  In this work, we propose an optimization framework for estimating a sparse
robust one-dimensional subspace. Our objective is to minimize both the
representation error and the penalty, in terms of the l1-norm criterion. Given
that the problem is NP-hard, we introduce a linear relaxation-based approach.
Additionally, we present a novel fitting procedure, utilizing simple ratios and
sorting techniques. The proposed algorithm demonstrates a worst-case time
complexity of $O(n^2 m \log n)$ and, in certain instances, achieves global
optimality for the sparse robust subspace, thereby exhibiting polynomial time
efficiency. Compared to extant methodologies, the proposed algorithm finds the
subspace with the lowest discordance, offering a smoother trade-off between
sparsity and fit. Its architecture affords scalability, evidenced by a 16-fold
improvement in computational speeds for matrices of 2000x2000 over CPU version.
Furthermore, this method is distinguished by several advantages, including its
independence from initialization and deterministic and replicable procedures.
Furthermore, this method is distinguished by several advantages, including its
independence from initialization and deterministic and replicable procedures.
The real-world example demonstrates the effectiveness of algorithm in achieving
meaningful sparsity, underscoring its precise and useful application across
various domains.
