Abstract A recently proposed SLOPE estimator [6] has been shown to adaptively achieve the minimax $\ell _2$ estimation rate under high-dimensional sparse linear regression models [25]. Such minimax optimality holds in the regime where the sparsity level $k$, sample size $n$ and dimension $p$ satisfy $k/p\rightarrow 0, k\log p/n\rightarrow 0$. In this paper, we characterize the estimation error of SLOPE under the complementary regime where both $k$ and $n$ scale linearly with $p$, and provide new insights into the performance of SLOPE estimators. We first derive a concentration inequality for the finite sample mean square error (MSE) of SLOPE. The quantity that MSE concentrates around takes a complicated and implicit form. With delicate analysis of the quantity, we prove that among all SLOPE estimators, LASSO is optimal for estimating $k$-sparse parameter vectors that do not have tied nonzero components in the low noise scenario. On the other hand, in the large noise scenario, the family of SLOPE estimators are sub-optimal compared with bridge regression such as the Ridge estimator.
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Optimal Estimation of Sparse Topic Models
Topic models have become popular tools for dimension reduction and exploratory analysis of text data which consists in observed frequencies of a vocabulary of p words in n documents, stored in a p×n matrix. The main premise is that the mean of this data matrix can be factorized into a product of two non-negative matrices: a p×K word-topic matrix A and a K×n topic-document matrix W. This paper studies the estimation of A that is possibly element-wise sparse, and the number of topics K is unknown. In this under-explored context, we derive a new minimax lower bound for the estimation of such A and propose a new computationally efficient algorithm for its recovery. We derive a finite sample upper bound for our estimator, and show that it matches the minimax lower bound in many scenarios. Our estimate adapts to the unknown sparsity of A and our analysis is valid for any finite n, p, K and document lengths. Empirical results on both synthetic data and semi-synthetic data show that our proposed estimator is a strong competitor of the existing state-of-the-art algorithms for both non-sparse A and sparse A, and has superior performance is many scenarios of interest.
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- Award ID(s):
- 2015195
- NSF-PAR ID:
- 10274094
- Editor(s):
- Dalalyan, Aynak
- Date Published:
- Journal Name:
- Journal of machine learning research
- Volume:
- 21
- Issue:
- 177
- ISSN:
- 1532-4435
- Page Range / eLocation ID:
- 1 - 45
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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