User andrew mullhaupt - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T10:12:37Z http://mathoverflow.net/feeds/user/4061 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/15654/extreme-point-compact-convex-set Extreme point compact convex set. Andrew Mullhaupt 2010-02-18T01:52:43Z 2010-09-17T05:07:20Z <p>The Krein-Milman theorem shows that a compact convex set in a Hausdorff locally convex topological vector space is the convex hull of its extreme points.</p> <p>It seems this implies that a compact convex set in such a space must have an extreme point.</p> <p>I am interested in whether there is a very simple elementary argument that shows that a compact convex set must have an extreme point.</p> <p>I have such an argument, but since it uses compactness of the unit ball, it is not so good if the space is infinite dimensional.</p> <p>In point of fact, I am using this in R^n, but if there is a way to put it that can generalize to infinite dimensions then that would seem preferable for the students.</p> http://mathoverflow.net/questions/15836/oneupsmanship-and-publishing-etiquette/15837#15837 Answer by Andrew Mullhaupt for Oneupsmanship and Publishing Etiquette Andrew Mullhaupt 2010-02-19T19:24:42Z 2010-02-19T19:24:42Z <p>See if the other author wants to be a co-author. This is playing nice, and also, it eliminates the chance that they will be a referee.</p> http://mathoverflow.net/questions/15550/microarray-tesing-if-a-sample-is-the-same-with-high-variance-data/15826#15826 Answer by Andrew Mullhaupt for MicroArray, tesing if a sample is the same with high variance data. Andrew Mullhaupt 2010-02-19T17:35:27Z 2010-02-19T17:35:27Z <p>What you have is a classic case of a high dimensional, low signal to noise ratio signal. There are a lot of ways to proceed, but ultimately you will want to know about three different effects:</p> <ol> <li>Bayesian estimation.</li> <li>Dimension reduction.</li> <li>Superefficient estimation.</li> </ol> <p>These three ideas are frequently conflated by people with an informal understanding of them. So let's clear that up right away.</p> <ol> <li>The Bayesian estimation can always be applied. To estimate the probability of success p of independent flips of a possibly unfair coin you can always have a belief that the coin is fair or loaded, and this can lead to a different posterior distribution. You cannot reduce the dimension of the parameter space - unless you decide that you have what passes for 'revealed wisdom' as to that probability. It is also impossible to perform superefficient estimation in this case because the parameter space does not admit it.</li> </ol> <p>Note that in high dimensional cases, the Bayesian prior may reflect beliefs about noise, parameter spaces, and high dimensional estimation in general, as opposed to the traditional understanding that the Bayesian prior reflects belief about the parameter value. The farther you go into the high dimensional / low signal to noise regime, the less you expect to find anything passing for "expert opinion" that relates to the parameter and the more you expect to find all sorts of "expert opinion" about how hard it is to estimate the parameter. This is an important point: You can throw out the bathwater and keep the baby by knowing a lot about babies. But, and this is the one that most statisticians and engineers don't keep in the front window: You can ALSO throw out the bathwater and keep the baby by knowing a lot about bathwater. One of the strategies for exploiting the high availability of noise is that you can study the noise pretty easily. Another important point is that your willingness to believe that you do not have a good idea how to parameterize the probability density means that you will be considering things like Jeffreys' rule for a prior - the unique prior which provides expectations invariant under change of parameterization. Jeffreys' rule is very thin soup as Bayesian priors go - but in the high dimensional low signal to noise ratio it can be significant. It represents another very important principle of this sort of work: "Don't know? Don't Care". There are a lot of things you are not going to know in your situation; you should arrange as much as possible to line up what you are not going to know with what you will not care about. Don't know a good parameterization? Then appeal to a prior (e.g. Jeffreys' rule) which does not depend on the choice of parameterization.</p> <p>As an example, in the parameterization of poles and zeros, a finite dimensional linear time invariant system has the Jeffreys' prior given by the hyperbolic transfinite diameter of the set of poles and zeros. It turns out that poles and zeros is a provably exponentially bad general parameterization for the finite dimensional linear time invariant system in high dimensions. But you can use the Jeffreys' prior and know that the expectations you would compute with this bad parameterization will be the same (at least on the blackboard) as if you had computed them in some unknown 'good' parameterization.</p> <ol> <li><p>Dimension reduction. A high dimensional model is by definition capable of dimension reduction. One can, by various means, map the high dimensional parameter space to a lower dimensional parameter space. For example one can locally project onto the eigenspaces of the Fisher information matrix which have large eigenvalues. A lot of naive information theory is along the lines that "fewer parameters is better". It turns out that is false in general, but sometimes true. There are many "Information criteria" which seek to choose the dimension of the parameter space based on how much the likelihood function increases with the parameter. Be skeptical of these. In actual fact, every finite dimensional parameterization has some sort of bias. It is normally difficult to reduce the dimension intelligently unless you have some sort of side information. For example, if you have a translation invariant system, then dimension reduction becomes very feasible, although still very technical. Dimension reduction always interacts with choice of parameterization. Practical model reduction is normally an ad hoc approach. You want to cultivate a deep understanding of Bayesian and superefficient estimation before you choose a form of dimension reduction. However, lots of people skip that step. There is a wide world of ad hoc dimension reduction. Typically, one gets these by adding a small ad hoc penalty to the likelihood function. For example an L1 norm penalty tends to produce parameter estimates with many components equal to zero - because the L1 ball is a cross-polytope (e.g. octahedron) and the vertices of the cross polytope are the standard basis vectors. This is called compressed sensing, and it is a very active area. Needless to say, the estimates you get from this sort of approach depend critically on the coordinate system - it is a good idea to think through the coordinate system BEFORE applying compressed sensing. We will see an echo of this idea a bit later in superefficient estimation. However it is important to avoid confounding dimension reduction with superefficient estimation; so to prove that I give the example of a parameter with two real unconstrained components. You can do compressed sensing in two dimensions. You cannot do superefficient estimation in the plane. Another very important aspect here is "Who's Asking?". If you are estimating a high dimensional parameter, but the only use that will be made of that parameter is to examine the first component? Stop doing that, OK? It is very worthwhile to parameterize the DECISIONS that will be made from the estimation and then look at the preimage of the decisions in the parameter space. Essentially you want to compose the likelihood function (which is usually how you certify your belief about the observations that have been parameterized) with the decision function, and then maximize that (in the presence of your dimension reduction). You can consider the decision function another piece of the baby/bathwater separation, or you can also look at maximal application of don't care to relieve the pressure on what you have to know.</p> <ol> <li>Superefficient estimation. In the 1950s, following the discovery of the information inequality (formerly known as Cramer-Rao bound), statisticians thought that the best estimates would be unbiased estimates. To their surprise, embarassment, and dismay, they were able to show that in three dimensions and higher, this was not true. The basic example is James-Stein estimation. Probably because the word "biased" sounds bad, people adhere much much longer to "unbiased" estimation than they should have. Plus, the other main flavor of biased estimation, Bayesian estimation, was embroiled in an internecine philosophical war among the statisticians. It is appropriate for academic statisticians to attend to the foundations of their subject, and fiercely adhere to there convictions. That's what the 'academy' is for. However, you are faced with the high dimensional low signal to noise ratio case, which dissolves all philosophy. You are in a situation where unbiased estimation will do badly, and an expert with an informative Bayesian prior is not to be found. But you can prove that superefficient estimation will be good in some important senses (and certainly better than unbiased estimation). So you will do it. In the end, the easiest thing to do is to transform your parameter so that the local Fisher information is the identity (we call this the Fisher Unitary parameterization) and then apply a mild modification of James-Stein estimation (which you can find on Wikipedia). Yes, there are other things that you can do, but this one is as good as any if you can do it. There are some more ad hoc methods, mostly called "shrinkage" estimation. There is a large literature, and things like Beran's REACT theory are worth using, as well as the big back yard of wavelet shrinkage. Don't get too excited about the wavelet in wavelet shrinkage - it's just another coordinate transformation in this business (sorry, Harmonic Analysts). None of these methods can beat a Fisher unitary coordinate transformation IF you can find one. Oddly enough, a lot of the work that goes into having a Fisher unitary coordinate transformation is choosing a parameterization which affords you one. The global geometry of the parameter space and superefficient estimation interact very strongly. Go read Komaki's paper:</li> </ol></li> </ol> <p>KOMAKI, F. (2006) Shrinkage priors for Bayesian prediction. to appear in Annals of Statistics. (<a href="http://arxiv.org/pdf/math/0607021" rel="nofollow">http://arxiv.org/pdf/math/0607021</a>)</p> <p>which makes that clear. It is thought that if the Brownian diffusion (with Fisher information as Riemannian pseudo-metric) on your parameter space is transient, then you can do superefficient estimation, and not if it is recurrent. This corresponds to the heat kernel on the parameter space, etc. This is very well known in differential geometry to be global information about the manifold. Note that the Bayesian prior and most forms of model reduction are entirely local information. This is a huge and purely mathematical distinction. Do not be confused by the fact that you can show that after you do shrinkage, that there EXISTS a Bayesian prior which agrees with the shrinkage estimate; that just means that shrinkage estimates have some properties in common with Bayesian estimates (from the point of view of decisions and loss functions, etc.) but it does not give you that prior until you construct the shrinkage estimate. Superefficient estimation is GLOBAL.</p> <p>One other excruciatingly scary aspect of superefficient estimation is that it is extremely "non-unique". There is an inexplicable arbitrary choice; typically of a "point of superefficiency", but it is more like a measure of superefficiency that started out as an atomic measure. You have to choose this thing. There is no reason whatever for you to make one or another available choice. You might as well derive your choice from your social security number. And the parameter estimate that you get, as well as the performance of that estimate, depends on that choice. This is a very important reason that statisticians hated this kind of estimation. But they also hate the situation where you have tons of variance and grams of data, and that is where you are. You can prove (e.g. Komaki's paper gives an example of such proof) that your estimation will be better if you make this choice, so you're going to do it. Just don't expect to ever understand much about that choice. Apply the dont' know/don't care postulate - you will not know, so you're better off not caring. The defense of your estimation is the theorem that proves it's better.</p> <p>It should be very clear now that these three "nonclassical" effects in estimation theory are really distinct. I think most people don't really understand that. And to some extent it's easy to see why.</p> <p>Suppose you have an overparameterized generalized linear model (GLM), so your Fisher information is singular, but you do something like iteratively reweighted least squares for the Fisher scoring (because that's what you do with a GLM) and it turns out that the software you use solves with say Householder QR with column pivoting. It's down there under the covers enough that many statisticians performing this estimation would not necessarily know that is what was happening. But because the QR with column pivoting regularizes the system, effectively it is estimating the parameter in a reduced dimension system, where the reduction was done in a Fisher unitary coordinate system (because of the R in the QR factorization). It's really hard for people to understand what they are really doing when they are not aware of the effect of each step. We used to call this "idiot regularization" but I think "sleepwalker" is more accurate than "idiot". But what if the software package used modified Cholesky to solve the system? Well that actually amounts to a form of shrinkage (again in a Fisher unitary coordinate system), it can also be considered a form ("maximum a posteriori") Bayesian prior.</p> <p>So in order to sort out what these effects do, and what that means you should do, you need a reasonably deep understanding of the custody of your digits all the way from the data being observed through the final decision being taken (treatment, grant proposal, etc.).</p> <p>At this point (if you're still with me) you might want to know why didn't I just write a one line recommendation of some method suitable for beginners.</p> <p>Well there isn't one. If you really are in the high dimensional, low signal to noise case, then a high quality result is only available to someone who understands a pretty big part of estimation theory. This is because all three of these distinct effects can improve your result; but in each situation it is difficult to predict what combination will be the most successful. In fact the reason you cannot prescribe a method in advance is precisely because you are in the regime you are in. You can bail out and do something out of a canned program, but that has a good risk of leaving a good deal of the value on the table.</p> http://mathoverflow.net/questions/15780/can-we-extract-information-about-how-fast-a-function-decay-from-its-laplace-trans/15790#15790 Answer by Andrew Mullhaupt for Can we extract information about how fast a function decay from its Laplace transform? Andrew Mullhaupt 2010-02-19T05:05:04Z 2010-02-19T05:11:14Z <p>Can you control the oscillation of f(x) as x increases? If you can show that the ratio of f(x) to your 'simplified' form is 'slowly varying' then your asymptotics will probably work out.</p> <p>A simple example of what you cannot afford is a log-periodic oscillation; this is because the limits of oscillation of the function and Laplace transform need not agree. The simplest example of a log-periodic oscillation is a complex exponential:</p> <p>$\int_{0}^{\infty }e^{-st}t^{\alpha +i\beta }dt=\left[ \frac{\Gamma \left( \alpha +i\beta +1\right) }{\Gamma \left( \alpha +1\right) }e^{i\beta \log t}\right] \frac{\Gamma \left( \alpha +1\right) }{s}s^{-\alpha }$</p> <p>In a sense you can view the imaginary part of the exponential as a 'wobbly constant' which changes more and more slowly. The point is the amplitude of the wobble in the transform depends on β but not for the function.</p> <p>If you do have this sort of problem (it happens all the time in analysis of algorithms and chaotic dynamics) then you can for example resort to the 'gamma function method' of DeBruijn.</p> <p>The same thing holds true for the moment question. If you look up a counterexample for the moment problem, (e.g. Feller volume II p. 227) you see the ubiquitous log-periodic oscillation.</p> <p>Not surprisingly the log-periodic oscillation also shows up in convergence questions of Fourier series, but there it is not oscillating more and more slowly, but faster and faster.</p> http://mathoverflow.net/questions/13896/what-are-some-famous-rejections-of-correct-mathematics/15773#15773 Answer by Andrew Mullhaupt for What are some famous rejections of correct mathematics? Andrew Mullhaupt 2010-02-19T01:18:10Z 2010-02-19T01:18:10Z <p>It is only a slight exaggeration to say that the collected works of Hannes Alfven would fit this description.</p> http://mathoverflow.net/questions/15444/the-phenomena-of-eventual-counterexamples/15762#15762 Answer by Andrew Mullhaupt for The phenomena of eventual counterexamples Andrew Mullhaupt 2010-02-18T21:47:31Z 2010-02-19T00:59:01Z <p>R. M. Grassl and A. P. Mullhaupt, "Hook and Shifted Hook Numbers", Discrete Mathematics, Volume 79, Number 2, January (1990) pp. 153-167</p> <p>"An infinite number of counter examples is provided for the conjecture that a shifted tableau shape is uniquely determined by its multiset of shifted hook numbers. Nevertheless, the previous conjecture of the first author that there was only one example of nonuniqueness is discussed and it is shown that it is «almost» true, based on computer search."</p> <p>There were about five million examples before the counterexample, and approximately 1 mole of examples before the next counterexample is thought to occur.</p> http://mathoverflow.net/questions/14574/your-favorite-surprising-connections-in-mathematics/15763#15763 Answer by Andrew Mullhaupt for Your favorite surprising connections in Mathematics Andrew Mullhaupt 2010-02-18T21:56:09Z 2010-02-18T21:56:09Z <p>That the Jeffreys' prior in the pole-zero parameterization of a transfer function is the hyperbolic transfinite diameter of the support of the poles and zeros.</p> <p>It's my favorite because I just discovered it last month. I like laughing at my own jokes.</p> http://mathoverflow.net/questions/15614/interesting-applications-of-max-flow-and-linear-programming/15640#15640 Answer by Andrew Mullhaupt for Interesting applications of max-flow and linear programming Andrew Mullhaupt 2010-02-17T22:50:48Z 2010-02-17T22:50:48Z <p>You can prove the Birkhoff-von Neumann theorem directly with linear programming. Depending on your taste it is a quite elegant way to prove that result. There are basically two ways - one to use the conditions for a vertex of a polytope given by constraints to show that a doubly stochastic matrix which is a vertex of the Birkhoff polytope must have a row or column with only one nonzero entry, then induce. This does not use the full "fundamental theorem of linear programming".</p> <p>The other approach is to observe that at a vertex there is a full dimensional set of linear objectives for which the vertex is optimal, formulate the dual program and then show that the 2n unconstrained dual variables lie on an n dimensional space; complementary slackness then shows that the primal variable has only n nonzero elements, double stochasticity then guarantees there must be one in each row, one in each column, and each must be unity - therefore a permutation matrix. Obviously this approach really does exploit the linear program structure, if that is what you want to teach.</p> <p>I came up with this myself so don't know of an actual reference, but it should not be that novel.</p> <p>You can also prove Birkhoff-von Neumann are a max flow/min cut theorem (which is pretty well known) but I do not find that as elegant. However if you are emphasizing max flow/min cut as opposed to the linear programming structure, then you might want to do that one.</p> http://mathoverflow.net/questions/15577/using-wavelet-transforms-to-approximate-matrices/15639#15639 Answer by Andrew Mullhaupt for Using Wavelet Transforms to Approximate Matrices Andrew Mullhaupt 2010-02-17T22:35:10Z 2010-02-17T22:35:10Z <p>Wavelets could work, but they're not the only option. Have you had a go with bandwidth reduction reordering? (Or, more subtle - reordering to reduce "fill in" of the Cholesky factor?)</p> <p>Have you looked at the inverse? Have you looked for low grade (aka semi-separable) structure?</p> <p>Both of these could lead a bit far afield, but there are very powerful techniques of model reduction which can be applied; the way the inverse elements decay away from the main diagonal can be diagnostic for some of this.</p> <p>You might consider some sort of multipole acceleration if your matrix is big enough.</p> http://mathoverflow.net/questions/15654/extreme-point-compact-convex-set/39058#39058 Comment by Andrew Mullhaupt Andrew Mullhaupt 2010-09-17T08:48:48Z 2010-09-17T08:48:48Z The idea was to try and get a proof that a convex set has an extreme point without using any of the &quot;transcendental&quot; lemmata - like Axiom of Choice, Zorn's Lemma, Hahn-Banach, etc. In complete generality, it appears that Krein Milman requires at least one of these big guns. On the other hand, we also found that my simple proof works in most of the spaces I care about, and the other proof (construct any strictly convex function in the space) is even simpler and can work in all the spaces I care about; I used that approach in my notes. http://mathoverflow.net/questions/15836/oneupsmanship-and-publishing-etiquette/15837#15837 Comment by Andrew Mullhaupt Andrew Mullhaupt 2010-02-28T02:55:13Z 2010-02-28T02:55:13Z @Stankewicz: That's a good point. http://mathoverflow.net/questions/15836/oneupsmanship-and-publishing-etiquette/15837#15837 Comment by Andrew Mullhaupt Andrew Mullhaupt 2010-02-20T16:31:22Z 2010-02-20T16:31:22Z My suggestion was to &quot;SEE IF the other author WANTS to be a co-author&quot;. It seems people are wrongly assuming that this could only mean &quot;see if you can subscribe the other author to the existing paper&quot;. These are quite different things, and I don't think people have thought that through. Establishing a collaboration could have many different editorial results; in particular, one possibility is that the other author might suggest useful ways to improve the paper but not subscribe. Or other further significant results might be forthcoming from that author. That's why SEE IF and WANTS. http://mathoverflow.net/questions/15836/oneupsmanship-and-publishing-etiquette/15837#15837 Comment by Andrew Mullhaupt Andrew Mullhaupt 2010-02-19T23:57:27Z 2010-02-19T23:57:27Z I didn't know that community wiki had that property. Having my reputation drained on this wouldn't be unreasonable though. The acadmic thinking about publication is a lot more cutthroat than mine. But in industry, I put a premium on evidence of good collaboration skills when hiring mathematicians. In fact, the one thing I would say about the top tier tenured faculty I did hire at one point is that they did not collaborate anywhere near as well as people we hired as postdocs. http://mathoverflow.net/questions/15836/oneupsmanship-and-publishing-etiquette/15837#15837 Comment by Andrew Mullhaupt Andrew Mullhaupt 2010-02-19T22:04:43Z 2010-02-19T22:04:43Z I see by my draining reputation that this one is not too popular. But consider the original questioner has pointed out that his result is not completely new - he is simply patching up the lack of generality of previous work. Drawing lines in the sand about credit for that seems a bit of a stretch. http://mathoverflow.net/questions/13896/what-are-some-famous-rejections-of-correct-mathematics/15773#15773 Comment by Andrew Mullhaupt Andrew Mullhaupt 2010-02-19T20:34:50Z 2010-02-19T20:34:50Z I think the easiest way to fit that into the limited characters allowed here is to point at the wikipedia page for Alfven waves: <a href="http://en.wikipedia.org/wiki/Alfv%C3%A9n_wave" rel="nofollow">en.wikipedia.org/wiki/Alfv%C3%A9n_wave</a> This was the rule, not the exception for Alfven. Even after his Nobel he didn't get much respect, (except from guys like Fermi and Chandrasekhar - you could do worse). It's also worth looking at his wikipedia page: <a href="http://en.wikipedia.org/wiki/Hannes_Alfv%C3%A9n" rel="nofollow">en.wikipedia.org/wiki/Hannes_Alfv%C3%A9n</a> http://mathoverflow.net/questions/15577/using-wavelet-transforms-to-approximate-matrices/15639#15639 Comment by Andrew Mullhaupt Andrew Mullhaupt 2010-02-19T20:04:34Z 2010-02-19T20:04:34Z It's OK to show us a bit of the inverse. http://mathoverflow.net/questions/15654/extreme-point-compact-convex-set/15663#15663 Comment by Andrew Mullhaupt Andrew Mullhaupt 2010-02-19T18:01:28Z 2010-02-19T18:01:28Z This question was for a mathematical preliminary section in some lecture notes I am writing. If I use it, how should I cite the example you gave? http://mathoverflow.net/questions/15550/microarray-tesing-if-a-sample-is-the-same-with-high-variance-data/15826#15826 Comment by Andrew Mullhaupt Andrew Mullhaupt 2010-02-19T17:38:55Z 2010-02-19T17:38:55Z Mathoverflow insists on changing the display of numbers in my sections - I actually have typed in 1. 2. and 3. I do not know why they display as 1. 1. and 1. in the final. http://mathoverflow.net/questions/15444/the-phenomena-of-eventual-counterexamples/15762#15762 Comment by Andrew Mullhaupt Andrew Mullhaupt 2010-02-19T00:58:27Z 2010-02-19T00:58:27Z I think you're right. It's been a while since I looked at that; I'll edit my post. http://mathoverflow.net/questions/15614/interesting-applications-of-max-flow-and-linear-programming/15640#15640 Comment by Andrew Mullhaupt Andrew Mullhaupt 2010-02-18T21:41:39Z 2010-02-18T21:41:39Z Not off the top of my head, you can take any of the proofs of Birkhoff-von Neumann by Hall's Theorem (for example here: <a href="http://planetmath.org/?op=getobj&amp;from=objects&amp;id=3611" rel="nofollow">planetmath.org/&hellip;</a>) smash that together with a proof of Hall's Theorem by max flow/min cut (<a href="http://www.cs.umass.edu/~barring/cs611/lecture/11.pdf" rel="nofollow">cs.umass.edu/~barring/cs611/lecture/11.pdf</a>). I did see a max flow/min cut version of BvN on the internet a while back. Surprisingly, I haven't previously seen the versions I sketched in this thread. The vertex condition approach seems the most direct possible way to go. The dual version is microscopically less obvious. http://mathoverflow.net/questions/15654/extreme-point-compact-convex-set/15657#15657 Comment by Andrew Mullhaupt Andrew Mullhaupt 2010-02-18T21:11:56Z 2010-02-18T21:11:56Z @Gerald: HB does not imply AC, I infer from your remark that KM does not imply HB as well as KM does not imply AC. Without HB does &quot;nonempty compact convex set has an extreme point&quot; (EP) still imply KM? It appears KM always implies EP. My work does not avoid Hahn-Banach, I'm just asking out of curiosity. Having opened the can of worms, I might as well get to the bottom of it. http://mathoverflow.net/questions/15654/extreme-point-compact-convex-set Comment by Andrew Mullhaupt Andrew Mullhaupt 2010-02-18T03:44:14Z 2010-02-18T03:44:14Z Well I'll change my screen name if that is more comfortable here. http://mathoverflow.net/questions/15654/extreme-point-compact-convex-set/15663#15663 Comment by Andrew Mullhaupt Andrew Mullhaupt 2010-02-18T03:42:13Z 2010-02-18T03:42:13Z I should have been clear that I was looking for a simplicity/generality trade off. I just didn't want to use a needlessly finite dimensional proof just because it's a finite dimensional matter at hand, short of using high caliber results which are out of scope. I like your suggestion. http://mathoverflow.net/questions/15654/extreme-point-compact-convex-set/15657#15657 Comment by Andrew Mullhaupt Andrew Mullhaupt 2010-02-18T03:24:07Z 2010-02-18T03:24:07Z Yes I see your point that I don't need compactness of the unit ball. The infinite dimensional spaces that appear in this field are normally Hilbert spaces, Hardy spaces, and sometimes L^p, so maybe this argument is not a total loss in infinite dimensions.