Author: Ji Hyung Jung, Hye Won Chung, Ji Oon Lee
Conference and Year: ICML 2021
Keywords: Signal detection, Spiked Rectangular Model
We consider the problem of detecting signals in the rank-one signal-plus-noise data matrix models that generalize the spiked Wishart matrices. We show that the principal component analysis can be improved by pre-transforming the matrix entries if the noise is non-Gaussian. As an intermediate step, we prove a sharp phase transition of the largest eigenvalues of spiked rectangular matrices, which extends the BBP transition. We also propose a hypothesis test to detect the presence of signal with low computational complexity, based on the linear spectral statistics, which minimizes the sum of the Type-I and Type-II errors when the noise is Gaussian.
Figure1: We compare the reconstruction performance of the proposed PCA (top lines) and the standard PCA (bottom lines) for two FashionMNIST images, with the number of measurements N = [3136, 1568, 784, 588, 392] where the data dimension is M = 784. The left most column displays the original images for comparison.
Figure2: The histograms of the test statistic under null hypothesis H0 and alternative hypothesis H1, respectively, for the Gaussian noise with SNR ω = 0.35 and ω = 0.45. It can be shown that the difference of the means of the test statistic under H0 and H1 is larger for ω = 0.45.