1 research outputs found
Relative Error Tensor Low Rank Approximation
We consider relative error low rank approximation of with respect
to the Frobenius norm: given an order- tensor , output a rank- tensor for which
OPT, where OPT . Despite the success on obtaining relative error low rank
approximations for matrices, no such results were known for tensors. One
structural issue is that there may be no rank- tensor achieving the
above infinum. Another, computational issue, is that an efficient relative
error low rank approximation algorithm for tensors would allow one to compute
the rank of a tensor, which is NP-hard. We bypass these issues via (1)
bicriteria and (2) parameterized complexity solutions:
(1) We give an algorithm which outputs a rank
tensor for which OPT in time in the real RAM model. Here is the
number of non-zero entries in .
(2) We give an algorithm for any which outputs a rank tensor
for which OPT and runs in time in
the unit cost RAM model.
For outputting a rank- tensor, or even a bicriteria solution with
rank- for a certain constant , we show a
time lower bound under the Exponential Time Hypothesis.
Our results give the first relative error low rank approximations for tensors
for a large number of robust error measures for which nothing was known, as
well as column row and tube subset selection. We also obtain new results for
matrices, such as -time CUR decompositions, improving previous
-time algorithms, which may be of independent interest