Estimating the intrinsic dimension of datasets by a minimal neighborhood information (1803.06992v1)

Abstract: Analyzing large volumes of high-dimensional data is an issue of fundamental importance in data science, molecular simulations and beyond. Several approaches work on the assumption that the important content of a dataset belongs to a manifold whose Intrinsic Dimension (ID) is much lower than the crude large number of coordinates. Such manifold is generally twisted and curved, in addition points on it will be non-uniformly distributed: two factors that make the identification of the ID and its exploitation really hard. Here we propose a new ID estimator using only the distance of the first and the second nearest neighbor of each point in the sample. This extreme minimality enables us to reduce the effects of curvature, of density variation, and the resulting computational cost. The ID estimator is theoretically exact in uniformly distributed datasets, and provides consistent measures in general. When used in combination with block analysis, it allows discriminating the relevant dimensions as a function of the block size. This allows estimating the ID even when the data lie on a manifold perturbed by a high-dimensional noise, a situation often encountered in real world data sets. We demonstrate the usefulness of the approach on molecular simulations and image analysis.

• The paper introduces a minimal neighborhood TWO-NN estimator that accurately computes intrinsic dimensions using only the first two neighbor distances.
• It demonstrates theoretical exactness for uniform distributions and shows asymptotic convergence in diverse datasets, including molecular simulations and image data.
• The method's low computational cost and robust performance enable effective dimensionality reduction, offering actionable insights for high-dimensional data analysis.

Analyzing the Intrinsic Dimension of High-Dimensional Data with the TWO-NN Estimator

The estimation of the intrinsic dimension (ID) of datasets is a critical problem in the domain of data science and beyond, influencing several applications such as molecular simulations and image analysis. The paper by Facco et al. proposes a novel approach for estimating ID using a minimal neighborhood information strategy, which employs only the distances to the first and second nearest neighbors—the TWO-NN estimator. This approach aims to circumvent issues posed by twisted manifold geometry and non-uniform point distributions, commonly faced in high-dimensional data analysis.

The TWO-NN estimator offers several advantages over traditional methods that often require larger neighborhood sizes, thus making them susceptible to curvature and density variations. By confining the neighborhood to just two neighbors, the estimator achieves robustness against these variables and reduces computational costs. The paper further demonstrates that this method is theoretically exact for uniformly distributed datasets and shows consistency in more general conditions.

One of the core contributions of the paper is an equation derived for ID estimation, which remains independent of local density variations, a significant improvement over existing methods. The empirical performance of the TWO-NN estimator is thoroughly validated across diverse datasets, including uniform distributions on hypercubes, Swiss Roll datasets, and Cauchy distributions. The results provide compelling evidence of the estimator's efficacy, demonstrating asymptotic convergence to the true dimension as dataset size increases. This behavior is crucial for practical applications where sample sizes are large but finite.

The paper also explores scale-dependent intrinsic dimensions by linking the estimator’s locality to its ability to handle multiscale problems effectively. This is achieved through a block analysis approach that detects meaningful dimensions corresponding to real signal directions amidst high-dimensional noise. Experimental results reveal that the ID of complex real-world datasets, such as configurations from molecular dynamics and image datasets like MNIST and Isomap faces, can be reliably estimated with TWO-NN.

In practical terms, the implications of employing the TWO-NN estimator are profound. It provides a robust and computationally efficient strategy for dimensionality reduction, thereby facilitating the application of numerous machine learning and data analysis techniques. This is particularly valuable in scenarios where the dataset's dimensionality hampers algorithmic performance due to the curse of dimensionality.

Theoretical implications extend to the mathematical understanding of manifold structures within high-dimensional spaces. By reducing the dependency on local point density, the TWO-NN estimator offers insights into the geometric properties of data distributions, enabling more precise characterizations of data manifold structures.

Looking forward, potential developments could include the extension of this approach to more complex data structures and the integration of TWO-NN within pipeline processes for automated feature extraction and dimensionality reduction. Additionally, further exploration into hybrid approaches combining TWO-NN with other dimensionality reduction techniques might yield superior performance across a broader scope of applications.

In conclusion, Facco et al.'s TWO-NN estimator presents a significant advancement in the field of intrinsic dimension estimation. Its minimalistic yet powerful approach underscores the potential for innovative solutions to longstanding challenges in high-dimensional data analysis. As such, it holds promise for widespread application and further research in both theoretical and applied data sciences.