id: 06008051
dt: j
an: 06008051
au: Lerman, Gilad; Zhang, Teng
ti: Robust recovery of multiple subspaces by geometric $l_{p}$ minimization.
so: Ann. Stat. 39, No. 5, 2686-2715 (2011).
py: 2011
pu: Institute of Mathematical Statistics, Beachwood, OH
la: EN
cc: 
ut: detection; clustering; hybrid linear modeling; optimization of
    Grassmannians; robustness; geometric probability; high-dimensional data
ci: 
li: doi:10.1214/11-AOS914
ab: Summary: We assume i.i.d. data sampled from a mixture distribution with $K$
    components along fixed $d$-dimensional linear subspaces and an
    additional outlier component. For $p > 0$, we study the simultaneous
    recovery of the $K$ fixed subspaces by minimizing the $l_{p}$-averaged
    distances of the sampled data points from any $K$ subspaces. Under some
    conditions, we show that if $0 < p \leq 1$, then all underlying
    subspaces can be precisely recovered by $l_{p}$ minimization with
    overwhelming probability. On the other hand, if $K > 1$ and $p > 1$,
    then the underlying subspaces cannot be recovered or even nearly
    recovered by $l_{p}$ minimization. The results of this paper partially
    explain the successes and failures of the basic approach of $l_{p}$
    energy minimization for modeling data by multiple subspaces.
rv: