Interpreting the Curse of Dimensionality from Distance Concentration and Manifold Effect

The characteristics of data like distribution and heterogeneity, become more complex and counterintuitive as the dimensionality increases. This phenomenon is known as curse of dimensionality, where common patterns and relationships (e.g., internal and boundary pattern) that hold in low-dimensional space may be invalid in higher-dimensional space. It leads to a decreasing performance for the regression, classification or clustering models or algorithms. Curse of dimensionality can be attributed to many causes. In this paper, we first summarize five challenges associated with manipulating high-dimensional data, and explains the potential causes for the failure of regression, classification or clustering tasks. Subsequently, we delve into two major causes of the curse of dimensionality, distance concentration and manifold effect, by performing theoretical and empirical analyses. The results demonstrate that nearest neighbor search (NNS) using three typical distance measurements, Minkowski distance, Chebyshev distance, and cosine distance, becomes meaningless as the dimensionality increases. Meanwhile, the data incorporates more redundant features, and the variance contribution of principal component analysis (PCA) is skewed towards a few dimensions. By interpreting the causes of the curse of dimensionality, we can better understand the limitations of current models and algorithms, and drive to improve the performance of data analysis and machine learning tasks in high-dimensional space.

Scalable manifold learning by uniform landmark sampling and constrained locally linear embedding

As a pivotal approach in machine learning and data science, manifold learning aims to uncover the intrinsic low-dimensional structure within complex nonlinear manifolds in high-dimensional space. By exploiting the manifold hypothesis, various techniques for nonlinear dimension reduction have been developed to facilitate visualization, classification, clustering, and gaining key insights. Although existing manifold learning methods have achieved remarkable successes, they still suffer from extensive distortions incurred in the global structure, which hinders the understanding of underlying patterns. Scalability issues also limit their applicability for handling large-scale data. Here, we propose a scalable manifold learning (scML) method that can manipulate large-scale and high-dimensional data in an efficient manner. It starts by seeking a set of landmarks to construct the low-dimensional skeleton of the entire data, and then incorporates the non-landmarks into the learned space based on the constrained locally linear embedding (CLLE). We empirically validated the effectiveness of scML on synthetic datasets and real-world benchmarks of different types, and applied it to analyze the single-cell transcriptomics and detect anomalies in electrocardiogram (ECG) signals. scML scales well with increasing data sizes and embedding dimensions, and exhibits promising performance in preserving the global structure. The experiments demonstrate notable robustness in embedding quality as the sample rate decreases.

MeanCut: A Greedy-Optimized Graph Clustering via Path-based Similarity and Degree Descent Criterion

As the most typical graph clustering method, spectral clustering is popular and attractive due to the remarkable performance, easy implementation, and strong adaptability. Classical spectral clustering measures the edge weights of graph using pairwise Euclidean-based metric, and solves the optimal graph partition by relaxing the constraints of indicator matrix and performing Laplacian decomposition. However, Euclidean-based similarity might cause skew graph cuts when handling non-spherical data distributions, and the relaxation strategy introduces information loss. Meanwhile, spectral clustering requires specifying the number of clusters, which is hard to determine without enough prior knowledge. In this work, we leverage the path-based similarity to enhance intra-cluster associations, and propose MeanCut as the objective function and greedily optimize it in degree descending order for a nondestructive graph partition. This algorithm enables the identification of arbitrary shaped clusters and is robust to noise. To reduce the computational complexity of similarity calculation, we transform optimal path search into generating the maximum spanning tree (MST), and develop a fast MST (FastMST) algorithm to further improve its time-efficiency. Moreover, we define a density gradient factor (DGF) for separating the weakly connected clusters. The validity of our algorithm is demonstrated by testifying on real-world benchmarks and application of face recognition. The source code of MeanCut is available at https://github.com/ZPGuiGroupWhu/MeanCut-Clustering.

A Robust and Efficient Boundary Point Detection Method by Measuring Local Direction Dispersion

Boundary points pose a significant challenge for machine learning tasks, including classification, clustering, and dimensionality reduction. Due to the similarity of features, boundary areas can result in mixed-up classes or clusters, leading to a crowding problem in dimensionality reduction. To address this challenge, numerous boundary point detection methods have been developed, but they are insufficiently to accurately and efficiently identify the boundary points in non-convex structures and high-dimensional manifolds. In this work, we propose a robust and efficient method for detecting boundary points using Local Direction Dispersion (LoDD). LoDD considers that internal points are surrounded by neighboring points in all directions, while neighboring points of a boundary point tend to be distributed only in a certain directional range. LoDD adopts a density-independent K-Nearest Neighbors (KNN) method to determine neighboring points, and defines a statistic-based metric using the eigenvalues of the covariance matrix of KNN coordinates to measure the centrality of a query point. We demonstrated the validity of LoDD on five synthetic datasets (2-D and 3-D) and ten real-world benchmarks, and tested its clustering performance by equipping with two typical clustering methods, K-means and Ncut. Our results show that LoDD achieves promising and robust detection accuracy in a time-efficient manner.

Uncertainty-Aware Multi-View Visual Semantic Embedding

The key challenge in image-text retrieval is effectively leveraging semantic information to measure the similarity between vision and language data. However, using instance-level binary labels, where each image is paired with a single text, fails to capture multiple correspondences between different semantic units, leading to uncertainty in multi-modal semantic understanding. Although recent research has captured fine-grained information through more complex model structures or pre-training techniques, few studies have directly modeled uncertainty of correspondence to fully exploit binary labels. To address this issue, we propose an Uncertainty-Aware Multi-View Visual Semantic Embedding (UAMVSE)} framework that decomposes the overall image-text matching into multiple view-text matchings. Our framework introduce an uncertainty-aware loss function (UALoss) to compute the weighting of each view-text loss by adaptively modeling the uncertainty in each view-text correspondence. Different weightings guide the model to focus on different semantic information, enhancing the model's ability to comprehend the correspondence of images and texts. We also design an optimized image-text matching strategy by normalizing the similarity matrix to improve model performance. Experimental results on the Flicker30k and MS-COCO datasets demonstrate that UAMVSE outperforms state-of-the-art models.

Forecast Network-Wide Traffic States for Multiple Steps Ahead: A Deep Learning Approach Considering Dynamic Non-Local Spatial Correlation and Non-Stationary Temporal Dependency

Obtaining accurate information about future traffic flows of all links in a traffic network is of great importance for traffic management and control applications. This research studies two particular problems in traffic forecasting: (1) capture the dynamic and non-local spatial correlation between traffic links and (2) model the dynamics of temporal dependency for accurate multiple steps ahead predictions. To address these issues, we propose a deep learning framework named Spatial-Temporal Sequence to Sequence model (STSeq2Seq). This model builds on sequence to sequence (seq2seq) architecture to capture temporal feature and relies on graph convolution for aggregating spatial information. Moreover, STSeq2Seq defines and constructs pattern-aware adjacency matrices (PAMs) based on pair-wise similarity of the recent traffic patterns on traffic links and integrate it into graph convolution operation. It also deploys a novel seq2sesq architecture which couples a convolutional encoder and a recurrent decoder with attention mechanism for dynamic modeling of long-range dependence between different time steps. We conduct extensive experiments using two publicly-available large-scale traffic datasets and compare STSeq2Seq with other baseline models. The numerical results demonstrate that the proposed model achieves state-of-the-art forecasting performance in terms of various error measures. The ablation study verifies the effectiveness of PAMs in capturing dynamic non-local spatial correlation and the superiority of proposed seq2seq architecture in modeling non-stationary temporal dependency for multiple steps ahead prediction. Furthermore, qualitative analysis is conducted on PAMs as well as the attention weights for model interpretation.

A Hybrid Traffic Speed Forecasting Approach Integrating Wavelet Transform and Motif-based Graph Convolutional Recurrent Neural Network

Traffic forecasting is crucial for urban traffic management and guidance. However, existing methods rarely exploit the time-frequency properties of traffic speed observations, and often neglect the propagation of traffic flows from upstream to downstream road segments. In this paper, we propose a hybrid approach that learns the spatio-temporal dependency in traffic flows and predicts short-term traffic speeds on a road network. Specifically, we employ wavelet transform to decompose raw traffic data into several components with different frequency sub-bands. A Motif-based Graph Convolutional Recurrent Neural Network (Motif-GCRNN) and Auto-Regressive Moving Average (ARMA) are used to train and predict low-frequency components and high-frequency components, respectively. In the Motif-GCRNN framework, we integrate Graph Convolutional Networks (GCNs) with local sub-graph structures - Motifs - to capture the spatial correlations among road segments, and apply Long Short-Term Memory (LSTM) to extract the short-term and periodic patterns in traffic speeds. Experiments on a traffic dataset collected in Chengdu, China, demonstrate that the proposed hybrid method outperforms six state-of-art prediction methods.