2021/03/20 by Amit Deshpande, Deshpande, Amit, Rameshwar Pratap +1
Computer Science · Engineering · Mathematics · #Complexity and Algorithms in Graphs #Computational Geometry (cs.CG) #Computational Geometry and Mesh Generation #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Machine Learning and Algorithms #Sparse and Compressive Sensing Techniques #Spectral Theory (math.SP) #cs.CG #cs.LG #math.SP #stat.ML
paper · pdf · doi:10.48550/arxiv.2103.11107
arxiv created 2021/03/20 · openalex publication_date 2021/03/20 · arxiv updated 2021/03/23 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
We consider the problem of subset selection for ℓp subspace approximation, i.e., given n points in d dimensions, we need to pick a small, representative subset of the given points such that its span gives (1+ε) approximation to the best k-dimensional subspace that minimizes the sum of p-th powers of distances of all the points to this subspace. Sampling-based subset selection techniques require adaptive sampling iterations with multiple passes over the data. Matrix sketching techniques give a single-pass (1+ε) approximation for ℓp subspace approximation but require additional passes for subset selection. In this work, we propose an MCMC algorithm to reduce the number of passes required by previous subset selection algorithms based on adaptive sampling. For p=2, our algorithm gives subset selection of nearly optimal size in only 2 passes, whereas the number of passes required in previous work depend on k. Our algorithm picks a subset of size poly(k/ε) that gives (1+ε) approximation to the optimal subspace. The running time of the algorithm is nd + d~poly(k/ε). We extend our results to the case when outliers are present in the datasets, and suggest a two pass algorithm for the same. Our ideas also extend to give a reduction in the number of passes required by adaptive sampling algorithms for ℓp subspace approximation and subset selection, for p ≥ 2.