Hierarchical Nearest Neighbor Graph Embedding for Efficient Dimensionality Reduction (2203.12997v3)

Abstract: Dimensionality reduction is crucial both for visualization and preprocessing high dimensional data for machine learning. We introduce a novel method based on a hierarchy built on 1-nearest neighbor graphs in the original space which is used to preserve the grouping properties of the data distribution on multiple levels. The core of the proposal is an optimization-free projection that is competitive with the latest versions of t-SNE and UMAP in performance and visualization quality while being an order of magnitude faster in run-time. Furthermore, its interpretable mechanics, the ability to project new data, and the natural separation of data clusters in visualizations make it a general purpose unsupervised dimension reduction technique. In the paper, we argue about the soundness of the proposed method and evaluate it on a diverse collection of datasets with sizes varying from 1K to 11M samples and dimensions from 28 to 16K. We perform comparisons with other state-of-the-art methods on multiple metrics and target dimensions highlighting its efficiency and performance. Code is available at https://github.com/koulakis/h-nne

• The paper introduces h-NNE, a deterministic, optimization-free dimensionality reduction technique that leverages hierarchical 1-NN graphs.
• The method builds a hierarchical tree of centroids and employs a streamlined PCA with hierarchical point translation to preserve local and global structures.
• Experimental results show h-NNE achieves competitive trustworthiness and efficiency, outperforming traditional methods on large-scale datasets.

Hierarchical Nearest Neighbor Graph Embedding for Efficient Dimensionality Reduction

This essay provides an expert analysis of the research paper titled "Hierarchical Nearest Neighbor Graph Embedding for Efficient Dimensionality Reduction" by Sarfraz et al. The paper presents a novel dimension reduction method, h-NNE, which seeks to address the challenges of scalability and complexity posed by contemporary dimensionality reduction techniques. In this analysis, we will explore its methodological approaches, experimental evaluations, and its implications in the field of machine learning and data visualization.

Methodological Overview

The research introduces h-NNE, a technique grounded in hierarchical clustering derived from 1-nearest neighbor graphs (1-NNGs). The central innovation lies in eschewing optimization-based approaches common in methods like t-SNE and UMAP, opting instead for a deterministic clustering-based technique that provides a computationally efficient alternative. The authors construct a hierarchical tree structure from 1-NNGs, capturing both the local and global structure of the data without requiring the costly pairwise distance calculations typical of existing methods.

The process consists of a three-step algorithm:

1. Building a Hierarchical Structure: The data is represented through a hierarchy of centroids using a recursive application of 1-NNG to capture the data's clustering properties at various levels.
2. Preliminary Linear Projection: The algorithm employs a streamlined version of PCA by utilizing centroids to initialize a low-dimensional representation, reducing the associated computational complexity.
3. Hierarchical Point Translation: This final step refines the projection using the hierarchical data structure to preserve nearest neighbor relationships.

This optimization-free approach not only renders h-NNE suitable for large datasets but also bypasses the common need for hyperparameter tuning, thereby streamlining the dimensionality reduction process.

Experimental Results

The empirical evaluations carried out by the authors effectively highlight h-NNE’s computational efficiency and competitive performance. It is showcased across several datasets varying in size and dimensions, from COIL20 with 1,440 samples to the massive HIGGS dataset comprising 11 million entries.

Key performance metrics included:

• Trustworthiness: Demonstrating h-NNE’s capability to maintain local neighborhood structures, the method achieves comparable results to t-SNE and other state-of-the-art algorithms.
• Centroid Triplet Accuracy: Underlining its global structure preservation, h-NNE maintains competitive scores, indicating its robustness in maintaining relative distances between the centroids in the lower-dimensional space.

Perhaps the most striking aspect is the runtime comparison. h-NNE consistently outperforms traditional methods by orders of magnitude in terms of speed, clocking in performance speeds that make it a viable choice for real-time data analysis and visualization applications.

Theoretical and Practical Implications

The introduction of h-NNE has profound implications. Theoretically, it challenges the prevailing reliance on optimization-driven approaches, offering a simpler yet effective alternative that leverages fundamental properties of nearest neighbor graphs for dimensionality reduction. Practically, its efficiency and scalability are transformative for large-scale data applications where computational resources and time are constraints.

Furthermore, its hierarchy-based nature provides an intuitive alignment with clustering-based analysis, making it potentially attractive for exploratory data analysis tasks in domains such as bioinformatics, image processing, and natural language processing.

Future Directions

Future research could explore extending h-NNE through the integration of hybrid approaches that blend traditional methods with hierarchical structures to enhance robustness and applicability. Additionally, adapting h-NNE for other dimensionality reduction tasks, such as feature selection and manifold learning, could expand its utility within broader contexts.

In conclusion, this paper offers a significant advancement in dimensionality reduction methodologies. The balance it achieves between simplicity and performance marks it as a notable contribution to the repertoire of techniques available to researchers and practitioners. Its emphasis on scalability without sacrificing structural preservation is particularly noteworthy, promising considerable potential for future investigations and applications.