基于SDP的过完备独立成分分析
Overcomplete Independent Component Analysis via SDP
摘要：
提出了一种新的过完备独立分量分析（ICA）算法，其中潜在源的数目k超过观测变量的维数p。以往的算法要么计算复杂度高，要么对混合矩阵的形式做了很强的假设。我们的算法不做任何稀疏性假设，但享有良好的计算和理论性能。我们的算法包括两个主要步骤：（a）估计的累积量生成函数的海森（相对于大多数算法使用的四阶和高阶累积量）和（b）一个新的半定规划（SDP）松弛恢复混合成分。我们表明，这种松弛可以有效地解决与投影加速梯度下降法，这使得整个算法计算实用。此外，我们推测，所提出的程序恢复的混合成分的速率k < p^2/4，并证明了混合成分可以恢复的概率高，当k <（2 -n）p log p时，原始成分是均匀随机采样的超球。实验上提供的合成数据和CIFAR-10数据集的真实图像。
英文摘要：
We present a novel algorithm for overcomplete independent components analysis (ICA), where the number of latent sources k exceeds the dimension p of observed variables. Previous algorithms either suffer from high computational complexity or make strong assumptions about the form of the mixing matrix. Our algorithm does not make any sparsity assumption yet enjoys favorable computational and theoretical properties. Our algorithm consists of two main steps: (a) estimation of the Hessians of the cumulant generating function (as opposed to the fourth and higher order cumulants used by most algorithms) and (b) a novel semi-definite programming (SDP) relaxation for recovering a mixing component. We show that this relaxation can be efficiently solved with a projected accelerated gradient descent method, which makes the whole algorithm computationally practical. Moreover, we conjecture that the proposed program recovers a mixing component at the rate k < p^2/4 and prove that a mixing component can be recovered with high probability when k < (2 - epsilon) p log p when the original components are sampled uniformly at random on the hyper sphere. Experiments are provided on synthetic data and the CIFAR-10 dataset of real images.
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