De Rerum Natura

Our technical report on how to recover precision error estimates with ${\ell }_1$-minimization has been accepted by the 2008 International Conference on Machine Learning.

The paper originally got three anonymous reviews. Two were positive, one strongly negative. In our response to the reviews, we agreed with the general criticism by the reviewers that one experimental demonstration is not enough. In our precision error papers so far, we have only been using one dataset — aerial photographs from the Twenty-Nine Palms region in California. So we are going to include some results from North Carolina forest data to show that our technique works for all sorts of images.

Readers of previous posts may note that besides maps, the precision error has been recovered for questions in a multiple-choice-quesiton (MCQ) exam. It would be nice to include this in our ICML paper, but the title of the paper is “Autonomous geometric precision error estimation in low-level computer vision tasks” so it seems incongruous to do so.

The paper was submitted in early January. Afterwards, we realized that our precision error technique for elevation errors in maps applies to any set of models that make scalar predictions about multiple entities. We are now working on a draft for a Science magazine article that will combine the examples from maps and exams to illustrate the wide applicability of our technique.

I’ve been nibbling on a bunch of books for the past week. They are, in no particular order:

Mirage: Napoleon’s Scientists and the Unveiling of Egypt deals with the scientific side of Napoleon’s famous imperialistic debacle — the 1798 invasion of Egypt. We tend to think of historical knowledge as continuous in time. If we know something now, everyone in the past must have known it. This book shatters that illusion. Ancient Egypt had been lost to humanity for centuries. The savants in the expedition started the recovery of this lost civilization. One young scientist that participated in the expedition was Joseph Fourier. I have read many biographical sketches of Fourier but I do not recall ever reading that he was part of Napoleon’s Egypt “expedition”. One of the categories in this blog, “Fourier analysis”, is named after him. We can thank Fourier for many things but one that comes immediately to mind is the MP3 music file standard.

The World Without Us has been getting a lot of press. It clearly deserves it. The premise — what would happen to the world if we just disappeared overnight — forms a great hook on which to hang all sorts of scientific observations about biology, the durability of materials, the relentless march of entropy, human evolution, and much more. I highly recommend this book.

Author Stewart Mader makes a convincing case in his wikipatterns book that wikis are a powerful collaboration tool. I have dabbled briefly with wikis. I’ve come to rely more and more on wikipedia to understand technical terms quickly. I just don’t practice collaboration with them.

This may change if a grant that we currently have pending with the NSF is approved. We proposed a collaboration with New Mexico scientists at the Jornada Experimental Range to develop a photogrammetric system for UAV images. Collaboration management was a mandated section of the proposal and we included a slew of tools that we currently use — revision control, emails, issue tracking — as well as the MediaWiki software in a list of tools we intend to use to facilitate managing the work related to the collaboration.

The more tools like Wikis become part of our scientific practice, the more I wondered how anybody got things done in the past. How did scientists communicate before emails? I know letters were written. I wrote a few of them back in graduate school. But it seems so strange now to think of writing a letter to someone instead of sending an email.

The scientist Faraday was self-educated. As a young man he was an apprentice to a bookbinder. He read many of the scientific works he bound. It is said that his notebooks were beautifully bound by him. I visited the Faraday museum in 2004 but the notebooks were only accessible to scholars and did not form part of the public display. I, too, have notebooks of my scientific work and because of this I have picked up little historical tidbits about the usefulness and devotion of scientist to their notebooks.

I am now reading The Telephone Gambit. At its center is an abrupt time gap in Alexander Graham Bell’s notebook just before his invention of the telephone. I have not finished the book yet. But the author makes a convincing case that Bell stole the idea from a patent by Elisha Gray.

In a previous post I erroneously claimed that the mathematician Hilbert advocated teaching geometry in the dark. Hilbert’s “Foundations of Geometry” axiomatized the subject and carried out its exposition without a single diagram. I found the correct attribution yesterday while re-reading Hofstadter’s foreword to “King of Infinite Space”, a biography of geometer David Coxeter.

Hofstadter does not say for certain who advocated the practice of geometry in the dark and can only recollect that it was some 19th century geometer — possibly Steiner, Plucker, von Staudt, or Feuerbach.

The Archimedes Codex is turning out to be a great read on the importance of diagrams in Greek mathematics, the transmission of ancient knowledge to present times and modern document forensic techniques. The book is written by two principals of the Archimedes Palimpsest Project. I just finished reading Reviel Netz’s explanation of why visual thinking has become so reviled in modern mathematics and it was so simple to understand that I want to share it with readers

As Netz explains it, the problem with diagrammatic proofs hinges on the fact that diagrams do not have the generality of language. For example, if one wants to discuss triangles in general, a diagram of a triangle thwarts that generality since, by construction, it represents a specific triangle. The ambiguity of language is turned to good use by turning it into an encompassing generality.

Netz argues that Greek diagrams are schematic not illustrative. Evidence from the Palimpsest and later medieval documents strongly suggests that Archimedes drew a polygon inscribed inside a circle with circular arcs rather than straight lines in his “Spheres and Cylinders”. He also drew straight lines for sections of a spiral in “On Spirals”. All of this is the reverse of what modern diagrams in editions of Archimedes would do. Our diagrams are illustrative, theirs were schematic.

This schematic versus illustrative distinction is Netz’s explanation for why the Greeks never made a logical mistake in their mathematical works even though the diagrams are central to their exposition.

This bias in 20th-century mathematics to visual proofs (Coxeter mentions that Hilbert thought geometry should be taught in a darkened room!) is now creating a backlash that could bring diagrams back into the heart of proofs. Take a look at Euclid and His Twentieth Century Rivals: Diagrams in the Logic of Euclidean Geometry for how that most non-geometric machine — the computer — is making visual proofs rigorous.

I have started to work thru Anton Deitmar’s “A First Course in Harmonic Analysis”. This is background reading for my perusal of the Compressed Sensing literature. On the first chapter he wants to prove the convergence in the $L^2$-norm of Fourier series for periodic functions. To do this he asserts (offhandedly to my untrained eyes) the equality
$$\sum _{k=1}^n\cos (2\pi kx)=\frac{\sin ((2n+1)\pi x)}{2\sin (\pi x)}-\frac{1}{2}.$$

It is embarrassing how long it has taken me to prove this on my own. In the end, Euler’s formula,
$$e^{i\theta }=\cos \theta +i\sin \theta ,$$
ended up being the right thing to use since it turns the proof into checking the equality of polynomials. My proof is inductive. First prove the $n=1$ case,
$$\cos (2\pi x)=\frac{\sin (3\pi x)}{2\sin (\pi x)}-\frac{1}{2},$$
by rewriting it as,
$$\sin (\pi x)\cos (2\pi x)=\frac{1}{2}(\sin (3\pi x)-\sin (\pi x))$$
$$\left(\frac{e^{i\theta }-e^{-i\theta }}{2i}\right)\left(\frac{e^{i2\theta }+e^{-i2\theta }}{2}\right)=\frac{1}{2}(\frac{e^{i3\theta }-e^{-i3\theta }}{2i}-\frac{e^{i\theta }-e^{-i\theta }}{2i}),$$
the demonstration of the equality becomes a polynomial check. Induction to the case $n+1$ by using the same trick with Euler’s formula leads to the general proof of the formula.