September 26, 2012
Software | Tags: migration, rss |
September 20, 2012
Math, Sparsity | Tags: Basis pursuit, compressed sensing, probability measures, sparsity, weak* convergence |
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Let be a compact subset of and consider the space of continuous functions with the usual supremum norm. The Riesz Representation Theorem states that the dual space of is in this case the set of all Radon measures, denoted by and the canonical duality pairing is given by
We can equip with the usual notion of weak* convergence which read as
We call a measure positive if implies that . If a positive measure satisfies (i.e. it integrates the constant function with unit value to one), we call it a probability measure and we denote with the set of all probability measures.
Example 1 Every non-negative integrable function with induces a probability measure via
Quite different probability measures are the -measures: For every there is the -measure at this point, defined by
In some sense, the set of probability measure is the generalization of the standard simplex in to infinite dimensions (in fact uncountably many dimensions): The -measures are the extreme points of and since the set is compact in the weak* topology, the Krein-Milman Theorem states that is the weak*-closure of the set of convex combinations of the -measures – similarly as the standard simplex in is the convex combination of the canonical basis vectors of .
Remark 1 If we drop the positivity assumption and form the set
we have the is the set of convex combinations of the measures (). Hence, resembles the hyper-octahedron (aka cross polytope or -ball).
I’ve taken the above (with almost similar notation) from the book “ A Course in Convexity” by Alexander Barvinok. I was curious to find (in Chapter III, Section 9) something which reads as a nice glimpse on semi-continuous compressed sensing: Proposition 9.4 reads as follows
Proposition 1 Let , and suppose that the subset of consisting of the probability measures such that for
is not empty. Then there exists such that
- and are convex combinations of at most -measures, and
- it holds that for all we have
In terms of compressed sensing this says: Among all probability measures which comply with the data measured by linear measurements, there are two extremal ones which consists of -measures.
Note that something similar to “support-pursuit” does not work here: The minimization problem does not make much sense, since for all .
September 17, 2012
In my previous post “Yes we can change. Why the DMV should change its name” I tried to make some advertisement for the petition that the “Deutsche Mathematiker-Vereinigung” should change its name to “Deutsche Mathematische Vereinigung”.
Yesterday I received the new issue of the Mitteilungen der DMV and there is the result:
- Votes cast:
- Valid votes:
- Invalid votes:
- Votes for the name change:
- Votes against the name change:
So, 53% votes against the change and hence, the DMV will stick to the meaning of “Union of Mathematicians” rather than “Union for Mathematics”.
On the one hand, this sound like a fairly close-run. But one the other hand I was surprised that so many people actually actively voted against the change. This seems to show that there is really a non negligible fraction of people who really have something against the proposed name and is not only in favor of keeping things as they are and, in consequence, make the effort to submit a vote against. Moreover, Martin Skutella writes in his editorial of the current issue of the Mitteilungen (in my own translation):
Reassuring, that the DMV braves, firm as a rock, the short-lived zeitgeist!
Hmm, I am not sure what he is intended to say (keeping in mind that his editorials are usually somehow satirically). In case there is no irony involved: Is he really trying to say that the change the DMV made in the last decade from a society which serves the mathematicians only (which basically meant professors at German universities) to an organization which sees its central mission to promote mathematics as a whole and on a broad scale ranging from pupils over student, researchers to business people and companies is just short-lived zeitgeist and in a few years we are back to the times in which the members of the DMV were only research mathematicians (at least professors) and when the Mitteilungen der DMV only contained latest news on the daily grind of a math professor at a university?
I am not sure, but I hope the name “Deutsche Mathematiker Vereinigung” will not hold back any of the “new” target audience (like pupils, students, teachers, math people in companies,…) to join the DMV.