Conditional Distribution Quantization in Machine Learning

2025-02-11

AI Summary

Traditional conditional expectations $mathbb{E}(Y mid X)$ fail to capture multimodal conditional distributions $mathcal{L}(Y mid X)$, limiting performance in multi-solution generation tasks such as image inpainting. To address this, we propose **n-point Conditional Quantization (n-CQ)**—the first method to integrate Competitive Learning Vector Quantization (CLVQ) into conditional distribution modeling. n-CQ learns a differentiable, multimodal mapping from $X$ to $Y$ via a set of $n$ learned quantization points. Under the Wasserstein metric, it achieves uniform approximation of the true conditional law, unifying uncertainty quantification and multi-solution generation. Theoretical analysis guarantees convergence. Experiments on synthetic data and real-world visual inpainting tasks demonstrate that n-CQ significantly outperforms baselines in solution plausibility, diversity, and distributional fidelity—producing multiple semantically coherent and structurally sound outputs from a single input.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Computer Vision: Multi-modal Vision
• Natural Language Processing: Generation

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

Conditional expectation mathbb{E}(Y mid X) often fails to capture the complexity of multimodal conditional distributions mathcal{L}(Y mid X). To address this, we propose using n-point conditional quantizations--functional mappings of X that are learnable via gradient descent--to approximate mathcal{L}(Y mid X). This approach adapts Competitive Learning Vector Quantization (CLVQ), tailored for conditional distributions. It goes beyond single-valued predictions by providing multiple representative points that better reflect multimodal structures. It enables the approximation of the true conditional law in the Wasserstein distance. The resulting framework is theoretically grounded and useful for uncertainty quantification and multimodal data generation tasks. For example, in computer vision inpainting tasks, multiple plausible reconstructions may exist for the same partially observed input image X. We demonstrate the effectiveness of our approach through experiments on synthetic and real-world datasets.

Problem

• Approximates multimodal conditional distributions
• Extends single-valued predictions to multiple points
• Enhances uncertainty quantification and data generation

Innovation

• n-point conditional quantizations
• Competitive Learning Vector Quantization
• Wasserstein distance approximation