输入稀疏时间的总体最小二乘回归
Total Least Squares Regression in Input Sparsity Time
摘要：
在总最小二乘问题中，给出一个$m \times n$矩阵$A$和一个$m \times d$矩阵$B$，并试图“校正”$A$和$B$，获得矩阵$\hat{A}$和$\hat{B}$，从而存在满足等式$\hat{A}X = \hat{B}$的$X$。通常情况下，问题是过约束的，这意味着$m \gg \max(n,d)$。解决方案$\hat{A}, \hat{B}$的成本由$\|A-\hat{A}\|_F^2 + \|B - \hat{B}\|_F^2$给出。我们给出了一个算法来寻找一个解决方案$X$的线性系统$\hat{A}X=\hat{B}$的成本$\|A-\hat{A}\|_F^2 + \|B-\hat{B}\|_F^2$最多是一个乘法$(1+ε)$的因素倍的最佳成本，添加剂的错误$η$，可能是一个任意小的功能$n$。重要的是，我们的运行时间是$\tilde{O}( \mathrm{nnz}(A) + \mathrm{nnz}(B) ) + \mathrm{poly}(n/ε) \cdot d$，其中对于矩阵$C$，$\mathrm{nnz}(C)$表示其非零条目的数量。重要的是，我们的运行时间并不直接依赖于大参数$m$。由于已知总最小二乘回归可通过低秩近似求解，因此自然的方法是调用用于近似低秩近似的快速算法，从该低秩近似获得矩阵$\hat{A}$和$\hat{B}$，然后求解$X$，使得$\hat{A}X = \hat{B}$。然而，现有算法不适用，因为在总最小二乘法中，低秩近似的秩需要是$n$，因此已知方法的运行时间将至少是$mn^2$。相比之下，我们能够通过从不显式地形成方程$\hat{A} X = \hat{B}$，而是求解作为隐式这样的方程的解的$X$，来实现更快的运行时间以寻找$X$。最后，我们将我们的算法推广到正则化的总体最小二乘问题。
英文摘要：
In the total least squares problem, one is given an $m \times n$ matrix $A$, and an $m \times d$ matrix $B$, and one seeks to "correct" both $A$ and $B$, obtaining matrices $\hat{A}$ and $\hat{B}$, so that there exists an $X$ satisfying the equation $\hat{A}X = \hat{B}$. Typically the problem is overconstrained, meaning that $m \gg \max(n,d)$. The cost of the solution $\hat{A}, \hat{B}$ is given by $\|A-\hat{A}\|_F^2 + \|B - \hat{B}\|_F^2$. We give an algorithm for finding a solution $X$ to the linear system $\hat{A}X=\hat{B}$ for which the cost $\|A-\hat{A}\|_F^2 + \|B-\hat{B}\|_F^2$ is at most a multiplicative $(1+ε)$ factor times the optimal cost, up to an additive error $η$ that may be an arbitrarily small function of $n$. Importantly, our running time is $\tilde{O}( \mathrm{nnz}(A) + \mathrm{nnz}(B) ) + \mathrm{poly}(n/ε) \cdot d$, where for a matrix $C$, $\mathrm{nnz}(C)$ denotes its number of non-zero entries. Importantly, our running time does not directly depend on the large parameter $m$. As total least squares regression is known to be solvable via low rank approximation, a natural approach is to invoke fast algorithms for approximate low rank approximation, obtaining matrices $\hat{A}$ and $\hat{B}$ from this low rank approximation, and then solving for $X$ so that $\hat{A}X = \hat{B}$. However, existing algorithms do not apply since in total least squares the rank of the low rank approximation needs to be $n$, and so the running time of known methods would be at least $mn^2$. In contrast, we are able to achieve a much faster running time for finding $X$ by never explicitly forming the equation $\hat{A} X = \hat{B}$, but instead solving for an $X$ which is a solution to an implicit such equation. Finally, we generalize our algorithm to the total least squares problem with regularization.
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