De Rerum Natura

I have applied the autonomous difference equations to test the quality of ten out of twenty questions I used in a Physics exam I gave in the Spring of 2006 to an introductory class for engineering students. That dark square in position six of the matrix corresponds to the question least likely to be answered correctly. Only 64 students out of 250 answered correctly. The reason this happened was that I gave a very clever wrong answer that attracted most of the students (the correct answer but forgetting to take a square root). I have used the precision error covariance matrix to assess the test maker not the students!

This example also highlights the general applicablity of the precision error covariance matrix. There now exist two experimental verifications of its usefulness: digital elevation maps and test assessment.

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.

Today I was able to look at the precision error equations for the digital elevation models and see for the first time that they can be trivially generalized to other machine learning fields like information retrieval, information extraction, and bioinformatics. This is so embarrassingly simple that I cannot stop coming up with new variants every hour or so! The pattern is easy to explain, so I think any reader of this should be able to come up with a variant that applies to their area of work. Please let me know if you do so, I would like to start a catalogue of the many ways this can be done.

Here is the pattern. I’ll start with information retrieval. Assume you have a system that creates a relevance model of a corpus of documents. These relevance models are judgments of the form $r(d,q)=\{\mathrm{0,1}\}$. The notation is meant to capture that the relevance judgment is a binary decision on whether a particular document $d$ is relevant to a query $q$. Maybe you made that relevance model using maximum likelihood estimates, or maybe you used Latent Dirichlet Allocation. Each way you calculate that relevance judgment is a model. Assume that you have used different algorithms, or different parameter settings, or whatever, to come up with $n$ different relevance judgments for a set of test queries.

Each relevance judgment of a specific model $i$ can be written as
$$r_{\text{estimated}}(d,q{)}_i=r_{\text{true}}(d,q{)}_i+\delta (d,q{)}_i$$

Now consider the following quantities that can be calculated with these many relevance models
$$\sum _{q,d}(\frac{1}{E}\sum _{i=1}^E{r}_i(d,q))–(\frac{1}{M}\sum _{j=1}^M{r}_j(d,q))$$
These quantities would not include the $r_{\text{true}}$ value. It would cancel out. So the above equation can be written as
$$\sum _{q,d}(\frac{1}{E}\sum _{i=1}^E{\delta }_i(d,q))–(\frac{1}{M}\sum _{j=1}^M{\delta }_j(d,q)).$$ These equations would allow you to recover the precision errors $\{{\delta }_i\}$ for a collection of information retrieval models!

The pattern can now be generalized ad-infinitum to any machine learning task! You pick the model prediction for which you want to measure the precision error.

I’m writing a talk for the Machine Learning Lunch at UMass/Amherst. It has got me thinking about the issue of sparsity in the precision error. Here are the technical details. The precision error when using $n$ models leads to a $n^2$ covariance matrix. Because the matrix is symmetric, it really has $n(n+1)/2$ independent components. You can think of the covariance matrix as a data structure that lives in a $n(n+1)/2$. It does not inhabit the full space because cross-correlations must be less than the variance of the cross-correlated variables.

The autonomous difference equations that were first publishd in the Swiss paper lead to $n(n+1)/2–n$ independent equations. So the linear algebra system for the precision error covariance matrix is always under-determined by $n$ equations. The ${\ell }_1$ minimization technique advocated by Donoho would be able to solve this system if enough of the entries in the matrix were zero. This is what I mean by: is the precision error covariance matrix sparse enough?

You can calculate that the percentage of equations you have as the following fraction less than one
$$\frac{n(n+1)/2-n}{n(n+1)/2}.$$
This number is equal to
$$1–\frac{2}{n+1}$$
so it approaches a well-determined system (the value one) with a term that dies as order $1/n$. Doesn’t this mean that if you have enough models ($n$ large enough) you would be able to become sparse enough to reconstruct the covariance matrix?

This means that sparsity of the error of the signal behaves empirically different than the sparsity of the signal. This has profound implications for the wide-applicability of the theory Howard Schultz and I have developed.

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.

I’ve been sick all week but today has been the worst. In between my sleeping hallucinations I have been thinking a lot about a proposal I’m currently writing on the use of non-commutative harmonic analysis to study mapping error patterns. It has become clear that the approach we are advocating at the AIRS lab is applicable to other areas of machine learning. Before I describe in additional detail what I mean by this let me present a graphic that abstractly represents the scientific enterprise:
The work I am describing here lies at the bottom of the picture. It is an algorithmic prescription for understanding the precision of our models given a dataset used to construct those models. I present this diagram to make clear the limitations of our work. It is not a description or explanation of errors in general. It is a technique for probing the error patterns in your system. The hard work is still left to you on how to apply it for a specific system that constructs models from data, and its usefulness is not guaranteed. The technique may tell you nothing interesting about your system.

We can view model creation as a black box. It takes data inputs and produces a model. The crucial point is that in some machine learning situations the number of models we can build is very large. Data is model-redundant.The redundancy is sometimes continuous, for example, the initial position of a camera or the weight of a Lagrangian term. But it can also be discrete — a finite set of documents or photographs. Changing the data inputs can then be used to probe the variation of a system’s model predictions. These variations will not be completely random (i.e. patternless). Some documents are more informative, some photographs give us a better view. Therefore we can use the symmetry groups associated with our data inputs to Fourier analyze the model variation. This model variation or model precision is informative about the quality of the data and can be used to reject bad data or discount lower quality inputs.

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.

In a previous post I mentioned a way of Fourier analyzing the geometric precision error of DEMs. Today I realized that the scheme I proposed can only account for part of the error signal. The approach I proposed is correct but it can only capture one particular aspect of the total error. The simplest way of seeing this is to consider the $S_2$ symmetry group. This would be the one to use for $p=2$ photographs. From two photographs I can produce two DEMs: $A\to B$ and $B\to A$. The covariance matrix for these two DEMS would be a 2×2 covariance matrix of the form:
$$\left(\begin{array}{c}ab\\ bc\end{array}\right).$$
But the representation induced by $S_2$ on these two DEMs generates the matrices:
$$\left(\begin{array}{c}10\\ 01\end{array}\right)\text{and}\left(\begin{array}{c}01\\ 10\end{array}\right)$$
These two matrices cannot capture the three independent degrees of freedom in the 2×2 covariance matrix. Therefore, the induced representation cannot capture all of the possible errors that are observed when two photograps are used to produce two maps. But the representation would allow you to project out that component of the error that is explained by permutations of the images.

You would need at least $p=7$ photograps to have enough members in $S_p$ to completely model the variation in the DEMs observed when you use two photographs to produce a map. To understand the error that cannot be explained by the permutation group you would need to use three photographs to create a DEM. For $p$ photographs this would create $p*(p-1)*(p-2)$ DEMs. Since we can produce as much or even more than $p!$ DEMs from $p$ photographs, at some point we will always overwhelm the representational power of the symmetry group of $p$ objects (in this case, $p$ photographs). What error remains after we project out the component that can be modelled by $S_n$? I hypothesize that it would be error that can be further Fourier analyzed by using the symmetry group associated with the orientation and positions of the cameras. These parameters are themselves error prone and would, by virtue of their geometry, only induce certain error patterns.

This viewpoint of the errors would therefore view the observed error as one that can be captured by a succesive series of symmetry groups. One component would be that related to the finite group of $S_p$. Another component would be that one induced by translations and rotations of the camera positions and orientations. Like any real theory of errors, this approach would only peel away layers of error — always remaining would be a nugget of error that would require more and more complex models to decompose. The second law of thermodynamics is not violated!

The metaphor to x-raying in the title of this post comes from using Fourier analysis to study X-ray diffraction photographs by crystals. Crystals induce a certain periodicity on the scattered X-rays even when the sample is crushed into a powder. In other words, the randomly scattered blocks of crystal in the powder individually send a perfect difraction pattern. But the X-ray photograph records the mismash of the signals — the picture is blurry. Nonetheless, the bluriness has a symmetry component that comes from the periodic structure of the crystals and therefore Fourier analysis is able to pick the symmetry in the x-ray caused by the crystal periodicity. The Fourier decompositions for geometric errors are doing the same thing. There are many sources of errors in DEMs from aerial photographs. Some come from the fact that you used individual photographs to create the maps. This component of the error can therefore be accounted by studying representations of the symmetry group of p objects. Others come from uncertainty in the position or orientation of the camera when it took the photograph. These are explained by induced representations of non-abelian Lie groups like 3-D rotations in the space of covariance matrices.

I submitted my paper on autonomous precision error estimation in 3-D models to the 2008 International Conference on Machine Learning yesterday. One week early, too, a first for me! The format for the paper is the standard double column format and this makes it very hard to have complex equations in the paper. One mathematical object that is hard to display are the covariance matrices for the DEM errors that I keep talking about in these posts. These are nxn matrices of real numbers. One particular example I use comes from images of a desert terrain in the Twenty-Nine Palms area in California. We have four photographs and can therefore produce $12=4*3$ DEMs. Because of mistakes, two of the DEMs have to be dropped so I end up with 10 DEMs. The resulting covariance matrices are then 10×10 matrices — a hard thing to display in the double-column format since now you have to present 10 numbers in row. So I have hit upon a simple graphical way to present them that saves space but also ends up being more informative to the reader (or me) about the structure of the matrix.

The idea is to turn the 10×10 matrix into a 10×10 pixel image. Each pixel is now a shade of gray. The highest value in the matrix gets the darkest shade, the lowest gets the lighest. Here is an example that illustrates our correlated-pair error model The only terms that are “turned on” are those along the diagonal. In contrast, here is the covariance matrix when you do ${\ell }_1$-minimization and do not assume beforehand that certain DEMs are uncorrelated with each other. So the correlated-pair error model is close to the actual covariances but we see that there are some cross-correlations off the diagonal that are on, albeit weaker than those on the block-diagonal defined by the asymmetric DEM pairs.

I apologize for the strange layout of the mages relative to the text of this post but my WordPress instalation does not save changes that I make to the img tag to identify it as requiring it to have text flow around it.
In any case, I hope this illustration makes clear some of the more abstract ideas I have been discussing about errors in DEMs.