Embedding Dimension Research Articles

On bounds for the atom bond sum connectivity index of graphs associated with symmetric numerical semigroups

The computation of the clique number of a graph is a fundamental problem in graph theory, which has many applications in computational chemistry, bioinformatics, computer, and social networking. A subset S of non-negative integers N 0 is called a numerical semigroup if it is a submonoid of N 0 and has a finite complement in N 0 . The graph associated with numerical semigroup S is denoted by G [ S ] and is defined by the vertex set { x : x ∈ g ( S ) } and the edge set { xy ⇔ x + y ∈ S } . In this article, we compute the clique number and the minimum degree of those graphs, which can be associated with symmetric numerical semigroups of embedding dimension 2. Moreover, on this basis, we give some bounds for the atom-bond sum-connective index of graphs G [ S ] in terms of the harmonic index, the first Zagreb index, the sum-connectivity index, the maximum degree, the minimum degree, the chromatic number, and the clique number.

Influence maximization under imbalanced heterogeneous networks via lightweight reinforcement learning with prior knowledge

Influence Maximization (IM) stands as a central challenge within the domain of complex network analysis, with the primary objective of identifying an optimal seed set of a predetermined size that maximizes the reach of influence propagation. Over time, numerous methodologies have been proposed to address the IM problem. However, one certain network referred to as Imbalanced Heterogeneous Networks (IHN), which widely used in social situation, urban and rural areas, and merchandising, presents challenges in achieving high-quality solutions. In this work, we introduce the Lightweight Reinforcement Learning algorithm with Prior knowledge (LRLP), which leverages the Struc2Vec graph embedding technique that captures the structural similarity of nodes to generate vector representations for nodes within the network. In details, LRLP incorporates prior knowledge based on a group of centralities, into the initial experience pool, which accelerates the reinforcement learning training for better solutions. Additionally, the node embedding vectors are input into a Deep Q Network (DQN) to commence the lightweight training process. Experimental evaluations conducted on synthetic and real networks showcase the effectiveness of the LRLP algorithm. Notably, the improvement seems to be more pronounced when the the scale of the network is larger. We also analyze the effect of different graph embedding algorithms and prior knowledge on algorithmic results. Moreover, we conduct an analysis about some parameters, such as number of seed set selections T, embedding dimension d and network update frequency C. It is significant that the reduction of number of seed set selections T not only keeps the quality of solutions, but lowers the algorithm’s computational cost.

Enhanced HS Code Classification for Import and Export Goods via Multiscale Attention and ERNIE-BiLSTM

Accurate classification of import and export goods’ Harmonized System (HS) codes is essential for ensuring tax security. Applying text classification technologies for HS code classification can significantly enhance the prevention and control of customs tax risks. However, the goods text is a semi-structured one that involves multi-domain Chinese professional vocabulary, which poses challenges for current classification models. These models often suffer from inadequate text representation and imprecise feature extraction. To address these challenges, we propose a novel classification model ERNIE-BiLSTM-Channel attention–Spatial attention (EBLCS). This model integrates ERNIE (Enhanced Representation through Knowledge Integration) with a Bidirectional Long Short-Term Memory Network (BiLSTM) and employs multi-scale attention mechanisms. The ERNIE-BiLSTM model provides a more comprehensive and accurate representation of the goods text, effectively capturing the global features of the text. By introducing channel attention and spatial attention mechanisms, greater weights are assigned to important words and word embedding dimensions, significantly enhancing the model’s ability to perceive key information. The experimental results on a customs dataset demonstrate that the EBLCS model consistently outperforms various baseline models across all evaluation metrics, effectively enhancing the performance of HS code classification.

Crumpled-to-flat transition of quenched disordered membranes at two-loop order.

We investigate the effects of quenched elastic disorder on the nature of the crumpling-to-flat transition of D-dimensional polymerized membranes using a two-loop computation near the upper critical dimension D_{c}=4. While the pure system undergoes fluctuation-induced first-order transitions below D_{c} and for an embedding dimension d<d_{c,pure}≃218.2, one observes, in thepresence of disorder, the emergence of various regions of second order governed by a disordered stable fixed point for d<d_{c1}∼d_{c,pure}. This opens the possibility of a new universality class associated with the crumpling-to-flat transition of disordered membranes in d=3.

Complexity of electrodermal activity to mental stress is changed during adolescent age-period.

Complexity characterizes behaviour of all physiological systems whose components interact in multiple ways usually quantified by entropy techniques. However, complexity analysis regarding electrodermal activity (EDA)-related sympathetic cholinergic nervous system is rare. Thus, we aimed to study EDA dynamics complexity changes from aspect of various embedding dimensions (m) and timescales (τ) (sample entropy (SampEn) with m ∈ <2,7>, and multiscale entropy (MSE) in τ ∈ <1,20>) in association with traditionally used EDA indices (skin conductance level (SCL) and non-specific skin conductance responses (NS.SCRs)) to mental stress (mental arithmetic test - MAT) in healthy participants at critical adolescent age. The cohort (total group) consisted of 60 adolescents (17.5 ± 0.5 yrs) divided into three groups: Group-1: early (13.1 ± 0.3 yrs), Group-2: middle (16.6 ± 0.2 yrs) and Group-3: late (22.9 ± 0.1 yrs) adolescence. SampEn (m > 2) and MSE (for all τ) were significantly higher during MAT than baseline in total group and Group-2 (p < 0.05). Index MSE for all τ was significantly higher during MAT than baseline in total group, and Group-2; for τ ∈ <2,13> in Group-1 (p < 0.05). Additionally, while SCL was significantly higher during MAT than baseline in all groups, NS.SCRs was lower during stress only in Group-3 (p < 0.05). In conclusion, this study revealed distinct EDA complexity characteristics in individual examined groups indicating importance of complexity evaluation in stress-related sympathetic regulatory mechanisms within individual adolescent age ranges.

Bounds on Gorenstein dimensions and exceptional complete intersection maps

We prove that if f : R → S is a local homomorphism of noetherian local rings of finite flat dimension and M is a nonzero finitely generated S-module whose Gorenstein flat dimension over R is bounded by the difference of the embedding dimensions of R and S, then M is a totally reflexive S-module and f is an exceptional complete intersection map. This is an extension of a result of Brochard, Iyengar, and Khare to Gorenstein flat dimension. We also prove two analogues involving Gorenstein injective dimension.

A verification of Wilf's conjecture up to genus 100

For a numerical semigroup S⊆N, let m,e,c,g denote its multiplicity, embedding dimension, conductor and genus, respectively. Wilf's conjecture (1978) states that e(c−g)≥c. As of 2023, Wilf's conjecture has been verified by computer up to genus g≤66. In this paper, we extend the verification of Wilf's conjecture up to genus g≤100. This is achieved by combining three main ingredients: (1) a theorem in 2020 settling Wilf's conjecture in the case e≥m/3, (2) an efficient trimming of the tree T of numerical groups identifying and cutting out irrelevant subtrees, and (3) the implementation of a fast parallelized algorithm to construct the tree T up to a given genus. We further push the verification of Wilf's conjecture up to genus 120 in the particular case where m divides c. Finally, we unlock three previously unknown values of the number ng of numerical semigroups of genus g, namely for g=73,74,75.

Correlation dimension for temperature and other micrometeorological variables in different ground cover from six biomes of the Amazon basin

Correlation dimension () for embedding dimensions, varying between 3 to 11, were estimated for a 1- or 2-month time series of air temperature and other micrometeorological variables, such as energy fluxes, solar radiation and concentration, in six ecosystems in Amazon and the central region of Brazil. Dc remained within the 1.5 - 3.0 range, for variables locally coupled to ecosystems, and outside the range for variables not locally coupled, such as precipitation and wind velocity. Results indicate that the feature of the variables does not have an important stochastic component. There is a small amount of data available concerning estimates of correlation dimension () of micrometeorological variables. The measurement of for several variables from six Brazilian biomes indicates that this parameter is sensitive to environmental conditions as the rainfall. The obtained results also indicates that the natural processes in the ecosystems remain in a non-chaotic regime. In certain circumstances, as in conditions of high humidity, natural ecosystems may stay inside a limit which is close but lower than chaos. Moreover, an important stochastic component was not observed.

Knowledge Graph Embedding for Hierarchical Entities Based on Auto-Embedding Size

Knowledge graph embedding represents entities and relations as low-dimensional continuous vectors. Recently, researchers have attempted to leverage the potential semantic connections between entities with hierarchical relationships in the knowledge graph. However, existing knowledge embedding methods that model entity hierarchical structures face the following challenges: (1) Setting the same embedding dimension for entities at different hierarchical levels can lead to overfitting issues and the waste of storage and computational resources; (2) Manually searching the embedding dimension in discrete space makes it easy to miss the optimal parameters and increases training costs. Therefore, we propose a knowledge embedding method, HEAES, that automatically learns the embedding dimensions for entities based on their hierarchical structure. Specifically, we first propose a new modeling method to capture the hierarchical relationships of entities, and we then train the model on a triple classification task. Then, we adaptively learn the pruning threshold to trim the embedding vectors, automatically learning different embedding dimensions for entities at different hierarchical levels. Experiments on the YAGO26K and DB111K datasets verify that introducing entity hierarchical information helps boost the model performance.

Grain storage temperature prediction based on chaos and enhanced RBF neural network

Grain storage has very strict temperature requirements. Aiming at the problems of nonlinear characteristics and poor prediction accuracy of temperature parameters in grain storage, a combination of chaos theory and enhanced radial basis neural network is proposed as a temperature prediction model for grain storage (C-ERBF). The model first determines the embedding dimension and time delay of the grain storage temperature sequence using chaos theory. It then calculates the Lyapunov exponent to confirm its chaotic properties and reconstructs the sequence in the phase space to extract the hidden dynamic information and structure behind the sequence. Furthermore, the q-Normalized Least Mean Square Fourth (qXE-NLMF) algorithm is designed to enhance the radial basis function (RBF) neural network model for weight updating, to improve its prediction accuracy, and to accelerate the training speed of the model. As verified by the simulation experiments of Mackey–Glass chaotic time series prediction, the enhanced RBF (ERBF) network has faster convergence speed and lower steady-state error compared to the traditional RBF network. Finally, the optimized dataset from chaos theory is input into the model to achieve accurate predictions of grain storage temperature series. The experimental results show that the proposed C-ERBF model has higher prediction accuracy compared to other time series prediction methods. It can realize the grain pile temperature in advance, and take control measures in advance. This proactive approach significantly reduces the consumption of stored grain and prevents issues before they arise.

Correlation Fuzzy measure of multivariate time series for signature recognition.

Distinguishing different time series, which is determinant or stochastic, is an important task in signal processing. In this work, a correlation measure constructs Correlation Fuzzy Entropy (CFE) to discriminate Chaos and stochastic series. It can be employed to distinguish chaotic signals from ARIMA series with different noises. With specific embedding dimensions, we implemented the CFE features by analyzing two available online signature databases MCYT-100 and SVC2004. The accurate rates of the CFE-based models exceed 99.3%.

Legendre Polynomial Fitting-Based Permutation Entropy Offers New Insights into the Influence of Fatigue on Surface Electromyography (sEMG) Signal Complexity.

In a recently published work, we introduced local Legendre polynomial fitting-based permutation entropy (LPPE) as a new complexity measure for quantifying disorder or randomness in time series. LPPE benefits from the ordinal pattern (OP) concept and incorporates a natural, aliasing-free multiscaling effect by design. The current work extends our previous study by investigating LPPE's capability to assess fatigue levels using both synthetic and real surface electromyography (sEMG) signals. Real sEMG signals were recorded during biceps brachii fatiguing exercise maintained at 70% of maximal voluntary contraction (MVC) until exhaustion and were divided into four consecutive temporal segments reflecting sequential stages of exhaustion. As fatigue levels rise, LPPE values can increase or decrease significantly depending on the selection of embedding dimensions. Our analysis reveals two key insights. First, using LPPE with limited embedding dimensions shows consistency with the literature. Specifically, fatigue induces a decrease in sEMG complexity measures. This observation is supported by a comparison with the existing multiscale permutation entropy (MPE) variant, that is, the refined composite downsampling (rcDPE). Second, given a fixed OP length, higher embedding dimensions increase LPPE's sensitivity to low-frequency components, which are notably present under fatigue conditions. Consequently, specific higher embedding dimensions appear to enhance the discrimination of fatigue levels. Thus, LPPE, as the only MPE variant that allows a practical exploration of higher embedding dimensions, offers a new perspective on fatigue's impact on sEMG complexity, complementing existing MPE approaches.

Fractal-based supervised approach for dimensionality reduction of hyperspectral images

Dimensionality reduction is one of the most challenging and crucial issues apart from data mining, security, and scalability, which have retained much traction due to the ever-growing need to analyze the large volumes of data generated daily. Fractal Dimension (FD) has been successfully used to characterize data sets and has found relevant applications in dimension reduction. This paper presents an application of the FD Reduction (FDR) Algorithm on geospatial hyperspectral data, examining its usefulness for data sets with a relatively high embedding dimension. We examine the algorithm at two levels. First is the conventional FDR approach (unsupervised) at the image level. Alternatively, we propose a pixel-level supervised approach for band reduction based on time-series complexity analysis. Techniques for determining an optimal intrinsic dimension for the dataset using these two techniques are examined. We also develop a parallel GPU-based implementation for the unsupervised image-level FDR algorithm, reducing the run-time by nearly 10 times. Furthermore, both approaches use a support vector machine classifier to compare the classification performance of the original and reduced image obtained.

The ratio-covariety of numerical semigroups having maximal embedding dimension with fixed multiplicity and Frobenius number

In this paper we will show that $\MED(F,m)=\{S\mid S \mbox{ is a numeri-}\\ \mbox{cal semigroup with maximal embedding dimension, Frobenius number $F$ and }\\ \mbox{multiplicity }m\}$ is a ratio-covariety. As a consequence, we present two algorithms: one that computes $\MED(F,m)$ and another one that calculates the elements of $\MED(F,m)$ with a given genus. If $X\subseteq S\backslash (\langle m \rangle \cup \{F+1,\rightarrow\})$ for some $S\in \MED(F,m)$, then there exists the smallest element of $\MED(F,m)$ containing $X$. This element will be denoted by $\MED(F,m)[X]$ and we will say that $X$ one of its $\MED(F,m)$-system of generators. We will prove that every element $S$ of $\MED(F,m)$ has a unique minimal $\MED(F,m)$-system of generators and it will be denoted by $\MED(F,m)\msg(S).$ The cardinality of $\MED(F,m)\msg(S)$, will be called $\MED(F,m)$-$\rank$ of $S.$ We will also see in this work, how all the elements of $\MED(F,m)$ with a fixed $\MED(F,m)$-$\rank$ are.

Adaptive variable sampling T-SSD algorithm

Addressing the challenges of non-unique decomposition outcomes and prolonged decomposition durations in the fault feature adaptive extraction algorithm based on tensor decomposition, this paper presents a novel algorithm called the adaptive variable sampling tensor singular spectrum decomposition (T-SSD) algorithm. The proposed approach centers on decomposing multichannel time series with adaptive sampling frequency, leveraging tensor singular value decomposition. Initially, the embedding dimension and the number of resampling points were optimized by power spectral density analysis and adaptive sampling algorithm. Subsequently, a third-order tensor is constructed based on the principle of tensor-tensor-preserving order multiplication, combining trajectory tensor construction and embedding dimension. Finally, the signal decomposition and reconstruction of multichannel component signals are achieved through adaptive sampling, interpolation, and complementary back steps. Experimental signal analysis indicates that this algorithm can better extract fault characteristics from signals compared to common signal processing algorithms. In comparison with the traditional T-SSD algorithm, this method significantly improves decomposition efficiency, particularly for low-frequency components. It effectively tackles the efficiency challenge caused by data redundancy, enabling the organic fusion and adaptive decomposition of multichannel signals.

Analysis of human ambulation as a chaotic time-series: with nonlinear dynamics tools.

The aim of the present study is to investigate the complexity and stability of human ambulation and the implications on robotic prostheses control systems. Fourteen healthy individuals participate in two experiments, the first group run at three different speeds. The second group ascended and descended stairs of a five-level building block at a self-selected speed. All participants completed the experiment with seven inertial measurement units wrapped around the lower body segments and waist. The data were analyzed to determine the fractal dimension, spectral entropy, and the Lyapunov exponent (LyE). Two methods were used to calculate the long-term LyE, first LyE calculated using the full size of data sets. And the embedding dimensions were calculated using Average Mutual Information (AMI) and the False Nearest Neighbor (FNN) algorithm was used to find the time delay. Besides, a second approach was developed to find long-term LyE where the time delay was based on the average period of the gait cycle using adaptive event-based window. The average values of spectral entropy are 0.538 and 0.575 for stairs ambulation and running, respectively. The degree of uncertainty and complexity increases with the ambulation speed. The short term LyEs for tibia orientation have the minimum range of variation when it comes to stairs ascent and descent. Using two-way analysis of variance we demonstrated the effect of the ambulation speed and type of ambulation on spectral entropy. Moreover, it was shown that the fractal dimension only changed significantly with ambulation speed.

Classification of BCI-EEG Based on the Augmented Covariance Matrix.

Electroencephalography signals are recorded as multidimensional datasets. We propose a new framework based on the augmented covariance that stems from an autoregressive model to improve motor imagery classification. From the autoregressive model can be derived the Yule-Walker equations, which show the emergence of a symmetric positive definite matrix: the augmented covariance matrix. The state-of the art for classifying covariance matrices is based on Riemannian Geometry. A fairly natural idea is therefore to apply this Riemannian Geometry based approach to these augmented covariance matrices. The methodology for creating the augmented covariance matrix shows a natural connection with the delay embedding theorem proposed by Takens for dynamical systems. Such an embedding method is based on the knowledge of two parameters: the delay and the embedding dimension, respectively related to the lag and the order of the autoregressive model. This approach provides new methods to compute the hyper-parameters in addition to standard grid search. The augmented covariance matrix performed ACMs better than any state-of-the-art methods. We will test our approach on several datasets and several subjects using the MOABB framework, using both within-session and cross-session evaluation. The improvement in results is due to the fact that the augmented covariance matrix incorporates not only spatial but also temporal information. As such, it contains information on the nonlinear components of the signal through the embedding procedure, which allows the leveraging of dynamical systems algorithms. These results extend the concepts and the results of the Riemannian distance based classification algorithm.

Multi-Frequency Entropy for Quantifying Complex Dynamics and Its Application on EEG Data.

Multivariate entropy algorithms have proven effective in the complexity dynamic analysis of electroencephalography (EEG) signals, with researchers commonly configuring the variables as multi-channel time series. However, the complex quantification of brain dynamics from a multi-frequency perspective has not been extensively explored, despite existing evidence suggesting interactions among brain rhythms at different frequencies. In this study, we proposed a novel algorithm, termed multi-frequency entropy (mFreEn), enhancing the capabilities of existing multivariate entropy algorithms and facilitating the complexity study of interactions among brain rhythms of different frequency bands. Firstly, utilizing simulated data, we evaluated the mFreEn's sensitivity to various noise signals, frequencies, and amplitudes, investigated the effects of parameters such as the embedding dimension and data length, and analyzed its anti-noise performance. The results indicated that mFreEn demonstrated enhanced sensitivity and reduced parameter dependence compared to traditional multivariate entropy algorithms. Subsequently, the mFreEn algorithm was applied to the analysis of real EEG data. We found that mFreEn exhibited a good diagnostic performance in analyzing resting-state EEG data from various brain disorders. Furthermore, mFreEn showed a good classification performance for EEG activity induced by diverse task stimuli. Consequently, mFreEn provides another important perspective to quantify complex dynamics.

Understanding and mitigating dimensional collapse of Graph Contrastive Learning: A non-maximum removal approach

Graph Contrastive Learning (GCL) generates graph-level embeddings by maximizing Mutual Information between different augmented views of the same graph (positive pairs), and shows promising performance in graph representation learning (GRL) without the supervision of manual annotations. However, GCL suffers from a dimensional collapse problem, i.e., embedding vectors reside in a restricted low-dimensional subspace, curtailing the expressiveness of certain embedding dimensions. In this paper, we present a theoretical analysis identifying the smoothing effect of graph pooling and graph convolution’s implicit regularization as principal causes of dimension collapse in graph contrastive learning. To mitigate the above effects, we propose a Non-Maximum Removal Graph Contrastive Learning (nmrGCL) approach, which removes “prominent” dimensions (those significantly contributing to the similarity measure) for positive pairs within the pretext task. Comprehensive experiments on multiple benchmark datasets are conducted and the results show that the proposed nmrGCL outperforms the state-of-the-art methods.

A class of integer 3 × 3 matrices related to numerical semigroups with embedding dimension 4

The aim of this paper is to present a class of integer [Formula: see text] matrices related to numerical semigroups with embedding dimension 4, generated by four integers [Formula: see text] and [Formula: see text] such that [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. For a matrix [Formula: see text] of this class, we describe the Apéry set with respect to [Formula: see text], of the numerical semigroup related to [Formula: see text].