Recently, I have read a paper titled as "Robust Sparse Regression under Adversarial Corruption" appeared in ICML 2013, which is one of the most famous and top conference in machine learning（actually, many papers are talking about applied mathematics）.
This paper proposed a method to solve sparse regression problem in a robust scenario when there exists some corruptions in $X$. The mean idea is very simple, i.e., replacing the ordinary inner product as trimmed inner product, and the definition is trimmed inner product $\left< a,b\right>_{n_1}$ described as algorithm to compute trimmed inner product.
- Input: $a,b\in \mathbb{R}^N$ and $n_1$,
- Compute $q_i = a_ib_i, i = 1,...,N.$
- Sort $\{|q_i|\}$ and select the smallest $(N − n_1)$ ones. Let $\Omega$ be the set of selected indices.
- Output: $h = \sum_{i\in\Omega} q_i$.
So my question is:
Is the definition of trimmed inner product new, are there some similar ideas in statistics or some other fields? Why is this more robust when facing corruptions? Could any one provide some intuitions?