Comments on: Amplification, arbitrage, and the tensor power trick http://terrytao.wordpress.com/2007/09/05/amplification-arbitrage-and-the-tensor-power-trick/ Updates on my research and expository papers, discussion of open problems, and other maths-related topics. By Terence Tao Wed, 19 Jun 2013 02:43:54 +0000 hourly 1 http://wordpress.com/ By: Strengthening inequalities: a mathematical trick | Abstract Art http://terrytao.wordpress.com/2007/09/05/amplification-arbitrage-and-the-tensor-power-trick/#comment-228788 Sat, 11 May 2013 06:06:42 +0000 http://terrytao.wordpress.com/2007/09/05/amplification-arbitrage-and-the-tensor-power-trick/#comment-228788 […] [1.9] Amplification, arbitrage, and the tensor power trick […]

]]> By: Bird’s-eye views of Structure and Randomness (Series) | Abstract Art http://terrytao.wordpress.com/2007/09/05/amplification-arbitrage-and-the-tensor-power-trick/#comment-228782 Sat, 11 May 2013 06:05:08 +0000 http://terrytao.wordpress.com/2007/09/05/amplification-arbitrage-and-the-tensor-power-trick/#comment-228782 […] Strengthening inequalities: a mathematical trick (adapted from “Amplification, arbitrage, and the tensor power trick“) […]

]]> By: Denis Serre http://terrytao.wordpress.com/2007/09/05/amplification-arbitrage-and-the-tensor-power-trick/#comment-218555 Mon, 04 Mar 2013 14:45:22 +0000 http://terrytao.wordpress.com/2007/09/05/amplification-arbitrage-and-the-tensor-power-trick/#comment-218555 Hi Terry,

Great post! I like your style and try to follow it closely.

Here are two examples of amplification, where the amplification trick gets rid of constants.
1. The maximum principle for holomorphic functions in a disk D (which could be another domain). Let z\in D and f:D\rightarrow C be a holomorphic function. Then Cauchy’s integral formula gives you an inequality |f(z)|\leq C_z\|f\|, with respect to the sup-norm over the boundary. Apply the inequality to f^k, then take the k-th root. The constant is changed into C_z^{1/k}, which tends to 1 as k tends to infinity.
2. In matrix analysis, let \|\cdot\| be an algebra norm over M_n(R), that is \|AB\|\leq\|A\|\cdot\|B\|. Let \rho(A) denote the spectral radius of a matrix A (which involves the modulus of complex eigenvalues too). Then \rho(A)\le\|A\|. This would be obvious if the norm was subordinated to a complex norm (take an eigenvector associated with an eigenvalue of maximal modulus, bla-bla). If not, take any such subordinated norm N, we thus have \rho(A)\le N(A). By equivalence of norms, we obtain \rho(A)\le C\|A\|. Apply this to A^k, use \|A^k\|\le\|A\|^k, then take the k-th root and let k tend to infinity.

All the best,

Denis

]]> By: Jorn http://terrytao.wordpress.com/2007/09/05/amplification-arbitrage-and-the-tensor-power-trick/#comment-213578 Thu, 17 Jan 2013 14:15:28 +0000 http://terrytao.wordpress.com/2007/09/05/amplification-arbitrage-and-the-tensor-power-trick/#comment-213578 Another great example of amplification is the proof of Chernoff’s bound from Markov’s inequality in probability theory, which stunned me when I first came across it. Markov’s inequality states that $P(X \ge a) \le E(X)/a$ for non-negative random variables $X$. Using that $P(X \ge a) = P(\exp(\theta X) \ge \exp(\theta a))$ for any $\theta > 0$, we find that $P(X \ge a) \le E(\exp(\theta X)) \exp(-\theta a)$, in which the parameter $\theta > 0$ can be chosen freely.

]]> By: A mathematical formalisation of dimensional analysis « What’s new http://terrytao.wordpress.com/2007/09/05/amplification-arbitrage-and-the-tensor-power-trick/#comment-211409 Sat, 29 Dec 2012 21:04:55 +0000 http://terrytao.wordpress.com/2007/09/05/amplification-arbitrage-and-the-tensor-power-trick/#comment-211409 [...] to amplify a hybrid inequality into a dimensionally pure one by optimising over all rescalings; see this previous blog post for a discussion of this trick (which, among other things, amplifies the inhomogeneous Sobolev [...]

]]> By: Terence Tao http://terrytao.wordpress.com/2007/09/05/amplification-arbitrage-and-the-tensor-power-trick/#comment-135669 Thu, 29 Mar 2012 18:36:52 +0000 http://terrytao.wordpress.com/2007/09/05/amplification-arbitrage-and-the-tensor-power-trick/#comment-135669 Sorry, by “constant” I meant “constant in i” rather than “absolute constant”. The point is that the arithmetic mean-geometric mean inequality k^k x_1 \ldots x_k \leq (x_1 + \ldots + x_k)^k is close to equality when the x_i are comparable, no matter how large the x_i are. In this particular application, we want to reverse the AM-GM inequality with x_i := N_i |A+B_i|, so the optimisation proceeds by setting the x_i to be comparable, e.g. setting N_i := \lfloor L / |A+B_i| \rfloor for some large parameter L.

]]> By: Tomer Shalev http://terrytao.wordpress.com/2007/09/05/amplification-arbitrage-and-the-tensor-power-trick/#comment-135667 Thu, 29 Mar 2012 17:50:59 +0000 http://terrytao.wordpress.com/2007/09/05/amplification-arbitrage-and-the-tensor-power-trick/#comment-135667 Hi Prof Tao,

regarding amplification of (17).

can you elaborate on the optimization of N_1,...,N_k
it seems strange to pick a near global constant when one
knows that |A + B_i| can be big.

]]> By: Montgomery’s uncertainty principle « What’s new http://terrytao.wordpress.com/2007/09/05/amplification-arbitrage-and-the-tensor-power-trick/#comment-119928 Sun, 01 Jan 2012 20:16:03 +0000 http://terrytao.wordpress.com/2007/09/05/amplification-arbitrage-and-the-tensor-power-trick/#comment-119928 [...] which one dilates to (and replaces each frequency by their roots), and then sending (cf. the tensor product trick). But we will not need this refinement [...]

]]> By: AJ http://terrytao.wordpress.com/2007/09/05/amplification-arbitrage-and-the-tensor-power-trick/#comment-42651 Wed, 02 Dec 2009 22:48:42 +0000 http://terrytao.wordpress.com/2007/09/05/amplification-arbitrage-and-the-tensor-power-trick/#comment-42651 Are the techniques posted here of any relevance to entropy power inequalities arising in network information theory?

]]> By: Vorlesung Funktionalanalysis: Erste Etappe « UGroh's Weblog http://terrytao.wordpress.com/2007/09/05/amplification-arbitrage-and-the-tensor-power-trick/#comment-42419 Wed, 18 Nov 2009 14:31:52 +0000 http://terrytao.wordpress.com/2007/09/05/amplification-arbitrage-and-the-tensor-power-trick/#comment-42419 [...] die Cauchy-Schwarz Ungleichung. In diesem Zusammenhang möchte ich auf den Interessanten Artikel Amplification, Arbitrage and the Tensor Power Trick von T. Tao hinweisen, in dem die Methodik des Beweises der Cauchy-Schwarz Ungleichung in [...]

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