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「Ostrowski lecture: The uniform uncertainty principle and compressed sensing」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

For much of last week I was in Leiden, Holland, giving one of the Ostrowski prize lectures at the annual meeting of the Netherlands mathematical congress . My talk was not on the subject of the prize (arithmetic progressions in primes), as this was covered by a talk of Ben Green there, but rather on a certain “uniform uncertainty principle” in Fourier analysis, and its relation to compressed sensing; this is work which is joint with Emmanuel Candes and also partly with Justin

As mentioned in the previous post here , compressed sensing is a relatively new measurement paradigm which seeks to capture the “essential” aspects of a high-dimensional object using as few measurements as possible. There are many contexts in which compressed sensing is potentially useful, but for this talk (which is focussed on theory rather than applications) I will just consider a single toy model arising from Fourier analysis. Specifically, the object we seek to measure w

已知结果和反例

We suppose that we can measure some (but perhaps not all) of the Fourier coefficients of f, and ask whether we can reconstruct f from this information; the objective is to use as few Fourier coefficients as possible. More specifically, we fix a set of “observable” frequencies, and pose the following two questions:

For instance, if is the whole set of frequencies, i.e. , then the answer to Q1 is “yes” (because the Fourier transform is injective), and an answer to Q2 is provided by the Fourier inversion formula

证明或构造的主线

which can be computed quite quickly, for instance by using the fast Fourier transform .

Now we ask what happens when is a proper subset of . Then the answer to Q1, as stated above, is “no” (and so Q2 is moot). One can see this abstractly by a degrees-of-freedom argument: the space of all functions f on N points has N degrees of freedom, but we are only making measurements, thus leaving remaining degrees of freedom in the unknown function f. If is strictly less than N, then there are not enough measurements to pin down f precisely. More concretely, we can easily

阅读时建议盯住的点

However, we can hope to recover unique solvability for this problem by making an additional hypothesis on the function f. There are many such hypotheses one could make, but for this toy problem we shall simply assume that f is sparse . Specifically, we fix an integer S between 1 and N, and say that a function f is S-sparse if f is non-zero in at most S places, or equivalently if the support has cardinality less than or equal to S. We now ask the following modified versions of

Note that while we know how sparse f is, we are not given to know exactly where f is sparse – there are S positions out of the N total positions where f might be non-zero, but we do not know which S positions these are. The fact that the support is not known a priori is one of the key difficulties with this problem. Nevertheless, setting that problem aside for the moment, we see that f now has only S degrees of freedom instead of N, and so by repeating the previous analysis o

值得单独记下的条目

• Let N be a known integer, let be an unknown function, let a known set of frequencies, and let be a sequence of known Fourier coefficients of f for all . Is it possible to reconstruct f uniquely from this information?
• If so, what is a practical algorithm for finding f?
• Let S and N be known integers, let be an unknown S-sparse function, let a known set of frequencies, and let be a sequence of known Fourier coefficients of f for all . Is it possible to reconstruct f uniquely from this information?
• If so, what is a practical algorithm for finding f?

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

龙虾PRO做 OpenClaw 落地时，数学笔记最有用的部分往往是「可检验的步骤」：定义、反例、引理边界。智能体适合帮忙展开计算和检索，不适合代替你决定哪条假设能扔。

本文侧重全链路风控方法论。落地时请用自身业务单据做回放验证，不要把示例阈值直接当生产策略。 相关：风控体检 · 方案资源