Non-asymptotic analysis of the performance of the penalized least trimmed squares in sparse models

The least trimmed squares (LTS) estimator is a renowned robust alternative to the classic least squares estimator and is popular in location, regression, machine learning, and AI literature. Many studies exist on LTS, including its robustness, computation algorithms, extension to non-linear cases, asymptotics, etc. The LTS has been applied in the penalized regression in a high-dimensional real-data sparse-model setting where dimension $p$ (in thousands) is much larger than sample size $n$ (in tens, or hundreds). In such a practical setting, the sample size $n$ often is the count of sub-population that has a special attribute (e.g. the count of patients of Alzheimer's, Parkinson's, Leukemia, or ALS, etc.) among a population with a finite fixed size N. Asymptotic analysis assuming that $n$ tends to infinity is not practically convincing and legitimate in such a scenario. A non-asymptotic or finite sample analysis will be more desirable and feasible. This article establishes some finite sample (non-asymptotic) error bounds for estimating and predicting based on LTS with high probability for the first time.