[Paper Review] A New Method for Performance Analysis in Nonlinear Dimensionality Reduction

Jiaxi Liang, Shojaeddin Chenouri+1 more|arXiv (Cornell University)|Nov 16, 2017

Face and Expression Recognition22 references3 citations

TL;DR

This paper introduces a local rank correlation measure to evaluate nonlinear dimensionality reduction methods, offering a robust, interpretable performance metric that handles skewed nearest-neighbor distributions in high dimensions. The method effectively identifies optimal tuning parameters and estimates intrinsic dimensionality, with strong alignment to visual assessment and quantitative validation on benchmark datasets like Swiss roll and sculpture face data.

ABSTRACT

In this paper, we develop a local rank correlation measure which quantifies the performance of dimension reduction methods. The local rank correlation is easily interpretable, and robust against the extreme skewness of nearest neighbor distributions in high dimensions. Some benchmark datasets are studied. We find that the local rank correlation closely corresponds to our visual interpretation of the quality of the output. In addition, we demonstrate that the local rank correlation is useful in estimating the intrinsic dimensionality of the original data, and in selecting a suitable value of tuning parameters used in some algorithms.

Motivation & Objective

• To develop a performance metric for nonlinear dimensionality reduction that is interpretable and robust to skewed nearest-neighbor distributions in high-dimensional data.
• To address the lack of a universal, unsupervised performance measure for dimensionality reduction algorithms.
• To enable reliable estimation of intrinsic dimensionality and optimal tuning parameters (e.g., neighborhood size K) in manifold learning algorithms.
• To provide a metric that correlates strongly with visual quality of low-dimensional embeddings.

Proposed method

• Proposes a local rank correlation measure based on ranking nearest neighbors in input and output spaces to assess neighborhood preservation.
• Uses both Spearman’s rho and Kendall’s tau as correlation metrics between local neighborhood structures in original and reduced spaces.
• Defines two components: $G^I_J$ (intrinsic correlation) and $G^O_J$ (outlier correlation), capturing local structure fidelity and noise sensitivity.
• Applies the measure across varying dimensionality and tuning parameters to evaluate algorithm performance.
• Employs the measure in reverse scree-plot fashion to detect intrinsic dimensionality by observing stability in correlation values beyond the true dimension.
• Validates the method on benchmark datasets (e.g., Swiss roll, sculpture face) using ISOMAP, LTSA, MVU, and others.

Experimental results

Research questions

• RQ1 How can we quantify the performance of nonlinear dimensionality reduction methods in the absence of ground-truth labels?
• RQ2 Can a rank-based local correlation measure provide a robust and interpretable alternative to distance-based metrics in high-dimensional, skewed data?
• RQ3 To what extent does the local rank correlation align with visual assessment of embedding quality?
• RQ4 Can the local rank correlation be used to estimate the intrinsic dimensionality of a manifold?
• RQ5 Can the local rank correlation guide the selection of optimal tuning parameters, such as neighborhood size K, in algorithms like ISOMAP?

Key findings

• The local rank correlation closely matches human visual interpretation of embedding quality, with peak performance at K=13 in the Swiss roll example.
• A sharp drop in local rank correlation at K=14 indicates a failure in neighborhood preservation, correctly identifying overfitting or instability in the algorithm.
• For the sculpture face dataset, the local rank correlation stabilizes beyond q=3, correctly estimating the intrinsic dimensionality as 3.
• The method successfully identifies optimal tuning parameters, such as K=13 in ISOMAP, where performance peaks before degradation.
• The local rank correlation remains stable when the reduced dimension exceeds the true intrinsic dimension, confirming its utility in intrinsic dimension estimation.
• The use of rank-based correlation mitigates issues from skewed nearest-neighbor distributions, enhancing robustness in high-dimensional settings.