Thursday, May 21, 2009

Ida y Vuelta: a Tail?

It was all over the news and the net yesterday: the missing link has been found! Darwin has finally been proven right!

Hold on a minute. There's no "missing link"--we have example after example of the evolutionary forebearers of Homo sapiens. Evolution was not in question--we already knew Darwin was right. Ida (the name of the fossil found) is not a direct ancestor of humans--the fossil represents a transitional form between lemurs and other apes.

I do not mean to denigrate what is obviously a very important and exceptionally well-preserved fossil that may provide important insights into primate evolution. My objection is to the media frenzy that gives the impression that this is the final resolution to a scientific debate about evolution that a) never existed and b) would not have been resolved by the fossil in question if it did exist.

We have Neanderthals, Homo erectus, Homo habilis, Australopithecus afarensis, and more. Why is Ida, who isn't even our evolutionary forebearer, the "missing link"? I have a theory: it's about the tail. I think that somehow, through all the monkey-human-common-ancestor debate, through the discovery of early hominids, through all of the discussion about increasing cranial capacity and tool usage, everyone has really been wondering "what about the tail? What happened to the tail?" The media was holding out for something it could tout as a human ancestor (or something close enough) with a tail to finally declare the "missing link" found and the evolution debate settled.

You might say that this is all for the best--that the false debate, however belatedly, is finally being put to rest with one last hurrah. But here's a scenario that makes be cringe just to think about it: suppose it turns out that this fossil, which was recovered from a private collection, turns out to be something other than the specimen that the scientist in question thinks it is. Scientists makes mistakes--that's what peer review is for. Then suddenly we've breathed new life into a misconception about evolution that has been hanging around for way too long already.

Tuesday, May 12, 2009

Golomb Rulers/Squares/Rectangles

Yesterday I came across an interesting example of the isolation of academic fields from one another.

A common design parameter for antenna arrays is to try to obtain uniform coverage of the aperture plane to get as many independent measurements as possible. Interferometers sample the aperture plane at locations that correspond to the difference vectors between antenna elements. For example, in one dimension, if I put four antennas at locations (0,1,4,6), then that array would sample the difference set of those positions: (1,2,3,4,5,6). However, if I put antennas at (0,1,2,3), the difference set would only include (1,2,3) with 1 occuring 3 times and 2 occuring twice. For the purpose of uniformly sampling an aperture, redundant spacings are lost measurements. We're looking for a minimum-redundancy array.
There have been a few papers in radio astronomy on minimum-redundancy arrays for the more useful 2-dimensional case, including Golay (1971), Klemperer (1974), and Cornwell (1988).

Thinking that this might be a mathematical problem of interest, I ran it by a good friend of mine: Phil Matchett Wood--a mathematician at Princeton. He quickly uncovered the equivalent problem as formulated in math literature: Golomb rulers in 1-D and Golomb rectangles in 2-D. Some relevant papers on the subject are Shearer (1995), Meyer & Jaumard (2005), Robinson (1985), and Robinson (1997). These papers are on the exact problem and describe applications to "radar and sonar signal detection", obviously referring to the need for independent aperture samples in radar and sonar interferometers. Somehow, the differing nomenclature between these fields was never quite bridged, and so there has not been and cross-referencing between these two formulations of the same underlying problem. People like to talk about the possibility that relevant research in one field goes unnoticed by other fields. This is the first time I've across it myself, though.

Tuesday, May 5, 2009

Compressed Sensing and Wiener Filtering

Today I'm trying to expand my understanding of how we can best remove contaminant signals from the data we take with the Precision Array for Probing the Epoch of Reionization (PAPER). There is a specific problem I want to make sure we can solve for PAPER. Foregrounds to our signal, particularly synchrotron radiation, are expected to be very smooth with frequency. The idea put forth by the MWA and LOFAR groups is that by observing the same spatial harmonics at multiple frequencies, we should be able to remove such smooth components to suppress them relative to the cosmic reionization signal we are looking for. However, generating overlapping coverage of spatial harmonics as a function of frequency is expensive. My intuition is that since foregrounds do not have a spatial structure that changes dramatically with frequency, we shouldn't need to sample a given spatial harmonic very finely in frequency to get the suppression we want. This would allow us to spread our antennas out a little more and get measurements of the sky at a variety of spatial modes.

In many ways, our problem is analogous to what was done with the Cosmic Microwave Background (CMB). For foreground removal in CMB work, Tegmark and Efstathiou (1996) begin with an assumption that foregrounds can be described as the product of a spatial term and a spectral frequency term. This allows them to construct Wiener filters that use the internal degrees of freedom of their data, together with a model of their foreground and a weighting factor based on the noisiness of their data, to construct a filter for removing that foreground. For the most part, this is standard Wiener filtering, except they have to be careful about what they do to their power spectrum, so they apply a normalization factor to correct for a deficiency in Wiener filters. Tegmark (1998) goes on to generalize this technique for foregrounds that vary slowly with frequency. I'm in the process of wading through these papers, but they seem to be directly applicable to what we are doing, and seem to confirm my suspicions that synchrotron emission should be well-enough behaved to require only sparse frequency coverage of a wavemode in order to be suppressed.

Another tactic that I am investigating is that of compressed sensing which I was alerted to in talks by Scaife and Schwardt at the SKA Imaging Workshop in Socorro this last April. The landmark paper on this principle seems to be Donoho (2006), where it is shown that the compressibility of a signal (being sparse for some choice of coordinates) is a sufficient regularization criterion to faithfully reconstruct signals using a small number of samples. In a way, this technique has an element of Occam's Razor in it--it tries to find a solution, in some optimal basis, that needs the fewest non-zero numbers to agree with the measured data. At least, that's my take on it without having finished the paper.

The relevance of compressed sensing to image deconvolution is explored in Wiaux et al (2009), and it seems to be powerful. I'm excited by this deconvolution approach because it meshes well with the intuitive approach I've been taking to deconvolution, which was to use wavelets and a Markov Chain Monte Carlo optimizer to find the model with the fewest number of components that reproduces our data to within the noise. Compressed sensing seems to be exactly this idea, but is agnostic about the basis chosen, instead of mandating one like wavelets. Anyway, this technique may also be relevant to our foreground removal problem because we might be able to use it to construct the minimal foreground model implied by our data. For synchrotron emission, which should have smoothly varying spatial structure with frequency, I envision that this could construct a maximally smooth model that would allow us to use sparse frequency coverage to remove the foreground emission to the extent that it is possible to do so.

Monday, May 4, 2009

Is Tenure a Problem in Science Departments?

Somewhat belatedly, I wanted to comment on the New York Times Op-Ed by Mark Taylor that addresses some of the flaws of the current academic system and proposes some solutions. The central problems that Taylor highlights in his article are that academic departments are too isolated from one another and from the world, and that there aren't enough academic positions for all of the people who are getting doctoral degrees these days. Taylor makes some very good points, but his article is strongly influenced by his experiences in a humanities department, and I am not sure how relevant his suggestions are for a science department.

Astronomy suffers from many of the same problems Taylor describes. There are far more graduate students and post-doctoral researchers than there are tenured professorial positions, and yet students are trained as if they were all to be professors. However, science students are often not paying their own tuition (it is paid by the grant of a supporting professor) and there are more options for science students outside of tenure-track positions because of the many sources of external funding that support scientific research. Unlike the humanities, there are a variety of scientific programming and research positions for graduates who are not seeking professorial positions. These positions are aligned with the education students receive through their doctoral research.

This isn't to say that there is not a major problem in scientific disciplines concerning the ratio of student positions to professional positions. Rather, it is that the problem may not be as closely tied to tenure and the longevity of tenured professors as in the humanities. The problem may be that professional positions available to graduating students are being occupied by the students themselves. Scientific research in the United States relies on a large pool of skilled labor. Currently, this labor is being bought at well under market price in the form of cheap graduate student researchers. If more of these positions were filled by full-time research professionals, we might have a healthier employment system for scientific academia.

The problem is that this raises the price of research in the United States and may result in a reduction in the total number of projects (and therefore, researchers) that can be supported. From the perspective of researchers, this may be a healthier state of affairs--to not be misled into spending 5-7 years underpaid as a graduate student only to find that the only way to continue to do what you've been trained to do is to continue to be underpaid. But unlike in the humanities, science graduate students usually have not accumulated debt beyond their undergraduate education and they have been supported (however cheaply) through this process. The solution, then, may simply be to ensure that prospective graduate students in science are well-informed about what employment prospects they should expect after they file their dissertation.