A Modified Levenberg-Marquardt Algorithm For Tensor CP Decomposition in Image Compression

This paper proposes a new variant of the Levenberg-Marquardt (LM) algorithm used for third order tensor Canonical Polyadic (CP) decomposition with an emphasis on image compression and reconstruction. The optimization problem related to the CP decomposition can be formulated as follows:{equation*} {min }_{{a_r},{b_r},{c_r}} 1/2\| {{X} - ∑r = 1^R {{a_r}} ^∘ {b_r}^∘ {c_r}} \|_F^2{1}{equation*}where a <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</inf> ○ b <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</inf> ○ c <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</inf> is a rank one tensor via an outer product and ∥ · ∥ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</inf> represents the Frobenius norm. In this study, we formulate (1) as a nonlinear least squares optimization problem. Then, we present an iterative Levenberg-Marquardt (LM)-based algorithm for computing the CP decomposition. Our approach addresses the computational intensity typically associated with the Jacobian matrix related to the nonlinear least squares of (1) by making use of the current Jacobian to predict the Jacobian for future steps. This significantly reduces the time since calculating the Jacobian matrix is prohibitively expensive when the dimension of the data is large. Ultimately, we test the algorithm on various datasets, including randomly generated tensors and RGB images. The proposed method proves to be both efficient and effective, offering a reduced computational burden compared to the traditional LM technique. A more detailed version of this article can be found on arXiv. 1