3 research outputs found
Iteratively Reweighted Least Squares Algorithms for L1-Norm Principal Component Analysis
Principal component analysis (PCA) is often used to reduce the dimension of
data by selecting a few orthonormal vectors that explain most of the variance
structure of the data. L1 PCA uses the L1 norm to measure error, whereas the
conventional PCA uses the L2 norm. For the L1 PCA problem minimizing the
fitting error of the reconstructed data, we propose an exact reweighted and an
approximate algorithm based on iteratively reweighted least squares. We provide
convergence analyses, and compare their performance against benchmark
algorithms in the literature. The computational experiment shows that the
proposed algorithms consistently perform best
Optimization for L1-Norm Error Fitting via Data Aggregation
We propose a data aggregation-based algorithm with monotonic convergence to a
global optimum for a generalized version of the L1-norm error fitting model
with an assumption of the fitting function. The proposed algorithm generalizes
the recent algorithm in the literature, aggregate and iterative disaggregate
(AID), which selectively solves three specific L1-norm error fitting problems.
With the proposed algorithm, any L1-norm error fitting model can be solved
optimally if it follows the form of the L1-norm error fitting problem and if
the fitting function satisfies the assumption. The proposed algorithm can also
solve multi-dimensional fitting problems with arbitrary constraints on the
fitting coefficients matrix. The generalized problem includes popular models
such as regression and the orthogonal Procrustes problem. The results of the
computational experiment show that the proposed algorithms are faster than the
state-of-the-art benchmarks for L1-norm regression subset selection and L1-norm
regression over a sphere. Further, the relative performance of the proposed
algorithm improves as data size increases