We describe a method for unmixing mixtures of freely independent random variables in a manner analogous to the independent component analysis (ICA)-based method for unmixing independent random variables from their additive mixtures. Random matrices play the role of free random variables in this context so the method we develop, which we call free component analysis (FCA), unmixes matrices from additive mixtures of matrices. Thus, while the mixing model is standard, the novelty and difference in unmixing performance comes from the introduction of a new statistical criteria, derived from free probability theory, that quantify freeness analogous to how kurtosis and entropy quantify independence. We describe the theory, the various algorithms, and compare FCA to vanilla ICA which does not account for spatial or temporal structure. We highlight why the statistical criteria make FCA also vanilla despite its matricial underpinnings and show that FCA performs comparably to, and sometimes better than, (vanilla) ICA in every application, such as image and speech unmixing, where ICA has been known to succeed. Our computational experiments suggest that not-so-random matrices, such as images and short-time Fourier transform matrix of waveforms are (closer to being) freer “in the wild” than we might have theoretically expected.

我们描述了一种以类似于基于独立分量分析 (ICA) 的方法从它们的加性混合物中分离独立随机变量的方法来分离自由独立随机变量的混合物的方法。随机矩阵在这种情况下扮演自由随机变量的角色，因此我们开发的方法，我们称之为自由分量分析 (FCA)，将矩阵从矩阵的加性混合中分离出来。因此，虽然混合模型是标准的，但分解性能的新颖性和差异来自引入了一种新的统计标准，该标准源自自由概率论，该标准量化自由度，类似于峰度和熵如何量化独立性。我们描述了理论、各种算法，并将 FCA 与不考虑空间或时间结构的普通 ICA 进行了比较。我们强调了为什么统计标准使 FCA 也成为香草，尽管它有矩阵基础，并表明 FCA 在每个应用程序（例如图像和语音分离）中的表现与（香草）ICA 相当，有时甚至更好，而 ICA 在这些应用程序中众所周知是成功的。我们的计算实验表明，非随机矩阵，例如图像和波形的短时傅立叶变换矩阵，比我们理论上预期的“在野外”（更接近于）更自由。