1 research outputs found
Optimal Analysis of Subset-Selection Based L_p Low Rank Approximation
We study the low rank approximation problem of any given matrix over
and in entry-wise
loss, that is, finding a rank- matrix such that is
minimized. Unlike the traditional setting, this particular variant is
NP-Hard. We show that the algorithm of column subset selection, which was an
algorithmic foundation of many existing algorithms, enjoys approximation ratio
for and for . This improves
upon the previous bound for
\cite{chierichetti2017algorithms}. We complement our analysis with lower
bounds; these bounds match our upper bounds up to constant when .
At the core of our techniques is an application of \emph{Riesz-Thorin
interpolation theorem} from harmonic analysis, which might be of independent
interest to other algorithmic designs and analysis more broadly.
As a consequence of our analysis, we provide better approximation guarantees
for several other algorithms with various time complexity. For example, to make
the algorithm of column subset selection computationally efficient, we analyze
a polynomial time bi-criteria algorithm which selects columns. We
show that this algorithm has an approximation ratio of for
and for . This improves over the
best-known bound with an approximation ratio. Our bi-criteria
algorithm also implies an exact-rank method in polynomial time with a slightly
larger approximation ratio.Comment: 20 pages, accepted by NeurIPS 201