Weighted quantization using MMD: From mean field to mean shift via gradient flows

2026-04-24 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences · Depth: Expert, extended

Summary

This research introduces novel algorithms, Weighted Quantization using MMD (WFR) and Mean Shift Interacting Particles (MSIP), for approximating probability distributions with a finite set of weighted particles, a fundamental task in machine learning and statistics. Unlike prior work that often relies on Wasserstein distance or fixed particle weights, this study minimizes Maximum Mean Discrepancy (MMD) within the Wasserstein–Fisher–Rao (WFR) geometry, allowing for variable particle weights. The WFR gradient flow yields an Ordinary Differential Equation (ODE) system, from which MSIP is derived as a fixed-point algorithm. MSIP extends the non-interacting mean shift algorithm and can be interpreted as preconditioned gradient descent, acting as a relaxation of Lloyd's algorithm for clustering. Numerical experiments on synthetic multi-modal and high-dimensional datasets, including MNIST, demonstrate that WFR and MSIP consistently outperform existing quantization algorithms, exhibiting superior robustness to initialization and faster convergence to near-optimal MMD quantizations.

Key takeaway

For Machine Learning Engineers and Research Scientists working on probability distribution approximation or clustering, adopting the Mean Shift Interacting Particles (MSIP) or Wasserstein-Fisher-Rao (WFR) Flow algorithms can significantly improve performance. Your models will achieve more robust and efficient quantization, especially for multi-modal and high-dimensional data, even with adversarial initializations, surpassing traditional methods like k-means and other MMD gradient flows.

Key insights

MMD minimization via WFR geometry and MSIP offers robust, efficient probability distribution quantization with variable particle weights.

Principles

• MMD provides computational advantages over Wasserstein distances.
• WFR geometry offers flexibility for MMD minimization by allowing mass transport and variation.
• Fixed-point iterations can extend classical algorithms like mean shift for interacting particle systems.

Method

The method involves minimizing MMD via gradient flow in WFR geometry, yielding an ODE system. A fixed-point algorithm, MSIP, is derived from this ODE system to target steady-state solutions.

In practice

• Use MSIP for robust quantization of multi-modal distributions.
• Apply WFR ODEs for high-dimensional approximation problems.
• Consider MSIP for faster convergence in MMD quantization tasks.

Topics

• Weighted Quantization
• Maximum Mean Discrepancy
• Wasserstein–Fisher–Rao Geometry
• Mean Shift Interacting Particles
• Gradient Flow Optimization

Code references

• AyoubBelhadji/disruptive_quantization

Best for: AI Engineer, AI Scientist, Research Scientist, Machine Learning Engineer

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.