Measure concentration inequalities

The fancy name for probability inequalities

A corral captures the idea of concentration of measure; we have some procedure that guarantees that most of mass (of buffalos) is where we can handle it. Image: Kevin M Klerks, CC BY 2.0

Welcome to the probability inequality mines!

When something in your process (measurement, estimation) means that you can be pretty sure that a whole bunch of your stuff is particularly likely to be somewhere in particular, e.g. being 80% sure I am only 20% wrong.

As undergraduates we run into central limit theorems, but there are many more diverse ways we can keep track of our probability, or at least most of it. This idea is a basic workhorse in univariate probability, and turns out to be yet more essential in multivariate matrix probability, as seen in matrix factorisation, compressive sensing, PAC-bounds and suchlike.


Overviews include


For any nonnegative random variable \(X,\) and \(t>0\) \[ \mathbb{P}\{X \geq t\} \leq \frac{\mathbb{E} X}{t} \] Corollary: if \(\phi\) is a strictly monotonically increasing non-negative-valued function then for any random variable \(X\) and real number \(t\) \[ \mathbb{P}\{X \geq t\}=\mathbb{P}\{\phi(X) \geq \phi(t)\} \leq \frac{\mathbb{E} \phi(X)}{\phi(t)} \]


A corollary of Markov’s bound is with \(\phi(x)=x^{2}\) is Chebyshev’s: if \(X\) is an arbitrary random variable and \(t>0,\) then \[ \mathbb{P}\{|X-\mathbb{E} X| \geq t\}=\mathbb{P}\left\{|X-\mathbb{E} X|^{2} \geq t^{2}\right\} \leq \frac{\mathbb{E}\left[|X-\mathbb{E} X|^{2}\right]}{t^{2}}=\frac{\operatorname{Var}\{X\}}{t^{2}} \] More generally taking \(\phi(x)=x^{q}(x \geq 0),\) for any \(q>0\) we have \[ \mathbb{P}\{|X-\mathbb{E} X| \geq t\} \leq \frac{\mathbb{E}\left[|X-\mathbb{E} X|^{q}\right]}{t^{q}} \] We can choose \(q\) to optimize the obtained upper bound for the problem in hand.


Taking \(\phi(x)=e^{s x}\) where \(s>0,\) for any random variable \(X,\) and any \(t>0,\) we have \[ \mathbb{P}\{X \geq t\}=\mathbb{P}\left\{e^{s X} \geq e^{s t}\right\} \leq \frac{\mathbb{E} e^{s X}}{e^{s t}} \] Once again, we choose \(s\) to make the bound as tight as possible.




Are these precisely results arising from Stein’s method

Let \(g: \mathcal{X}^{n} \rightarrow \mathbb{R}\) be a real-valued measurable function of n variables. Efron-Stein inequalities concern the difference between the random variable \(Z=g\left(X_{1}, \ldots, X_{n}\right)\) and its expected value \(\mathbb{E Z}\) when \(X_{1}, \ldots, X_{n}\) are arbitrary independent random variables.

Define \(\mathbb{E}_{i}\) for the expected value with respect to the variable \(X_{i}\), that is, \(\mathbb{E}_{i} Z=\mathbb{E}\left[Z \mid X_{1}, \ldots, X_{i-1}, X_{i+1}, \ldots, X_{n}\right]\) Then \[ \operatorname{Var}(Z) \leq \sum_{i=1}^{n} \mathbb{E}\left[\left(Z-\mathbb{E}_{i} Z\right)^{2}\right] \]

Now, let \(X_{1}^{\prime}, \ldots, X_{n}^{\prime}\) be an independent copy of \(X_{1}, \ldots, X_{n}\). \[ Z_{i}^{\prime}=g\left(X_{1}, \ldots, X_{i}^{\prime}, \ldots, X_{n}\right) \] Alternatively, \[ \operatorname{Var}(Z) \leq \frac{1}{2} \sum_{i=1}^{n} \mathbb{E}\left[\left(Z-Z_{i}^{\prime}\right)^{2}\right] \] Nothing here seems to constrain the variables here to be real-valued, merely the function \(g\), but apparently they do not work for matrix variables as written — you need to see Matrix efron stein results for that.




For the Gaussian distribution. Filed there, perhaps?



E.g. Hanson-Wright.

Martingale bounds



Let us copy from wikipedia:

Heuristically: if we pick \(N\) complex numbers \(x_1,\dots,x_N \in\mathbb{C}\), and add them together, each multiplied by jointly independent random signs \(\pm 1\), then the expected value of the sum’s magnitude is close to \(\sqrt{|x_1|^{2}+ \cdots + |x_N|^{2}}\).

Let \({\varepsilon_n}_{n=1}^N\) i.i.d. random variables with \(P(\varepsilon_n=\pm1)=\frac12\) for \(n=1,\ldots, N\), i.e., a sequence with Rademacher distribution. Let \(0<p<\infty\) and let \(x_1,\ldots,x_N \in \mathbb{C}\). Then

\[ A_p \left( \sum_{n=1}^N |x_n|^2 \right)^{1/2} \leq \left(\operatorname{E} \left|\sum_{n=1}^N \varepsilon_n x_n\right|^p \right)^{1/p} \leq B_p \left(\sum_{n=1}^N |x_n|^2\right)^{1/2} \]

for some constants \(A_p,B_p>0\). It is a simple matter to see that \(A_p = 1\) when \(p \ge 2\), and \(B_p = 1\) when \(0 < p \le 2\).

Empirical process theory


Matrix concentration

If we fix our interest to matrices in particular, some fun things arise. See Matrix concentration inequalities


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