weights to every term across all documents, regardless of whether they include the search keywords. Consequently, the more documents there are, the more protracted the weighting phase becomes. To address this, the current research pre-filters documents prior to weighting, utilizing linear search and Django ORM as filters. Testing on 62,169 hadiths with 20 keywords revealed that the average VSM search duration was 51 seconds. However, with the implementation of linear and Django ORM filters, the times were reduced to 7.93 and 8.41 seconds, respectively. The recall@10 rates were 79% and 78.5%, with MAP scores of 0.819 and 0.814, accordingly.

Multi-stage MFA for evaluating sustainable waste potential for bioplastics conversion in the circular economy: An examination of UK wastes to produce cellulose nanofibre

The negative environmental effect of petrochemical plastics prompts a shift in commercial interest towards bioplastics. Bioplastics produced from wastes and residues, when compared to primary biomass, reduces costs and emissions. However, these wastes are fed in alternative applications, resisting their use for bioplastics. The transition of wastes from current practices to bioplastics calls for a sustainability assessment, which is a combined examination of the economic, environmental and technical aspects. Material flow analysis (MFA), quantifying the flows of materials within a given system, is often used for waste management. However, MFA, not capturing additional waste characteristics, falls short in evaluating sustainable waste diversion. Therefore, this study suggests a Multi-stage MFA technique, integrating Multi criteria analysis (MCA) with MFA, to link additional information to the material flows, enabling sustainability assessment of waste to bioplastics conversion. To demonstrate the proposed methodology, this paper combines MFA with two particular MCA techniques, Analytic Hierarchy Process and Vector Space Theory, to analyse a UK case study of cellulose nanofibril (CNF) production. The findings indicate that, 11.6 Mt of the 18.04 dry Mt lignocellulosic wastes are sustainable for producing CNF films. However, this amount can change if the criteria or their relative importance in sustainability evaluation changes. The proposed approach is a generic tool for analysing sustainability of wastes to bioplastics conversion under a given geographic region to underpin circular economy.

Detecting damage in composites using volume decomposition analysis of tomographic data

Detection of damage in a single tow ceramic matrix composite specimen has been achieved using orthogonal decomposition of volumetric tomographic datasets collected at four tensile loads. This decomposition approach has been applied at two different length scales: (i) individual fibres and (ii) bulk volumes containing fibres and matrix material. Volumes were first decomposed to feature vectors, orders of magnitude smaller than the original volume they describe, and then comparisons between datasets at different load levels were made in feature vector space. The results show quantitative measurements of damage location, damage morphology and the relative growth of this damage with increased load when compared with a dataset with less or no damage. No prior knowledge of the dataset or training of algorithms is required for damage to be detected, it is only necessary that at least two datasets are available for comparison, e.g. from in situ or repeated scanning measurements. Results are generated on significantly shorter timescales when compared with previous automated approaches to tomography data processing. This approach has the potential to be applied to damage detection in a range of materials through comparisons of volumetric datasets from a range of measurement or computational techniques.

Some Generator Subgraphs of the Square of a Cycle

Graphs considered in this paper are finite simple graphs, which has no loops and multiple edges. Let $G= (V(G),E(G))$ be a graph with $E(G) = \{e_{1}, e_{2},\ldots, e_{m}\},$ for some positive integer $m.$ The \textit{edge space} of $G,$ denoted by $\mathscr{E}(G),$ is a vector space over the field $\zn_{2}.$ The elements of $\mathscr{E}(G)$ are all the subsets of $E(G)$. Vector addition is defined as $X+Y = X ~\Delta~ Y,$ the symmetric difference of sets $X$ and $Y,$ for $X,Y \in \mathscr{E}(G).$ Scalar multiplication is defined as $1\cdot X =X$ and $0 \cdot X = \emptyset$ for $X \in \mathscr{E}(G).$ Let $H$ be a subgraph of $G.$ The \textit{uniform set of $H$} with respect to $G,$ denoted by $E_{H}(G),$ is the set of all elements of $\mathscr E(G)$ that induces a subgraph isomorphic to $H.$ The subspace of $\mathscr E(G)$ generated by $E_{H}(G)$ shall be denoted by $\mathscr E_{H}(G).$ If $E_H(G)$ is a generating set, that is $\mathscr E_{H}(G)= \mathscr E(G),$ then $H$ is called a \textit{generator subgraph} of $G.$ This paper determines some generator subgraphs of the square of a cycle. Moreover, this paper established sufficient conditions for the generator subgraphs of the square of a cycle.

All convex bodies are in the subdifferential of some everywhere differentiable locally Lipschitz function

AbstractWe construct a differentiable locally Lipschitz function in with the property that for every convex body there exists such that coincides with the set of limits of derivatives of sequences converging to . The technique can be further refined to recover all compact connected subsets with nonempty interior, disclosing an important difference between differentiable and continuously differentiable functions. It stems out from our approach that the class of these pathological functions contains an infinite‐dimensional vector space and is dense in the space of all locally Lipschitz functions for the uniform convergence.

Improving information theory of context-aware phrase embeddings in HR domain

The object of the study is context-aware phrase representations. The growing need to automate candidate recruitment and job recommendation processes has paved the way for the utilization of text embeddings. These embeddings involve translating the semantic essence of text into a continuous, high-dimensional vector space. By learning context-aware rich and meaningful representations of phrases within the human resource domain, the efficacy of similarity searches and matching procedures is enhanced, which contributes to a more streamlined and effective recruitment process. However, existing approaches do not take into account the context when modeling phrases. This necessitates the improvement of information technology analysis in this area. In this paper, it is proposed to mark the beginning and end of phrases in the text using special tokens. This made it possible to reduce the requirements for computing power by calculating all phrase representations present in the text simultaneously. The effectiveness of the improvement was tested on a new dataset to compare and evaluate the models in the task of modeling phrases in the field of human resources management. The proposed approach to modeling phrase representations with regard to context in the field of human resources management leads to an improvement in computational efficiency by up to 50 % and an increase in accuracy by up to 10 %. The architecture of the machine learning model for creating context-aware phrase representations is developed, which is characterized by the presence of blocks for taking into account phrase boundaries. Experiments and comparisons with existing approaches have confirmed the effectiveness of the proposed solution. In practice, the proposed information analysis technology can be used to automate the process of identifying and normalizing candidates' skills in online recruiting

A Genetic Algorithm to Search the Space of Ehrhart h*-Vectors

We describe a genetic algorithm to find candidates for h * -vectors satisfying given properties in the space of integers vectors of finite length. We use an implementation of such algorithm to find a 52-dimensional lattice polytope having a non-unimodal h * -vector which is the Cartesian product of two lattice polytopes having unimodal h * -vectors. This counterexample answers negatively to a question by Ferroni and Higashitani.

Properties of generalized polyhedral convex multifunctions

This paper presents a study of generalized polyhedral convexity under basic operations on multifunctions. We address the preservation of generalized polyhedral convexity under sums and compositions of multifunctions, the domains and ranges of generalized polyhedral convex multifunctions, and the direct and inverse images of sets under such mappings. Then we explore the class of optimal value functions defined by a generalized polyhedral convex objective function and a generalized polyhedral convex constrained mapping. The new results provide a framework for representing the relative interior of the graph of a generalized polyhedral convex multifunction in terms of the relative interiors of its domain and mapping values in locally convex topological vector spaces. Among the new results in this paper is a significant extension of a result by Bonnans and Shapiro on the domain of generalized polyhedral convex multifunctions from Banach spaces to locally convex topological vector spaces.

The Accelerating Universe in a NoncommutativeAnalytically Continued Foliated Quantum Gravity

Abstract Based on an analytically continued Riemannian foliated quantum gravity
super-Hamiltonian, known as Branch Cut Quantum Gravity (BCQG) we propose
a novel approach to investigating the effects of noncommutative geometry on a
minisuperspace of variables, influencing the acceleration behavior of the universe’s
wave function and the cosmic scale factor. Noncommutativity is introduced through
a deformation of the conventional Poisson algebra, enhanced with a symplectic
metric. The resulting symplectic manifold provides a natural setting that enables
an isomorphism between canonically conjugate dual vector spaces, spanning the
BCQG cosmic scale factor and its complementary quantum counterpart. Using this
formulation, we describe the dynamic evolution of the universe’s wave function, the
cosmic scale factor, and its complementary quantum image. Our results strongly
suggest that the noncommutative algebra induces late-time accelerated growth of the
wave function, the universe’s scale factor, and its complementary quantum counterpart,
offering a new perspective on explaining the accelerating cosmic expansion rate and
the inflationary period. In contrast to the inflationary model, where inflation requires
a remarkably fine-tuned set of initial conditions in a patch of the universe, analytically
continued non-commutative foliated quantum gravity captures short and long scales,
driving the evolutionary dynamics of the universe through a reconfiguration of the
primordial cosmic content of matter and energy. This reconfiguration is encapsulated
into a quantum field potential, which leads to the generation of relic gravitational
waves, a topic for future investigation. Graphical representations and contour plots
indicate a characteristic torsion (or twist) deformation of spacetime geometry. This
result introduces new speculative elements regarding the reconfiguration of matter and
energy as a driver of spacetime torsion deformation, generating relic gravitational waves
and serving as an alternative topological mechanism for the universe’s acceleration.
However, these assumptions require further investigation.

Intrinsically Hölder Sections in Metric Spaces

We introduce the notion of intrinsically Holder graphs in metric spaces that generalized the one of ¨ intrinsically Lipschitz sections. This concept is relevant because it has many properties similar to Holder maps but ¨ is profoundly different from them. We prove some relevant results as the Ascoli-Arzela compactness Theorem, ` Ahlfors-David regularity and the Extension Theorem for this class of sections. In the first part of this note, thanks to Cheeger theory, we define suitable sets in order to obtain a vector space over R or C, a convex set and an equivalence relation for intrinsically Holder graphs. These last three properties are new also in the Lipschitz case.

Analyzing the Connection of Linear Algebra: Enhancing Visual Reasoning through Vector Spaces

Analyzing the Connection of Linear Algebra: Enhancing Visual Reasoning through Vector Spaces

Hyperdimensional computing: a framework for stochastic computation and symbolic AI

Hyperdimensional Computing (HDC), also known as Vector Symbolic Architectures (VSA), is a neuro-inspired computing framework that exploits high-dimensional random vector spaces. HDC uses extremely parallelizable arithmetic to provide computational solutions that balance accuracy, efficiency and robustness. The majority of current HDC research focuses on the learning capabilities of these high-dimensional spaces. However, a tangential research direction investigates the properties of these high-dimensional spaces more generally as a probabilistic model for computation. In this manuscript, we provide an approachable, yet thorough, survey of the components of HDC. To highlight the dual use of HDC, we provide an in-depth analysis of two vastly different applications. The first uses HDC in a learning setting to classify graphs. Graphs are among the most important forms of information representation, and graph learning in IoT and sensor networks introduces challenges because of the limited compute capabilities. Compared to the state-of-the-art Graph Neural Networks, our proposed method achieves comparable accuracy, while training and inference times are on average 14.6× and 2.0× faster, respectively. Secondly, we analyse a dynamic hash table that uses a novel hypervector type called circular-hypervectors to map requests to a dynamic set of resources. The proposed hyperdimensional hashing method has the efficiency to be deployed in large systems. Moreover, our approach remains unaffected by a realistic level of memory errors which causes significant mismatches for existing methods.

From communities to interpretable network and word embedding: an unified approach

Abstract Modelling information from complex systems such as humans social interaction or words co-occurrences in our languages can help to understand how these systems are organized and function. Such systems can be modelled by networks, and network theory provides a useful set of methods to analyze them. Among these methods, graph embedding is a powerful tool to summarize the interactions and topology of a network in a vectorized feature space. When used in input of machine learning algorithms, embedding vectors help with common graph problems such as link prediction, graph matching, etc In Natural Language Processing (NLP), such a vectorization process is also employed. Word embedding has the goal of representing the sense of words, extracting it from large text corpora. Despite differences in the structure of information in input of embedding algorithms, many graph embedding approaches are adapted and inspired from methods in NLP. Limits of these methods are observed in both domains. Most of these methods require long and resource greedy training. Another downside to most methods is that they are black-box, from which understanding how the information is structured is rather complex. Interpretability of a model allows understanding how the vector space is structured without the need for external information, and thus can be audited more easily. With both these limitations in mind, we propose a novel framework to efficiently embed network vertices in an interpretable vector space. Our Lower Dimension Bipartite Framework (LDBGF) leverages the bipartite projection of a network using cliques to reduce dimensionality. Along with LDBGF, we introduce two implementations of this framework that rely on communities instead of cliques: SINr-NR and SINr-MF. We show that SINr-MF can perform well on classical graphs and SINr-NR can produce high-quality graph and word embeddings that are interpretable and stable across runs.

One-loop double copy relation from twisted (co)homology

We propose a geometric relation between closed and open string amplitudes at one-loop. After imposing a homological splitting on the world-sheet torus, twisted intersection theory is used to establish a one-loop double copy relation. The latter expresses a closed string amplitude by a pair of open string amplitudes and twisted intersection numbers. These inner products on the vector space of twisted differential forms are related to the twisted homology and cohomology groups associated with the Riemann-Wirtinger integral.

Distributional hypothesis as isomorphism between word-word co-occurrence and analogical parallelograms.

Most of the modern natural language processing (NLP) techniques are based on the vector space models of language, in which each word is represented by a vector in a high dimensional space. One of the earliest successes was demonstrated by the four-term analogical reasoning task: what is to C as B is to A? The trained word vectors form "parallelograms" representing the quadruple of words in analogy. This discovery in NLP offers us insight into our understanding of human semantic representation of words via analogical reasoning. Despite successful applications of the large-scale language models, it has not been fully understood why such parallelograms emerge by learning through natural language data. As the vector space model is not optimized to form parallelograms, the key structure related to geometric shapes of word vectors is expected to be in the data, rather than the models. In the present article, we test our hypothesis that such parallelogram arrangement of word vectors readily exists in the co-occurrence statistics of language. Our approach focuses more on the data itself, and it is different from the existing theoretical approach trying to find the mechanism of parallelogram formation in the algorithms and/or vector arithmetic operations on word vectors. First, our analysis suggested that analogical reasoning is possible by decomposition of the bigram co-occurrence matrix. Second, we demonstrated the formation of a parallelepiped, a more structured geometric object than a parallelogram, by creating a small artificial corpus and its word vectors. With these results, we propose a refined form of distributional hypothesis pointing out an isomorphism between a sort of symmetry or exchangeability and word co-occurrence statistics.

Inheritance of signs of parent species by hybrids of Russet, Yellow and Speckled ground squirrels (<i>Spermophilus</i>, Sciuridae, Rodentia)

The inheritance of size, color and bioacoustic characteristics of Russet (Spermophilus major), Yellow (S. fulvus) and Speckled (S. suslicus) ground squirrels in hybrids differing in their genetic status were studied. In a hybrid population of Russet and Yellow ground squirrels, 10 individuals of S. major, 10 individuals of S. fulvus and 40 hybrids of S. major × S. fulvus were analyzed; in a hybrid population of Russet and Speckled ground squirrels, 11 individuals of S. major, 11 individuals of S. suslicus and 24 hybrids of S. major × S. suslicus were analyzed. Hybrid individuals different in genetic status have been shown to demonstrate differentiation in the vector space of the variability range of the parental species. At the same time, both a shift in some categories of hybrids towards one of the hybridizing species and a significant deviation are noted. The results obtained indicate an increase in the level of variability in size, color and bioacoustic characters in hybrid populations, associated with the erosion of the divergent hiatus of specific features of the parental forms due to the formation of a differentiated hybrid population. As hybrid individuals with different combinations of parental traits accumulate in the population, the situation arises when a sufficient number of hybrids with an overall combination of parental traits (Hcomb) appear in the hybridogenic population. In the hybrid population of the Russet and Yellow ground squirrels, the proportion of such hybrids, according to our data, is 22.5% (n = 40), and in the hybrid population of the Russet and Speckled ground squirrels it is 12.5% (n = 24). This category of hybrid individuals can be considered as possible material for further microevolutionary events and the process of speciation.