Matrix Approximation and Projective Clustering via Iterative Sampling
نویسندگان
چکیده
We present two new results for the problem of approximating a given real m × n matrix A by a rank-k matrix D, where k < min{m, n}, so as to minimize ||A−D||F . It is known that by sampling O(k/ε) rows of the matrix, one can find a low-rank approximation with additive error ε||A||F . Our first result shows that with adaptive sampling in t rounds and O(k/ε) samples in each round, the additive error drops exponentially as ε; the computation time is nearly linear in the number of nonzero entries. This demonstrates that multiple passes can be highly beneficial for a natural (and widely studied) algorithmic problem. Our second result is that there exists a subset of O(k/ε) rows such that their span contains a rank-k approximation with multiplicative (1 + ε) error (i.e., the sum of squares distance has a small “core-set” whose span determines a good approximation). This existence theorem leads to a PTAS for the following projective clustering problem: Given a set of points P in R, and integers k, j, find a set of j subspaces F1, . . . , Fj , each of dimension at most k, that minimize ∑ p∈P mini d(p, Fi) .
منابع مشابه
Computer Science and Artificial Intelligence Laboratory Matrix Approximation and Projective Clustering via Iterative Sampling
We present two new results for the problem of approximating a given real m × n matrix A by a rank-k matrix D, where k < min{m,n}, so as to minimize ||A−D||F . It is known that by sampling O(k/ ) rows of the matrix, one can find a low-rank approximation with additive error ||A||F . Our first result shows that with adaptive sampling in t rounds and O(k/ ) samples in each round, the additive error...
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