Total Least Squares Regression in Input Sparsity Time

In the total least squares problem, one is given an m × n matrix A, and an m × d matrix B, and one seeks to "correct" both A and B, obtaining matrices hatA and hatB, so that there exists an X satisfying the equation hatAX = hatB. Typically the problem is overconstrained, meaning that m gg max(n,d). The cost of the solution hatA, hatB is given by |A-hatA|F2 + |B - hatB|F2. We give an algorithm for finding a solution X to the linear system hatAX=hatB for which the cost |A-hatA|F2 + |B-hatB|F2 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 tildeO( nnz(A) + nnz(B) ) + poly(n/ε) · d, where for a matrix C, 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 hatA and hatB from this low rank approximation, and then solving for X so that hatAX = hatB. 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 mn2. In contrast, we are able to achieve a much faster running time for finding X by never explicitly forming the equation hatA X = hatB, 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.