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

Ramin Goudarzi Karim, Dipak Dulal, Carmeliza Navasca

TL;DR

This paper tackles efficient CP decomposition for image compression by recasting CP as a nonlinear least squares problem. It proposes a Modified Levenberg-Marquardt algorithm that computes the LM direction $h_k$ and an approximate direction $ ilde{h}_k$ by solving $(J_k^T J_k + mu_k I) ilde{h}_k = - J_k^T F(y_k)$ with $y_k = x_k + h_k$, and forms $s_k = h_k + ilde{h}_k$ for acceptance via a gain ratio. The current Jacobian $J_k$ is reused to avoid recomputation, decreasing memory and time relative to the standard LM. Empirical results on RGB images (Rose, Pepper, Leopard) and random tensors show comparable reconstruction quality with substantially lower CPU time, using compression defined by $R(I+J+K)/(IJK)$. The work demonstrates a scalable approach for low-rank CP-based image compression and points to future work on sampling for massive mode dimensions and theoretical convergence guarantees.

Abstract

This paper explores a new version of the Levenberg-Marquardt algorithm used for Tensor Canonical Polyadic (CP) decomposition with an emphasis on image compression and reconstruction. Tensor computation, especially CP decomposition, holds significant applications in data compression and analysis. In this study, we formulate CP as a nonlinear least squares optimization problem. Then, we present an iterative Levenberg-Marquardt (LM) based algorithm for computing the CP decomposition. 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 when compared to the traditional Levenberg-Marquardt technique.

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

TL;DR

This paper tackles efficient CP decomposition for image compression by recasting CP as a nonlinear least squares problem. It proposes a Modified Levenberg-Marquardt algorithm that computes the LM direction and an approximate direction by solving with , and forms for acceptance via a gain ratio. The current Jacobian is reused to avoid recomputation, decreasing memory and time relative to the standard LM. Empirical results on RGB images (Rose, Pepper, Leopard) and random tensors show comparable reconstruction quality with substantially lower CPU time, using compression defined by . The work demonstrates a scalable approach for low-rank CP-based image compression and points to future work on sampling for massive mode dimensions and theoretical convergence guarantees.

Abstract

This paper explores a new version of the Levenberg-Marquardt algorithm used for Tensor Canonical Polyadic (CP) decomposition with an emphasis on image compression and reconstruction. Tensor computation, especially CP decomposition, holds significant applications in data compression and analysis. In this study, we formulate CP as a nonlinear least squares optimization problem. Then, we present an iterative Levenberg-Marquardt (LM) based algorithm for computing the CP decomposition. 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 when compared to the traditional Levenberg-Marquardt technique.
Paper Structure (4 sections, 1 theorem, 18 equations, 3 figures, 3 tables, 1 algorithm)

This paper contains 4 sections, 1 theorem, 18 equations, 3 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Problem Formulation
  2. A Modified Levenberg-Marquardt Algorithm for CP Decomposition
  3. Numerical Experiments
  4. Conclusion

Key Result

Lemma 1

acar2011scalable The Jacobian matrix, $J$, of the function $F$ in (nlsproblem) can be expressed as: Each component, $J_a$, $J_b$, and $J_c$, can be further decomposed as: The individual matrices, $J_a^r$, $J_b^r$, and $J_c^r$, are defined as:

Figures (3)

  • Figure 1: (a) The sparsity of the Jacobian matrix for a $3 \times 4 \times 5$ tensor with $R = 2$. The red points indicate the nonzero entries of $J$. (b) The symmetry and sparsity of the $J^TJ$ for a $3 \times 4 \times 5$ tensor with $R = 2$. The blue points indicate the nonzero entries.
  • Figure 2: Modified LM: The performance of the proposed algorithm in terms of accuracy of the residual error for (a) a randomly generated tensor ($45\times35\times 25),$ R=40, 89% compression and (b) Leopard image R = 50, 79% compression.
  • Figure 3: Modified LM: Rose($100\times 100\times 3$), Pepper ($168\times 168\times 3$) and Leopard($162\times 162\times 3$).

Theorems & Definitions (1)

  • Lemma 1