The Singular Value Decomposition (SVD) of matrices is a widely used tool in scientific computing. In many applications of machine learning, data analysis, signal and image processing, the large datasets are structured into tensors, for which generalizations of SVD have already been introduced, for various types of tensor-tensor products. In this article, we present innovative methods for approximating this generalization of SVD to tensors in the framework of the Einstein tensor product. These singular elements are called singular values and singular tensors, respectively. The proposed method uses the tensor Lanczos bidiagonalization applied to the Einstein product. In most applications, as in the matrix case, the extremal singular values are of special interest. To enhance the approximation of the largest or the smallest singular triplets (singular values and left and right singular tensors), a restarted method based on Ritz augmentation is proposed. Numerical results are proposed to illustrate the effectiveness of the presented method. In addition, applications to video compression and facial recognition are presented.

翻译：矩阵的奇异值分解是科学计算中广泛使用的工具。在机器学习、数据分析、信号与图像处理的许多应用中，大规模数据集被构建为张量形式，针对不同类型的张量-张量乘积，已有SVD的推广形式被提出。本文在Einstein张量积框架下，提出了逼近张量SVD推广形式的创新方法。这些奇异元素分别称为奇异值和奇异张量。所提出的方法采用应用于Einstein积的张量Lanczos双对角化。与矩阵情形类似，在大多数应用中，极端奇异值具有特殊重要性。为了增强对最大或最小奇异三元组（奇异值及左右奇异张量）的逼近，提出了基于Ritz增广的重启方法。数值实验结果验证了该方法的有效性。此外，还展示了该方法在视频压缩和人脸识别中的应用。