Interpretable Dimensionality Reduction by Feature Preserving Manifold Approximation and Projection

TL;DR

This work addresses the lack of interpretability in nonlinear dimensionality reduction by introducing featMAP, a method that preserves source features in embeddings through tangent-space analysis.featMAP constructs local tangent spaces via local SVD, aligns tangent bases between neighboring points, and embeds the tangent frame with an anisotropic projection to preserve local density. The approach yields interpretable embeddings by displaying feature loadings and feature importance in the embedding tangent space, with demonstrations on MNIST digits, COIL-20 objects, and MNIST adversarial examples, showing meaningful explanations for classifications and misclassifications while maintaining competitive structure preservation. Overall, featMAP provides a plug-in, interpretable extension to manifold learning that makes nonlinear DR more transparent and actionable for visualization and analysis.

Abstract

Nonlinear dimensionality reduction lacks interpretability due to the absence of source features in low-dimensional embedding space. We propose an interpretable method featMAP to preserve source features by tangent space embedding. The core of our proposal is to utilize local singular value decomposition (SVD) to approximate the tangent space which is embedded to low-dimensional space by maintaining the alignment. Based on the embedding tangent space, featMAP enables the interpretability by locally demonstrating the source features and feature importance. Furthermore, featMAP embeds the data points by anisotropic projection to preserve the local similarity and original density. We apply featMAP to interpreting digit classification, object detection and MNIST adversarial examples. FeatMAP uses source features to explicitly distinguish the digits and objects and to explain the misclassification of adversarial examples. We also compare featMAP with other state-of-the-art methods on local and global metrics.

Figures (10)

• Figure 1: FeatMAP preserving source features. FeatMAP embeds the digit $1$ group of MNIST to two-dimensional space (top-left). One randomly selected data point (in red) is associated with the embedding tangent space ($\mathit{span}(v_1, v_2)$) showing the source features (top-$10$ annotated) of every pixel (top-right). The feature importance computed by the feature loadings is mapped to the selected image (bottom-right), detecting the digit edge.
• Figure 2: The framework of featMAP. With $n$-dimensional data (top-left) as input, featMAP first constructs the topological space by $k$NN graph (step $1$, top-middle), followed by compute the tangent space by local SVD (step $2$, bottom-middle) and embed the tangent space to preserve alignment (\ref{['fig:gauge_emb']}). Along the embedding tangent space, featMAP applies anisotropic projection (step $3$) to embedding the data to low-dimensional space (right), which locally retains the source features in embedding tangent space. Data points with darker blue indicate denser patterns, and vice versa.
• Figure 3: Tangent space embedding. The tangent spaces of original data points $x_i$ and $x_j$ are associated with basis vectors ($v_1$ and $v_2$ annotated, on the left); the transformation from $i$ to $j$ consists of translation by geodesic distance $d_{M}(i,j)$ and rotation $O_{ij}$ with the general angle $\Theta_{ij}$ (bottom-left). The embedding tangent space (on the right) is computed by preserving the rotation angle $\Theta_{ij}$.
• Figure 4: FeatMAP on MNIST showing feature importance. FeatMAP embeds MNIST to $2$-dimensional space with $10$ different clusters (left). Digit images are randomly selected from each cluster illustrating the feature importance with corresponding original images (right). Darker red means larger feature importance.
• Figure 5: FeatMAP on Fashion MNIST and COIL-20 showing feature importance. For each dataset, the upper part is the saliency map by feature importance and the bottom are the original images.

Theorems & Definitions (2)

• Theorem 3.1: singer2012vectorlim2021tangent
• Definition 3.1: Feature importance