2025/01/17 by Xiaolong Liu, Liu, Xiaolong, Matey Neykov +1
Mathematics · #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (stat.ML) #Statistical Methods and Inference #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2501.10025
openalex publication_date 2025/01/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We establish a novel criterion for comparing the performance of two densities, g1 and g2, within the context of corrupted data. Utilizing this criterion, we propose an algorithm to construct a density estimator within a star-shaped density class, F, under conditions of data corruption. We proceed to derive the minimax upper and lower bounds for density estimation across this star-shaped density class, characterized by densities that are uniformly bounded above and below (in the sup norm), in the presence of adversarially corrupted data. Specifically, we assume that a fraction ε≤ (1)/(3) of the N observations are arbitrarily corrupted. We obtain the minimax upper bound max\ τ_J2, ε\ \wedge d2. Under certain conditions, we obtain the minimax risk, up to proportionality constants, under the squared L2 loss as max\ τ*2 \wedge d2, ε\wedge d2 \, where τ^* := sup\ τ: Nτ2 ≤ log MFloc(τ, c) \ for a sufficiently large constant c. Here, MFloc(τ, c) denotes the local entropy of the set F, and d is the L2 diameter of F.