Ganesh Swami

I’m now a proud owner of an Apple iPhone 3G. There are tons of reviews out there that you can lookup, but here’s mine with the purpose of getting-shit-done:

Pros:

• The easiest phone on the planet. This has got to be one of the most usable devices I’ve ever used. Everything from looking up contacts, syncing, WiFi is so easy.
• Native syncing with Exchange. Hate duplicate copies of my contacts.
• Inbuilt music player.
• Seamless integration with my Mac.
• Volume controls as physical buttons on the outside unlike the iPod Touch.

Cons:

• The touch screen is “cool” for a few minutes but not usable. I regularly lookup information on my phone while driving — this practice will now have to stop.
• Web browsing on the phone isn’t even marginally useful for me.
• GPS isn’t very good. That’s OK because I got a real GPS unit.
• 50% battery charge in two days. Wow. In a year, I’ll need to keep the phone plugged in at all times.
• Sluggish. Skips music when syncing contacts.
• The alarm module is less useful than my 4 year Nokia phone.

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…

I know some of my readers are deeply interested in high performance computing and computational physics: this is a post for them. The conference I had mentioned in my previous post was the GROMACS Workshop on Advanced Simulation Methods.

Gromacs is a high performance simulation engine primarily for solving Newtonian dynamics (it also does normal mode analysis, structure minimization and mixed molecular mechanics-quantum mechanics simulations.) It was an industry leader in terms of raw single processor performance for many years, until Desmond from D.E. Shaw Research took over with their super-scalable algorithms (I’ve written about this before.) With Gromacs 4.0, they’ve fixed the scalability problems and with a variety of other algorithmic fixes, they are the top dog once again. Disclaimer: these are all claims by relevant parties and I have not verified them myself, though I’d love to do so unencumbered. Though the Gromacs 4.0 paper is published, I’ll only be writing about it when the actual product is released.

The focus of the workshop was on algorithms, though there were some applications too. I’m sure an applications person would have felt out of place, but I felt I had something to contribute in almost every topic that was discussed. I’m archiving the list of topics here for posterity:

• The new domain decomposition parallelization in Gromacs 4.0, with some tips & tricks to get the most out of your hardware
• Different methods to perform free energy calculations. Slow-growth, perturbations, Bennett Acceptance Ratio. Which protocol is most efficient, and what new things will be in Gromacs 4.0?
• QM/MM. How do you mix Quantum Mechanics with Gromacs?
• Virtual sites for hydrogen motion removal and long time-steps
• Membrane protein simulations
• Replica exchange, and extracting kinetic data from it
• Local pressure extensions to Gromacs
• Gromacs source code walk-through

The take home message: strong coupling between various pieces of the algorithm is anti-thesis to parallel scalability. The CPU industry seems to have hit a brick wall in terms of improving raw computational speed: the future is in multi-core. Therefore, remove the coupling with better algorithms and you are on your way to highly scalable and by definition superbly fast algorithms.

The timestep used in an integrator while solving a set of equations inherently determines the speed of the algorithm. Big timesteps will make the algorithm unstable as you your trajectory will not be able to follow the phase space manifold accurately (as a side note Euler-type integrators also become unstable as you make the timestep smaller, but this is the least of your worries with a non-symplectic integrator.) The Nyquist theorem determines the sampling rate, so removing fast (or high frequency) degrees of motion such as hydrogen bond vibrations with constraints on them is required for a big timestep. Usual constraints algorithms are coupled leading to undesirable non-scalable algorithms. The Gromacs developers have solved this with a new constraints algorithm called P-LINCS.