We study two canonical planted average-case problems -- noisy $k\mathsf{\text{-}XOR}$ and Tensor PCA -- and relate their computational properties via poly-time average-case reductions. In fact, we consider a \emph{family of problems} that interpolates between $k\mathsf{\text{-}XOR}$ and Tensor PCA, allowing intermediate densities and signal levels. We introduce two \emph{densifying} reductions that increase the number of observed entries while controlling the decrease in signal, and, in particular, reduce any $k\mathsf{\text{-}XOR}$ instance at the computational threshold to Tensor PCA at the computational threshold. Additionally, we give new order-reducing maps (e.g., $5\to 4$ $k\mathsf{\text{-}XOR}$ and $7\to 4$ Tensor PCA) at fixed entry density.

翻译：我们研究了两个典型的植入式平均情形问题——带噪声的$k\mathsf{\text{-}XOR}$与张量PCA——并通过多项式时间平均情形归约建立其计算特性之间的关联。事实上，我们考虑了一个在$k\mathsf{\text{-}XOR}$与张量PCA之间连续过渡的\emph{问题族}，允许中间密度与信号强度。我们提出了两种\emph{稠密化}归约方法，能在控制信号衰减的同时增加观测条目数量；特别地，该方法可将计算阈值处的任意$k\mathsf{\text{-}XOR}$实例归约为计算阈值处的张量PCA问题。此外，我们在固定条目密度下给出了新的降阶映射（例如$5\to 4$ $k\mathsf{\text{-}XOR}$与$7\to 4$张量PCA）。