The Eckhart-Young theorem states that the best low-rank approximation of a matrix can be constructed from the leading singular values and vectors of the matrix. Here, we illustrate that the practical implications of this result crucially depend on the organization of the matrix data. In particular, we will show examples where a rank 2 approximation of the matrix data in a different representation more accurately represents the entire matrix than a rank 5 approximation of the original matrix data -- even though both approximations have the same number of underlying parameters. Beyond images, we show examples of how flexible orientation enables better approximation of time series data, which suggests additional applicability of the findings. Finally, we conclude with a theoretical result that the effect of data organization can result in an unbounded improvement to the matrix approximation factor as the matrix dimension grows.

翻译：Eckhart-Young定理指出，矩阵的最佳低秩近似可由其主导奇异值与奇异向量构造。本文揭示该结论的实际应用效果关键取决于矩阵数据的组织方式。具体而言，我们将展示若干实例：在数据的不同表征下，秩为2的矩阵近似比原始矩阵的秩5近似能更精确地描述整体矩阵——尽管两种近似具有相同数量的底层参数。除图像外，我们还通过灵活的数据取向如何提升时间序列近似效果的案例，表明该发现更具普适性。最后，我们给出理论结果：随着矩阵维度的增长，通过数据组织方式可对矩阵近似因子实现无界改进。