dimensionality reduction

In unsupervised anomaly detection on high-dimensional data, dimensionality reduction often obscures the geometric distribution of anomalies, hindering discriminative capability. Method: We propose a “manifold-in/manifold-out” binary anomaly classification framework. Innovatively adopting a manifold-aware paradigm, we characterize the intrinsic geometric nature of anomalies on the learned low-dimensional manifold and design a manifold-aligned multi-method fusion strategy that synergistically integrates models such as Isolation Forest. Contribution/Results: Our approach maintains precision while significantly improving recall—achieving a 16% gain over the best-performing single model (Isolation Forest) on MNIST. Extensive experiments demonstrate strong generalization and robustness in real-world high-dimensional scenarios. By explicitly leveraging manifold geometry, our work establishes a novel geometric-structure-driven paradigm for anomaly detection.

To address the challenge of detecting weak signals and rare anomalies in high-dimensional data, this paper proposes a sparse, self-organizing local kernel anomaly detection framework. The method adaptively partitions statistically imbalanced regions in the representation space, enabling efficient anomaly localization under extremely low supervision—e.g., fully unsupervised or minimally labeled settings. Innovatively integrating sparsity, locality, and competitive learning, it constructs an interpretable and scalable self-organizing kernel model. Technical components include Gaussian kernel ensembles, semi-supervised Neyman–Pearson learning, and local likelihood ratio modeling to precisely characterize statistical discrepancies between anomalous and normal samples. Experiments demonstrate that the framework achieves accurate identification of statistically significant anomalous regions in thousands-dimensional spaces using only a small number of kernels. It substantially outperforms state-of-the-art methods on scientific discovery, novelty detection, intrusion identification, and generative model validation tasks.

Existing hierarchical dimensionality reduction methods struggle to simultaneously preserve both local and global structures across multiple granularity levels while maintaining users’ mental map continuity. To address this, we propose HUMAP—the first hierarchical dimensionality reduction framework built upon UMAP. HUMAP achieves dual optimization of structural fidelity and mental map consistency through three core techniques: cross-scale neighborhood graph propagation, multi-granularity similarity modeling, and incremental embedding alignment. Compared to state-of-the-art methods, HUMAP significantly improves hierarchical structure preservation across multiple benchmark datasets. Furthermore, it demonstrates strong interpretability and interactive utility in real-world data labeling tasks. By unifying geometric structure preservation with cognitive consistency, HUMAP establishes a novel paradigm for multi-granularity visual analytics, enabling scalable, intuitive, and semantically grounded exploration of high-dimensional hierarchical data.

This work addresses anomaly detection on low-dimensional manifold-structured data. We propose an efficient isolation-based method that embeds data into a high-dimensional semantic-enhanced preference space and employs Locality-Sensitive Hashing (LSH) to accelerate sparse neighborhood estimation, thereby identifying the most isolated samples as anomalies. To our knowledge, this is the first approach to integrate LSH into a preference-space isolation framework, achieving both theoretical soundness and computational efficiency. Extensive experiments on multiple benchmark datasets demonstrate state-of-the-art detection performance, with inference speed improved by 3–5× over existing methods, alongside substantial reductions in time and memory overhead. The source code is publicly available.

This work addresses the lack of a universal statistical interpretation for the manifold hypothesis—that high-dimensional data approximately reside on low-dimensional manifolds. We propose the Latent Metric Model (LMM), a generative framework grounded in fundamental statistical concepts: latent variables, variable dependence, and stationarity—providing the first unified statistical justification for the manifold assumption. Methodologically, LMM integrates neighborhood graph construction, spectral analysis, and an interpretable inference framework to enable unsupervised manifold discovery and geometric structure recovery under weak priors. Experiments demonstrate that complex manifold geometries naturally emerge from minimal statistical mechanisms; LMM significantly reduces reliance on hand-crafted priors on both synthetic and real-world datasets, while enabling interpretable reconstruction of manifold dimensionality, curvature, and coordinate systems.

This work addresses the non-uniqueness inherent in dimensionality reduction of high-dimensional data, where a single embedding often fails to simultaneously preserve local structure, ensure interpretability, and align with external knowledge. To overcome this limitation, the paper proposes modeling the Rashomon set of dimensionality reduction—i.e., the collection of multiple equally valid, high-quality embeddings—and extracting stable adjacency relationships from this set to enhance embedding robustness. Methodologically, it introduces PCA-guided alignment to improve axis interpretability, incorporates concept-alignment regularization to integrate external semantic knowledge, and jointly optimizes the embedding geometry. The resulting visualizations maintain strong local and global structural fidelity while significantly improving interpretability and alignment with user-defined objectives.

Unsupervised anomaly detection faces three key challenges: conflicting distributional assumptions, low computational efficiency, and difficulty in unifying diverse anomaly types. To address these, we propose ISER—a novel framework that encodes local density via hypersphere radius and establishes a similarity-based anomaly scoring mechanism. ISER introduces a theoretically grounded anomaly reference pattern to ensure consistency in pattern evaluation and refines the Isolation Forest scoring function to mitigate axis-aligned bias and improve detection of local anomalies. Crucially, ISER achieves efficient and robust detection while maintaining linear time and constant space complexity. Extensive experiments across 22 real-world datasets demonstrate that ISER significantly outperforms 11 state-of-the-art baseline methods, validating its effectiveness, generality, and scalability.

High-dimensional anomaly detection suffers from the “curse of dimensionality,” rendering conventional methods ineffective; existing approaches often compromise interpretability or computational efficiency. This paper introduces a novel outlierness statistic based on “distance-to-distance,” leveraging the asymptotic concentration of pairwise distances and inner products in high-dimensional spaces—transforming the curse of dimensionality into a discriminative advantage that ensures asymptotic separability between anomalies and inliers. We theoretically establish the existence of a non-vanishing separation boundary for this statistic in high dimensions. Furthermore, we propose a distribution-free random rotation testing framework, requiring no parametric assumptions and exhibiting strong robustness. Experiments on synthetic data and diverse real-world high-dimensional datasets—including gene expression profiles and image features—demonstrate that our method significantly outperforms state-of-the-art baselines: it achieves substantially higher recall while maintaining low false positive rates, combining statistical rigor, computational feasibility, and result interpretability.

Existing anomaly detection methods typically assume that normal data occupy a non-zero volume in the ambient space, overlooking their intrinsic geometric structure as lying on a low-dimensional manifold, which limits performance. This work proposes a novel manifold projection paradigm: learning a projection operator that maps inputs onto the manifold of normal samples and using the projection residual as the anomaly criterion. By avoiding explicit modeling of the degenerate data distribution, the approach prevents misclassifying rare yet normal instances and provides a unified explanation for both the effectiveness and failure modes of reconstruction-based methods. Extensive experiments demonstrate that the proposed framework significantly outperforms conventional boundary-learning approaches and achieves state-of-the-art results across multiple benchmarks compared to existing reconstruction-based models.

Detecting novelties in high-dimensional, noisy scientific big data remains challenging, and existing anomaly detection methods often lack statistical rigor in declaring anomalies. Method: This paper proposes an end-to-end framework integrating contrastive representation learning with nonparametric two-sample hypothesis testing. It innovatively combines contrastive pre-trained embeddings with the New Physics Learning Machine (NPLM), augmented by simulation-based data augmentation and low-dimensional feature extraction. Contribution/Results: The framework enables highly sensitive detection of subtle anomalous signals and supports quantifiable, p-value-driven scientific discovery. Evaluated on real and synthetic datasets across astronomy, physics, and biology, it significantly outperforms state-of-the-art anomaly detection methods, demonstrating strong robustness to noise, interpretability via statistically grounded inference, and cross-domain generalizability.