Ganesh Swami

Someone came to me recently for some help on computing determinants. Determinants of 2×2 and 3×3 and maybe even 4×4 matrices are trivial and you can easily hard code the expanded symbolic form. Determinants of bigger matrices are much harder and require careful thought.

First some rules:

• Singular matrices have determinant zero. Simple, but often overlooked.
• If you require a spectral decomposition at a later stage, the easiest is the product of the eigenvalues (the matrix cookbook is a handy reference.)
• For triangular matrices, it’s the product of the diagonal entries.
• The determinant of the product of two matrices is the product of their determinants.

Using the above rules, the first step is to factor your matrix into two triangular matrices. The LU decomposition achieves this. In general, a permutation matrix allows all ones on the diagonal of L. If you only care about the absolute value of the determinant, simply computing the product of the diagonal values of U gets you the determinant.

If your matrix is symmetric and positive definite, then you can use a specialization known as Cholesky decomposition which makes L the transpose of U (doing half the work with half the storage.) The determinant is then the product of the square of the diagonal elements. Diffusion tensors are covariance matrices (symmetric, positive definite) and so this decomposition always applies.

The workshop was a blast! I had an incredible time getting up to speed on the latest and greatest in data analysis research. It was quite humbling to brush shoulders with some of top folks pushing the frontiers of science. There were also many opportunities to network where I could get a peek at the motivations behind some of the projects presented.

Each day had a theme:

• Data Analysis and Data Applications
• Networked Data and Algorithmic Tools
• Statistical, Geometric, and Topological Methods
• Machine Learning and Dimensionality Reduction

The breadth of topics was quite exhaustive. I mostly pushed my own agenda: streaming algorithms. There’s so much to write here that I won’t even attempt to.

Besides streaming algorithms, the presentations on mathematical topics were really interesting. Some of it I’ve previously seen from my day-to-day work, some of it was new. Of particular interest to me were the following:

• Graph Sparsification : Never seen anything like this before.
• Massive Terrain Data : Real smart use of offline datastrutures.
• Symmetries in point cloud data : I’m intimately familiar with this style of mathematics from my previous work.
• Pathway Analysis in Protein Folding : Puts bread on the table.
• Intersection SVMs : Didn’t know this was a well known concept in machine learning known as the kernel trick. Goes by Reproducing Kernel Hilbert Space in my neck of the woods and also a precursor to the above mentioned image matching algorithm.
• Manifold regularization : Fréchet means anyone?
• Sufficient Dimension Reduction
• Semi-definite programming : Some mathematical insights to a couple of engineering problems (where <1e-4 is good enough) that’s making my life difficult.
• Spectral Algorithms
• Matrix/Tensor Factorization
• Future of Parallel Linear Algebra

Thanks to my friend Krishna who let me sleep on the floor in his house, thereby saving me from the grips of boredom.

I’m excited about the Workshop on Modern Massive Data Sets I’ll be attending next week at Stanford.

The 2008 Workshop on Algorithms for Modern Massive Data Sets (MMDS 2008) will address algorithmic, mathematical, and statistical challenges in modern statistical data analysis. The goals of MMDS 2008 are to explore novel techniques for modeling and analyzing massive, high-dimensional, and nonlinearly-structured scientific and internet data sets, and to bring together computer scientists, statisticians, mathematicians, and data analysis practitioners to promote cross-fertilization of ideas.

These are my notes (disclaimer: I’m no expert, corrections are welcome.)

Most algorithm engineers thus far are happy with algorithms that are linear, i.e., in the number of data points in the dataset. With web-scale data, linear is not good enough: sub-linear is required. These streams of data are typically unbounded and do not fit in main memory. They cannot even be stored, as it is infeasible to go back and reload them.

The Frequency Problem is used as a model problem over data streams, for example, computing means/variances, medians, top-k frequent items, distinct elements etc. One approach is to approximate the computation over the stream via random sampling.

There are four topics that keep recurring when looking at proofs of streaming algorithms using randomization:

• Markov’s inequality
• Chebyschev’s inequality
• Holder’s inequality
• Chernoff bounds

This is really key. As an example, suppose we wanted to run some complicated algorithm (offline) over the packets flowing through a router. As this would a huge number of packets, we won’t have the luxury of storing all packets, but only a subset. By sampling the packets with probability p, we can estimate the amount of memory required to store the packets as the expectation value of the binomial variable with parameter p and n, where n is the number of packets going through the router.

The use of one of the bounds over the other is a matter of how tight the bound is. The Markov inequality only uses the expectation value of the random variable while the other use higher moments. They do this by using the Markov inequality to a moment generating function exp(tX) of the random variable X.

You’ll want to check the preliminary program for the kind of topics that are going to be covered. While you are at it, check out the workshop webpage from 2006. I’ll go find something to wipe the drool off my chin…