We show that in finite-dimensional nonlinear approximations, the best r-term approximant of a function f almost always exists over $\mathbb{C}$ but that the same is not true over $\mathbb{R}$, i.e., the infimum ${\inf }_{f_1,\dots ,f_r\in D}\Vert f-f_1-\dots -f_r\Vert$ is almost always attainable by complex-valued functions $f_1,\dots ,f_r$ in D, a set (dictionary) of functions (atoms) with some desired structures. Our result extends to functions that possess properties like symmetry or skew-symmetry under permutations of arguments. When D is the set of separable functions, this is the best rank-r tensor approximation problem. We show that over $\mathbb{C}$, any tensor almost always has a unique best rank-r approximation. This extends to other notions of ranks such as symmetric and alternating ranks, to best r-block-terms approximations, and to best approximations by tensor networks. Applied to sparse-plus-low-rank approximations, we obtain that for any given r and k, a general tensor has a unique best approximation by a sum of a rank-r tensor and a k-sparse tensor with a fixed sparsity pattern; a problem arising in covariance estimation of Gaussian model with k observed variables conditionally independent given r hidden variables. The existential (but not uniqueness) part of our result also applies to best approximations by a sum of a rank-r tensor and a k-sparse tensor with no fixed sparsity pattern, and to tensor completion problems.

我们表明，在有限维非线性逼近中，函数f的最佳r项逼近几乎总是存在于$\mathbb{C}$ 但是那不是真的 $\mathbb{[R}$，即 ${\mathrm{信息}}_{F_{\mathrm{1个}}，\dots ，F_{\mathrm{[R}}\in d}\Vert F-F_{\mathrm{1个}}-\dots -F_{\mathrm{[R}}\Vert$ 几乎总是可以通过复数值函数实现的 $F_{\mathrm{1个}}，\dots ，F_{\mathrm{[R}}$在D中，具有某些所需结构的一组功能（原子）（字典）。我们的结果扩展到在参数的排列下具有对称性或斜对称性等属性的函数。当d是一组可分离的功能，这是最好的秩[R张量逼近问题。我们证明了$\mathbb{C}$，任何张量几乎总是具有唯一的最佳秩r近似。这扩展到其他等级概念（例如对称和交替等级），最佳r -block-terms近似以及张量网络的最佳近似。应用于稀疏加低秩逼近，我们得到，对于任何给定的r和k，一般张量具有唯一的最佳逼近度，即秩r张量和k稀疏张量之和具有固定的稀疏性。高斯模型的协方差估计中出现的一个问题，其中k个观测变量有条件地独立于给定r隐藏的变量。我们的结果的存在性（但不是唯一性）部分也适用于秩r张量和k稀疏张量之和且没有固定稀疏性模式的最佳逼近，以及张量完成问题。