This paper studies linear regression models for high dimensional multi-response data with a hybrid quantum computing algorithm. We propose an intuitively appealing estimation method based on identifying the linearly independent columns in the coefficient matrix. Our method relaxes the low rank constraint in the existing literature and allows the rank to diverge with dimensions. The linearly independent columns are selected by a novel non-oracular quantum search (NQS) algorithm which is significantly faster than classical search methods implemented on electronic computers. Besides, NQS achieves a near optimal computational complexity as existing quantum search algorithms but does not require any oracle information of the solution state. We prove the proposed estimation procedure enjoys desirable theoretical properties. Intensive numerical experiments are also conducted to demonstrate the finite sample performance of the proposed method, and a comparison is made with some popular competitors. The results show that our method outperforms all of the alternative methods under various circumstances.
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Linear regression with partially mismatched data: local search with theoretical guarantees
Linear regression is a fundamental modeling tool in statistics and related fields. In this paper, we study an important variant of linear regression in which the predictor-response pairs are partially mismatched. We use an optimization formulation to simultaneously learn the underlying regression coefficients and the permutation corresponding to the mismatches. The combinatorial structure of the problem leads to computational challenges. We propose and study a simple greedy local search algorithm for this optimization problem that enjoys strong theoretical guarantees and appealing computational performance. We prove that under a suitable scaling of the number of mismatched pairs compared to the number of samples and features, and certain assumptions on problem data; our local search algorithm converges to a nearly-optimal solution at a linear rate. In particular, in the noiseless case, our algorithm converges to the global optimal solution with a linear convergence rate. Based on this result, we prove an upper bound for the estimation error of the parameter. We also propose an approximate local search step that allows us to scale our approach to much larger instances. We conduct numerical experiments to gather further insights into our theoretical results, and show promising performance gains compared to existing approaches.
more » « less- NSF-PAR ID:
- 10369868
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- Mathematical Programming
- Volume:
- 197
- Issue:
- 2
- ISSN:
- 0025-5610
- Page Range / eLocation ID:
- p. 1265-1303
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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