陶哲轩博客写数学问题时，通常先把对象定义清楚，再给直觉、反例和证明轮廓。把「The Dantzig selector: Statistical estimation when p is much larger than n」改写成可阅读的中文笔记，重点是：问题在问什么、已知到哪一步、下一步最容易走偏在哪。原站广告、分享条和导航已去掉。

问题在问什么

Over two years ago, Emmanuel Candés and I submitted the paper “ The Dantzig selector: Statistical estimation when is much larger than ” to the Annals of Statistics . This paper, which appeared last year , proposed a new type of selector (which we called the Dantzig selector , due to its reliance on the linear programming methods to which George Dantzig , who had died as we were finishing our paper, had contributed so much to) for statistical estimation , in the case when the

Our selection algorithm, inspired by our previous work on compressed sensing, chooses the estimated parameters to have minimal norm amongst all vectors which are consistent with the data in the sense that the residual vector obeys the condition

已知结果和反例

(one can check that such a condition is obeyed with high probability in the case that , thus the true vector of parameters is feasible for this selection algorithm). This selector is similar, though not identical, to the more well-studied lasso selector in the literature, which minimises the norm of penalised by the norm of the residual.

A simple model case arises when n=p and X is the identity matrix, thus the observations are given by a simple additive noise model . In this case, the Dantzig selector is given by the hard soft thresholding formula

证明或构造的主线

The mean square error for this selector can be computed to be roughly

and one can show that this is basically best possible (except for constants and logarithmic factors) amongst all selectors in this model. More generally, the main result of our paper was that under the assumption that the predictor matrix obeys the RIP, the mean square error of the Dantzig selector is essentially equal to (2) and thus close to best possible.

阅读时建议盯住的点

After accepting our paper, the Annals of Statistics took the (somewhat uncommon) step of soliciting responses to the paper from various experts in the field, and then soliciting a rejoinder to these responses from Emmanuel and I. Recently, the Annals posted these responses and rejoinder on the arXiv :

Finally, Candès and myself gave a rejoinder to these responses. Our main points were:

值得单独记下的条目

• Efron, Hastie, and Tibshirani performed numerics to compare the accuracy of the Dantzig selector and the lasso selector, the performance was broadly rather similar, but the Dantzig selector appeared to have some artefacts arising from the c
• Ritov raised a more philosophical point, as to whether the prediction error (which is essentially in this model) is a better indicator of accuracy than the loss .
• Friedlander and Saunders focused on the speed of implementation of the Dantzig selector, concluding that using general-purpose linear algebra solvers (e.g. the simplex method) was moderately computationally intensive in practice.

阅读和落地时建议先做的 5 件事

1. 用自己的语言重写定义和结论，不看原文能不能说清对象是什么。
2. 找一个最小反例或边界情形，确认假设少一条会怎样。
3. 把证明拆成可独立检验的引理，每步只保留一个新想法。
4. 若涉及计算或形式化，先写可复现的小例子，再谈一般情形。
5. 记下尚未解决的缺口：缺估计、缺构造，还是缺正确的范畴。

和智能体、形式化工具怎么接

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