β-VAE的隐变量维度可被调控，深度增加使前几维更高效但尾部重建变差。
$β$-VAEs as Effective Theories: Tolerance-Dependent Dimension
- 通过调节正则化强度实现隐变量的频谱截断，按重建效用排序坍缩低效维度
- 非线性结构使坍缩阈值与效用不再严格对应，但整体排序仍保持一致
- 深度增加导致前几维效率提升，但尾部细节恢复能力下降，存在头尾权衡
在β-VAE中，增强正则化强度会通过坍缩低效隐变量坐标起到频谱截断作用。在线性高斯VAE中，坍缩顺序与重建效用排名完全一致，因二者均由PCA谱决定。我们研究了在WorldClim数据集上训练的全连接非线性VAE中该图景是否依然成立。发现非线性交互使坍缩起始点发生位移和展宽，导致阈值与效用不再精确重合。然而，在已解析的秩范围内，共同排序关系仍被保留，因此频谱截断仍可视为效用截断，有效描述逻辑仍然成立。由此得到的有效维度曲线揭示了头尾权衡：增加深度使效用集中于前几维，但恶化尾部保真度。
原文摘要 · Abstract (English)
In a $β$-VAE, increasing the regularization strength acts as a spectral cutoff by collapsing low-utility latent coordinates. In the linear Gaussian VAE, the collapse order matches the ranking of reconstruction utilities exactly, because both are set by the PCA spectrum. We ask which parts of this picture survive in fully connected nonlinear VAEs trained on WorldClim. We find that nonlinear interactions shift and broaden collapse onsets, so thresholds no longer coincide exactly with utilities. However, the common ordering is preserved over the resolved ranks, so the spectral cutoff still acts as a utility cutoff and the effective-description logic carries through. The resulting effective-dimension curves reveal a head--tail tradeoff: increasing depth concentrates utility into the first few coordinates but worsens tail fidelity.
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