The Annals of Statistics published a “discussion paper” on Conditional growth charts by Y. Wei and and X. He, with 4 comment papers and a rejoinder from the authors. It’s reminiscent of the Crooked Timber book events (Iron Council was a particular favorite), only significantly more formal. It’s a new format to me for mathematical publication, and almost feels like “peer review in public,” with the mean comments taken out. Perhaps if I get some time I’ll read the paper for real and see what the fuss is about…
Prasad brought work to a halt earlier this week by issuing the following little conundrum. One indepdenent fair 6-sided die is rolled for each of n people. Each person’s number is written on a card and stuck on their head so that they can see everyone else’s number but their own. The people are not allowed to communicate after the numbers are assigned, and from just viewing the others’ numbers they must guess their own. They can come up with any rule that they like for doing this.
What rule should you choose to that is the probability that all of them guess correctly is maximized, and what is that probability?
This is a variation on the “colored hats game,” but the criterion you want to maximize is slightly different than in some instances of the game.
On Monday the SF Symphony Chorus had their first rehearsal with John Adams and Peter Sellars for A Flowering Tree. The opera, which received its premiere at the New Crowned Hope festival, is adapted from a Kannda folktale. The libretto is an amalgamation of the folktale text, Tamil love poems, and Virasaiva religious poems. Our rehearsals to this point have been somewhat routine — learning the notes, getting a handle on the tricky rhythms and meter changes, and using little tricks to help ourselves be heard over the orchestra.
Monday’s rehearsals catapulted us into the world of characterization and theater. Our performance will be “semi-staged,” from what I understand, and I saw bits of scenery backstage before the rehearsal. Turning a massive chorus into a dramatic agent is no small task, and Sellars was as clear and effective as any director I have seen. He knew at what level he had to talk to the chorus to get the effect he wanted, and we the change in effect from the first readthrough to the second (after some direction) was huge. Adams will be conducting us, and he is likewise clear and direct in his requests and his conducting. It’s a real treat working on a piece like this, and after Monday’s rehearsal I am certain that the audience will be wowed. Although I do have to say that next week will put my voice through the wringer, so I better stock up this weekend on lemon juice, honey, and ginger…
Sometime if I have time I’ll write about our first rehearsal with Adams and Sellars. I have also written a small note on some of the religious poetry used in the libretto.
A Flowering Tree
by John Adams
libretto by John Adams and Peter Sellars
John Adams, conductor
Peter Sellars, director
Jessica Rivera, soprano
Russell Thomas, tenor
Eric Owens, bass
SFS Chorus, chorus
America’s foremost living composer, John Adams, imagines rich and beautiful worlds. This SFS co-commission, inspired by The Magic Flute, is an escape into dream and myth and comes on the heels of Adams’s opera Doctor Atomic. Peter Sellars returns to direct this semi-staged production. The premiere of any new Adams work is an event not to be missed.
Thursday 3/1 — Saturday 3/3, 7:30 PM
Asymptotically Optimal Approximation of Multidimensional pdf’s by Lower Dimensional pdf’s
IEEE Transactions on Signal Processing, V. 55 No. 2, Feb. 2007, p. 725–729
The title kind of says it all. The main idea is that if you have a sufficient statistic, then you can create the true probability density function (pdf) of the data from the pdf of the sufficient statistic. However, if there is no sufficient statistic, you’re out of luck, and you’d like to create a low-dimensional pdf that somehow best captures the features you want from the data. This paper proves that a certain pdf created by a projection operation is optimal in that it minimizes the Kullback-Leibler (KL) divergence. Since the KL divergence dictates the error in many hypothesis tests, this projection operation is good in that decisions based on the projected pdf will be close to decisions based on the true pdf.
This is a correspondence item, so it’s short and sweet — equations are given for the projection and it is proved to minimize the KL divergence to the true distribution. Examples are given for cases in which sufficient statistics exist and do not exist, and an application to feature selection for discrimination is given. The benefit is that this theorem provides a way of choosing a “good” feature set based on the KL divergence, even when the true pdf is not known. This is done by estimating an expectation from the observed data (the performance then depends on the convergence speed of the empirical mean to the true mean, which should be exponentially fast in the number of data points).
The formulas are sometimes messy, but it looks like it could be a useful technique. I have this niggling feeling that a “bigger picture” view would be forthcoming from looking at information geometry/differential geometry viewpoint, but my fluency in those techniques is lacking at the moment.
Update: My laziness prevented me from putting up the link. Thanks, Cosma, for keeping me honest!
I finally watched this documentary that was sent to me a few weeks ago called Acting Like a Thief. It is about the a street theatre organization in India from the Chhara community, a group that was labeled by the British as a “criminal tribe.” The disctrimination continues to this day. This is what taking community action via theater is about.
Also related, the Human Rights Watch report on discrimination against the Dalit community in India.
Only this time, I am taking square to another dimension. Two baker’s dozen may have been the cat’s meow, but my new cubic nature will yield many sparkling treasures, I am sure.
Locating Mobile Nodes with EASE : Learning Efficient Routes from Encounter Histories Alone
M. Grossglauser and M. Vetterli
IEEE/ACM Transactions on Networking, 14(3), p. 457- 469
This paper deals with last encounter routing (LER), a kind of protocol in which location information for nodes is essentially computed “on the fly” and there is no need to disseminate a location table and update it. They consider a grid toplogy (a torus, actually) on which nodes do a simple random walk. The walk time is much slower than the transmission time, so at any moment the topology is frozen with respect to a single packet transmission. Every node i maintains a table of pairs (Pij, Aij) for each other node j where P is its last known position and A is the age of that information. In the Exponential Age SEarch protocol (EASE) and GReedy EASE (GREASE), a packet keeps in its header an estimate of the position of its destination, and rewrites that information when it meets a node with a closer and more recent estimate. Because the location information and mobility processes are local, these schemes performs order-optimally (but with worse constants) to routing with location information, and with no overhead in the cost.
In particular, EASE computes a sequence of anchors for the packet, each one of has an exponentially closer estimate of the destination in both time and space. For example, each anchor could halve the distance to the destination as well as the age of that location estimate. This is similar in spirit to Kleinberg’s small world graphs paper, in which routing in a small world graph halves the distance, leading to a log(n) routing time, which comes from long hops counting the same as local hops. Here long hops cost more, so you still need n1/2 hops. The paper comes with extensive simulation results to back up the analysis, which is nice.
What is nicer is the intuition given at the end:
Intuitively, mobility diffusion exploits three salient features of the node mobility processes: locality, mixing, and homogeneity…
Locality ensure[s] that aged information is still useful… Mixing of node trajectories ensures that position information diffuses around this destination node… Homogeneity ensure[s] that the location information spreads at least as fast as the destination moves.
The upshot is that location information is essentially free in an order sense, since the mobility process has enough memory to guarantee that suboptimal greedy decisions are not too suboptimal.
I used to write a lot about each book I read but of course I don’t have the time. And besides, I haven’t been reading as quickly as I used to. Unfortunately my dream of more theory/nonfiction hasn’t come to fruition. So besides The New Yorker and Harpers I’ve delved into some more reads:
The Thursday Next Novels : The Eyre Affair, Lost In a Good Book, The Well of Lost Plots, Something Rotten (Jasper Fforde) — a bit of brain candy, these, but a lovely bit of literary comedy. It’s like Terry Pratchett with an ear for the classics (which I haven’t really read, truth be told). The books follow a Special Operations agent named Thursday Next, who lives in a world in which the line between fiction and reality is permeable, and in which there is a special division for literary crimes. What fun! Actually, the best part about it is that it actually makes me want to go back and read Great Expectations and the like. Even Jane Austen seems like it could be fun after these books, despite my falling asleep the last time I tried to read her. Maybe I have become more sensitive over time…
The Big Over-Easy (Jasper Fforde) — A spin-off series from the above, with a similar sensibility. Who wouldn’t like a murder mystery set in a world in which nursery rhyme characters populate the town? It’s more like Who Framed Roger Rabbit? than The Thin Man, so noir fans beware.
Speaking of Siva (AK Ramanujan)) — This was a slim volume of Bhakti-movement poems (vacanas, or utterances) from the Virasaiva community and were translated from the original Kannada. The poems themselves are quite beautiful — like most Bhakti poems, they get at the heart of what love and God and the self are in a relatively un-self-conscious way. One of the poems, by Allama Prabhu, is used in A Flowering Tree, the new John Adams opera that I am singing as part of a semi-staged SF Symphony concert at the beginning of next month. As one of the few, if only, South Asian singers in that concert, I feel a particular need to educate myself about the textual underpinnings. The poem was translated into Spanish before being set into music, and I’m not sure the music is appropriate to the relgious/philosophical outlook of the poem. Adams is free to set the poem as he likes, and he isn’t trying to don some mantle of authenticity, so I find his musical choices interesting, but I think the text serves his end, rather than the reverse. Perhaps I will write more on that later.
I’m currently working on a number of books in parallel. More when I actually finish some of them.
The Bulwer-Lytton winners have been announced. The winner?
Detective Bart Lasiter was in his office studying the light from his one small window falling on his super burrito when the door swung open to reveal a woman whose body said you’ve had your last burrito for a while, whose face said angels did exist, and whose eyes said she could make you dig your own grave and lick the shovel clean.
I have no idea what kind of body would say that to me, but it would take more than a beautiful woman to stop me from eating burritos.