This paper studies a robust approach to the simultaneous approximate diagonalization (SADI) of symmetric third-order tensors. Because \(\ell_{2}\) -norm–based formulations ( \(\ell_{2}\) -SADI) are sensitive to outliers and non-Gaussian noise, we propose an \(\ell_{1}\) -norm–based model ( \(\ell_{1}\) -SADI) that enhances robustness by minimizing the \(\ell_{1}\) -norm of the off-diagonal entries. To solve the resulting nonsmooth optimization problem, a Riemannian smoothing conjugate gradient (RSCG) algorithm with adaptively decaying smoothing parameters is introduced. It is shown that any accumulation point of the sequence generated by RSCG satisfies the necessary conditions for local optimality of the original problem, with all such points being limiting stationary points, forming a subset of the Clarke stationary point set. The model and algorithm can be naturally extended to \(d\) -order tensors. Numerical experiments demonstrate that RSCG performs well in terms of efficiency, robustness, and accuracy in the presence of outliers and noise.
