Abstract
Recently, the sparse representation of multivariate data has gained great popularity in real-world applications like neural activity analysis. Many previous analyses for these data utilize sparse principal component analysis (SPCA) to obtain a sparse representation. However, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell _{0}$ </tex-math></inline-formula> -norm based SPCA suffers from non-differentiability and local optimum problems due to non-convex regularization. Additionally, extracting dependencies between task parameters and feature responses is essential for further analysis while SPCA usually generates components without demixing these dependencies. To address these problems, we propose a novel approach, demixed sparse principal component analysis (dSPCA), that relaxes the non-convex constraints into convex regularizers, e.g., <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell _{1}$ </tex-math></inline-formula> -norm and nuclear norm, and demixes dependencies of feature response on various task parameters by optimizing the loss function with marginalized data. The sparse and demixed components greatly improve the interpretability of the multivariate data. We also develop a parallel proximal algorithm to accelerate the optimization for hybrid regularizers based on our method. We provide theoretical analyses for error bound and convergency. We apply our method on simulated datasets to evaluate its time cost, the ability to explain the demixed information, and the ability to recover sparsity for the reconstructed data. Finally, we successfully separate the neural activity into different task parameters like stimulus or decision, and visualize the demixed components based on the real-world dataset.
Highlights
Multivariate data is often used to describe systems with multiple dimensions and with many real-world applications like describing neural firing rate in neuroscience [1]–[7] or weather changes in meteorology [8]
We propose a method called demixed principal component analysis, uses the demixing idea similar to dPCA, and relax the non-convex constraints into convex regularizers to reach the ideal sparsity of loadings
The parameter (λ, γ ) of demixed sparse principal component analysis (dSPCA) are choosen by grid search in
Summary
Multivariate data is often used to describe systems with multiple dimensions and with many real-world applications like describing neural firing rate in neuroscience [1]–[7] or weather changes in meteorology [8]. A promising way to analyze the sparse representation of multivariate data is referred to as sparse principal component analysis (SPCA). To improve the performance of the original SPCA based on the 0 norm when facing non-convex optimization, several methods are proposed [11]–[16]. We propose a novel method called dSPCA to simultaneously extract the demixed sparse and low-rank PCs from data. The sparse and low-rank representation with demixed dependencies on various task parameters explains the multivariate data in a more explicit way and we can better analyze the relationship between feature response and different task parameters. Different form the non-convex constrains resulting in messy local optimum problems, 1-norm and nuclear norm regularizers can offer an accurate and tractable approximation for different task parameters.
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