Local Component Analysis

Kernel density estimation, a.k.a. Parzen windows, is a popular density estimation method, which can be used for outlier detection or clustering. With multivariate data, its performance is heavily reliant on the metric used within the kernel. Most earlier work has focused on learning only the bandwidth of the kernel (i.e., a scalar multiplicative factor). In this paper, we propose to learn a full Euclidean metric through an expectation-minimization (EM) procedure, which can be seen as an unsupervised counterpart to neighbourhood component analysis (NCA). In order to avoid overfitting with a fully nonparametric density estimator in high dimensions, we also consider a semi-parametric Gaussian-Parzen density model, where some of the variables are modelled through a jointly Gaussian density, while others are modelled through Parzen windows. For these two models, EM leads to simple closed-form updates based on matrix inversions and eigenvalue decompositions. We show empirically that our method leads to density estimators with higher test-likelihoods than natural competing methods, and that the metrics may be used within most unsupervised learning techniques that rely on such metrics, such as spectral clustering or manifold learning methods. Finally, we present a stochastic approximation scheme which allows for the use of this method in a large-scale setting.

💡 Research Summary

The paper “Local Component Analysis” addresses a fundamental problem in unsupervised learning: the choice of a distance metric that governs many algorithms such as clustering, spectral embedding, and outlier detection. While supervised metric learning has been extensively studied, unsupervised methods typically rely on naïve choices like the Mahalanobis distance derived from Principal Component Analysis (PCA). PCA, however, whitens data globally and ignores local structure, which is crucial for methods that operate on neighborhoods.

The authors propose to learn a full Euclidean metric (i.e., a positive‑definite covariance matrix Σ) directly from the data by treating kernel density estimation (Parzen windows) as a probabilistic model and maximizing a leave‑one‑out log‑likelihood. Because the ordinary log‑likelihood would drive Σ to zero (over‑fitting to Dirac spikes), they derive a variational lower bound using Jensen’s inequality and introduce responsibility variables λ_{ij}. This yields an Expectation‑Maximization (EM) algorithm:

• E‑step: Compute λ_{ij}= N(x_i, x_j; Σ) / Σ_{k≠i} N(x_i, x_k; Σ), where N denotes a Gaussian kernel with covariance Σ. λ_{ij} measures the relative proximity of point j to i.
• M‑step: Update Σ as the weighted average of pairwise outer products: Σ = (1/n) Σ_{i,j} λ_{ij} (x_i−x_j)(x_i−x_j)^T.

These updates have a closed‑form and are guaranteed to increase the bound, converging to a stationary point of the leave‑one‑out objective. The resulting transformation makes the data locally isotropic, in contrast to PCA’s global isotropy. The authors call this procedure Local Component Analysis (LCA).

Recognizing that Parzen windows over‑fit in high dimensions, the paper introduces a semi‑parametric Gaussian‑Parzen model. The data are linearly transformed by an invertible matrix B =