提出可快速生成且可解释的新型生成模型，融合数学定理实现高效推断。
Kolmogorov-Arnold Energy Models: Fast, Interpretable Generative Modeling
- 基于柯尔莫哥洛夫-阿诺德定理构建一维潜在先验，支持快速精确推断。
- 在SVHN/CIFAR10/CelebA上达到与主流模型相当的生成质量，单次前向传播完成采样。
- 提供可解释的潜变量结构，适合需要透明性与高效推理的应用场景。
生成模型通常依赖简单的潜在先验（如变分自编码器，VAE），虽高效但能力受限；或采用表达能力强的迭代采样方法（如扩散模型和能量模型，EBM），却代价高昂且不透明。本文提出一种新的无监督模型——柯尔莫哥洛夫-阿诺德能量模型（KAEM），以弥合这一权衡并提升可解释性。基于柯尔莫哥洛夫-阿诺德表示定理的新颖适配，KAEM引入一维潜在先验，使逆变换法可实现快速、精确的推断。在小数据集上，重要性采样成为可行、无偏且单遍的后验推断方法。对于需探索的场景，提出基于种群的策略，将后验分解为一系列退火分布，作为解决能量模型混合不良的新方案。KAEM在SVHN、CIFAR10和CelebA上取得与潜先验模型相当的弗雷歇起始距离（FID）表现，采样仅需一次前向传播，成本低于迭代式EBM，并揭示由一维密度构成的可解释先验。
原文摘要 · Abstract (English)
Generative models typically rely on either simple latent priors (e.g., Variational Autoencoders, VAEs), which are efficient but limited, or expressive iterative samplers (e.g., Diffusion and Energy-based Models), which are costly and opaque. We introduce a new unsupervised model, the Kolmogorov-Arnold Energy Model (KAEM), to bridge this trade-off and provide new opportunities for interpretability. Based on a novel adaptation of the Kolmogorov-Arnold Representation Theorem, KAEM imposes a univariate latent prior, enabling fast and exact inference via the inverse transform method. On small datasets, we show that importance sampling becomes a tractable, unbiased, and single-pass posterior inference method. For settings requiring exploration, we propose a population-based strategy that decomposes the posterior into a sequence of annealed distributions, serving as a new remedy for poor mixing in Energy-based Models. KAEM attains competitive Fréchet Inception Distance among latent-prior models on SVHN, CIFAR10, and CelebA while sampling in a single forward pass at lower cost than iterative EBMs, and exposing an interpretable prior built from 1D densities.
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