There is a conjectured computational-statistical gap in terms of the number of samples needed to perform tensor estimation. In particular, for a low rank 3-order tensor with $\Theta(n)$ parameters, Barak and Moitra conjectured that $\Omega(n^{3/2})$ samples are needed for polynomial time computation based on a reduction of a specific hard instance of a rank 1 tensor to the random 3-XOR distinguishability problem. In this paper, we take a complementary perspective and characterize a subclass of tensor instances that can be estimated with only $O(n^{1+\kappa})$ observations for any arbitrarily small constant $\kappa > 0$, nearly linear. If one considers the class of tensors with constant orthogonal CP-rank, the "hardness" of the instance can be parameterized by the minimum absolute value of the sum of latent factor vectors. If the sum of each latent factor vector is bounded away from zero, we present an algorithm that can perform tensor estimation with $O(n^{1+\kappa})$ samples for a $t$-order tensor, significantly less than the previous achievable bound of $O(n^{t/2})$, and close to the lower bound of $\Omega(n)$. This result suggests that amongst constant orthogonal CP-rank tensors, the set of computationally hard instances to estimate are in fact a small subset of all possible tensors.

中文翻译：

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具有近线性样本的张量估计

就执行张量估计所需的样本数量而言，存在推测的计算统计差距。特别是，对于参数为 $\Theta(n)$ 的低秩 3 阶张量，Barak 和 Moitra 推测 $\Omega(n^{3/2})$ 样本需要用于基于约简的多项式时间计算1 阶张量的特定硬实例的随机 3-XOR 可区分性问题。在本文中，我们从互补的角度来描述一个张量实例的子类，对于任何任意小的常数 $\kappa > 0$，几乎是线性的，可以仅用 $O(n^{1+\kappa})$ 观察来估计这些实例. 如果考虑具有恒定正交 CP 秩的张量类，则可以通过潜在因子向量之和的最小绝对值来参数化实例的“硬度”。如果每个潜在因子向量的总和远离零，我们提出了一种算法，该算法可以使用 $O(n^{1+\kappa})$ 样本对 $t$-阶张量执行张量估计，显着小于$O(n^{t/2})$ 的先前可实现界限，并且接近 $\Omega(n)$ 的下限。这个结果表明，在恒定的正交 CP 秩张量中，要估计的一组计算困难的实例实际上是所有可能张量的一个小子集。