Tensor decomposition is a fundamentally challenging problem. Even the simplest case of tensor decomposition, the rank-1 approximation in terms of the Least Squares (LS) error, is known to be NP-hard. Here, we show that, if we consider the KL divergence instead of the LS error, we can analytically derive a closed form solution for the rank-1 tensor that minimizes the KL divergence from a given positive tensor. Our key insight is to treat a positive tensor as a probability distribution and formulate the process of rank-1 approximation as a projection onto the set of rank-1 tensors. This enables us to solve rank-1 approximation by convex optimization. We empirically demonstrate that our algorithm is an order of magnitude faster than the existing rank-1 approximation methods and gives better approximation of given tensors, which supports our theoretical finding.

中文翻译：

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通过KL散度最小化实现最佳1级张量逼近的封闭形式解决方案

张量分解是一个根本上具有挑战性的问题。即使是最简单的张量分解情况，以最小二乘（LS）误差表示的Rank-1逼近也是已知的NP-hard。在这里，我们表明，如果考虑KL散度而不是LS误差，我们可以分析得出秩1张量的闭式解，从而使给定正张量的KL散度最小。我们的主要见识是将正张量视为概率分布，并将等级1逼近的过程公式化为等级1张量集的投影。这使我们能够通过凸优化来求解秩1逼近。我们凭经验证明，我们的算法比现有的rank-1逼近方法快一个数量级，并且可以更好地逼近给定张量，