[Paper Review] Optimal quantization of the mean measure and applications to statistical learning

Frédéric Chazal, Clément Levrard+1 more|arXiv (Cornell University)|Feb 4, 2020

Topological and Geometric Data Analysis4 citations

TL;DR

This paper proposes optimal quantization of the mean measure of i.i.d. discrete measures (e.g., persistence diagrams or point processes) using batch and mini-batch algorithms that achieve almost minimax optimal convergence rates. It introduces a vectorization map into R^k based on quantized centroids, enabling provably effective clustering of measures in mixture models, with theoretical guarantees and empirical validation in topological data analysis and classification tasks.

ABSTRACT

This paper addresses the case where data come as point sets, or more generally as discrete measures. Our motivation is twofold: first we intend to approximate with a compactly supported measure the mean of the measure generating process, that coincides with the intensity measure in the point process framework, or with the expected persistence diagram in the framework of persistence-based topological data analysis. To this aim we provide two algorithms that we prove almost minimax optimal. Second we build from the estimator of the mean measure a vectorization map, that sends every measure into a finite-dimensional Euclidean space, and investigate its properties through a clustering-oriented lens. In a nutshell, we show that in a mixture of measure generating process, our technique yields a representation in $\\mathbb{R}^k$, for $k \\in \\mathbb{N}^*$ that guarantees a good clustering of the data points with high probability. Interestingly, our results apply in the framework of persistence-based shape classification via the ATOL procedure described in \\cite{Royer19}.

Motivation & Objective

• Address the challenge of clustering data represented as discrete measures (e.g., persistence diagrams, point patterns) rather than scalar or vector data.
• Develop a vectorization method that maps measures into a finite-dimensional Euclidean space while preserving cluster structure.
• Provide theoretical guarantees for clustering performance under a mixture model framework for measure-generating processes.
• Design computationally efficient algorithms for approximating the mean measure via optimal quantization, avoiding costly Wasserstein barycenter computation.
• Ensure the method applies broadly, including in topological data analysis via the ATOL procedure, with empirical validation on real and synthetic data.

Proposed method

• Use the sample mean measure $\bar{X}_n = \frac{1}{n}\sum_{i=1}^n X_i$ as a plug-in estimator for the true mean measure $\mathbb{E}(X)$.
• Apply batch and mini-batch quantization algorithms to compute a $k$-point codebook $\mathbf{c} = (c_1, \dots, c_k)$ that optimally approximates $\mathbb{E}(X)$ in the $L^2$-Wasserstein sense.
• Define a vectorization map that assigns each measure $X_i$ to a vector in $\mathbb{R}^k$ by evaluating kernel functions centered at the quantized points $c_j$, enabling finite-dimensional embedding.
• Establish theoretical optimality by proving the algorithms achieve almost minimax optimal rates for estimating the best $k$-point approximation of $\mathbb{E}(X)$, under structural assumptions on the mean measure.
• Leverage shattering and concentration arguments to prove that the resulting vectorization preserves cluster separation with high probability in mixture models.
• Connect the method to persistence-based topological data analysis by showing compatibility with the ATOL procedure and kernel-based vectorization schemes.

Experimental results

Research questions

• RQ1 Can we achieve minimax optimal estimation of the mean measure of a distribution of discrete measures using a finite-dimensional quantization scheme?
• RQ2 How can we design a vectorization map from measures to $\mathbb{R}^k$ that preserves cluster structure in a mixture model of measure-generating processes?
• RQ3 What theoretical guarantees can be provided for clustering performance when using quantized centroids as vectorization points?
• RQ4 How does the proposed batch and mini-batch quantization algorithm compare to classical $k$-means in terms of optimality and computational efficiency for measure-valued data?
• RQ5 To what extent does the method generalize to topological data analysis, particularly in the context of persistence diagram clustering?

Key findings

• The proposed batch and mini-batch quantization algorithms achieve almost minimax optimal convergence rates for estimating the best $k$-point approximation of the mean measure $\mathbb{E}(X)$, under mild structural assumptions.
• The vectorization map based on kernel evaluation at quantized centroids ensures that, in a mixture model, clusters in the original measure space are preserved in $\mathbb{R}^k$ with high probability.
• The method provides the first theoretical guarantee for a measure-based clustering algorithm in the context of topological data analysis, particularly for persistence diagrams.
• For the ATOL procedure, the kernel-based vectorization using $\Psi_{AT}$ is shown to be a $(p, r, 1)$-shattering map under appropriate conditions, ensuring cluster separation.
• Empirical results on synthetic and real datasets, including text and graph classification, confirm the effectiveness of the method in clustering and classification tasks.
• The theoretical framework also recovers the optimality of classical quantization techniques in the standard point-sample $k$-means setting.