We consider a linear autoencoder in which the latent variables are quantized, or corrupted by noise, and the constraint is Schur-concave in the set of latent variances. Although finding the optimal encoder/decoder pair for this setup is a nonconvex optimization problem, we show that decomposing the source into its principal components is optimal. If the constraint is strictly Schur-concave and the empirical covariance matrix has only simple eigenvalues, then any optimal encoder/decoder must decompose the source in this way. As one application, we consider a strictly Schur-concave constraint that estimates the number of bits needed to represent the latent variables under fixed-rate encoding, a setup that we call \emph{Principal Bit Analysis (PBA)}. This yields a practical, general-purpose, fixed-rate compressor that outperforms existing algorithms. As a second application, we show that a prototypical autoencoder-based variable-rate compressor is guaranteed to decompose the source into its principal components.

中文翻译：

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原理位分析：具有 Schur-Concave 损失的自动编码

我们考虑一个线性自编码器，其中潜在变量被量化或被噪声破坏，并且约束是潜在方差集中的 Schur-concave。尽管为此设置找到最佳编码器/解码器对是一个非凸优化问题，但我们表明将源分解为其主要组件是最佳的。如果约束是严格的 Schur-concave 并且经验协方差矩阵只有简单的特征值，那么任何最佳编码器/解码器都必须以这种方式分解源。作为一个应用，我们考虑了一个严格的 Schur-concave 约束，它估计在固定速率编码下表示潜在变量所需的位数，我们称之为 \emph{Principal Bit Analysis (PBA)} 的设置。这产生了一种实用的、通用的、固定速率的压缩器，其性能优于现有算法。