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First-Order Perturbation Analysis of Low-Rank Tensor Approximations Based on the Truncated HOSVD


The truncated version of the higher-order singular value decomposition (HOSVD) has a great significance in multi-dimensional tensor-based signal processing. It allows extracting the principal components from noisy observations in order to find a low-rank approximation of the multi-dimensional data. In this work, we address the question of how good the approximation is by analytically quantifying the tensor reconstruction error introduced by the truncated HOSVD. We present a first-order perturbation analysis of the truncated HOSVD to obtain analytical expressions for the signal subspace error in each dimension as well as the tensor reconstruction error induced by the low-rank approximation of the noise corrupted tensor. The results are asymptotic in the signal-to-noise ratio (SNR) and expressed in terms of the second-order moments of the noise, such that apart from a zero mean, no assumptions on the noise statistics are required. Empirical simulation results verify the obtained analytical expressions

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