Misapplied Math

# Twelve Days 2013: Probabilistic Data Structures

## Day Three: Probabilistic Data Structures and Algorithms

### TL/DR

Probabilistic data structures can deal with data sets too large to handle by conventional means, or offer massive speedups at the cost of some uncertainty. Skip lists make for nice lock-free priority queues (useful in schedulers and online sorting algorithms), Bloom filters are well suited for set membership queries over data sets of arbitrary size, and locality-sensitive hashing provides quick approximate nearest neighbor queries and dimensionality reduction. Probabilistic matrix algorithms can speed up matrix math by several orders of magnitude with tight error bounds.

## Case One: Probabilistic Matrix Math

Matrix decomposition and factorization techniques such as SVD and QR play an important role in many problems, from optimization and online control to machine learning. The new kid on the block, rank-revealing QR factorization provides a very efficient means of estimating matrix rank. All three benefit from quite recent work on probabilistic matrix algorithms. It's possible to significantly improve both parallelism and computational complexity while giving up very little in terms of accuracy. The paper above presents a method for reducing the complexity of finding the top $k$ components of an SVD from $\O(mnk)$ to $\O(mnlog(k))$ while admitting a natural parallel implementation. Two side notes:

1. Joel Tropp, one of the authors of the paper above and a former prof of mine, is a great guy to follow for this. He's a very good writer and his areas of expertise are quite interesting/relevant to data science: sparse approximation, compressed sensing, and randomized algorithms.
2. Spectral theory and random matrices are pretty fascinating fields (especially with regard to questions such as detecting spurious correlation and reasoning about the distribution of eigenvalues).

## Case Two: Skip lists

In many cases skip lists are a nice drop in replacement for balanced trees when concurrent access is needed. Their basic operations (search, insert, and delete) all have all of the same big $O$ performance characteristics but their structure makes concurrent programming much simpler. They avoid the rebalancing issues that come up in alternatives such as red-black trees (rebalancing is also what makes efficient concurrent implementations difficult), and they're pretty cache friendly. As a bonus, range queries, k-th largest, and proximity queries are all $\O(\log(n))$.

## Case Three: Bloom Filters

Bloom filters provide space efficient set membership queries. They work by using $k$ hash functions to determine $k$ indices in a boolean array. When an item is inserted, each of those $k$ positions is set to true. To query an item, end each of the $k$ positions is checked. If all of the positions are set to true, the item is possibly in the set, but if any one of them is false, it's definitely not. This works well provided that the array doesn't become saturated with true values, and that the $k$ hash functions are independent/do a good job distributing the key space over the hash space uniformly. They're also simple enough to prove bounds on, which is useful for parameter selection.

## Case Four: Locality-Sensitive Hashing

Nearest neighbor queries (where distance can be computed under an arbitrary metric function – not necessarily a spacial one) are both very useful, and very hard to do efficiently for high dimensional spaces. In practice, for 30+ dimensions, the search time for exact algorithms such as K-D trees is worse than linear search when you take into account cache effects/branch misprediction that comes about from a real world implementation.

Locality-sensitive hashing does the opposite of what most hash functions try to do in that it aims to maximize the probability of a collision for keys in close proximity. It's an example of dimensionality reduction and as such gets used and abused for unsupervised machine learning. For queries, I've seen it perform poorly on some geometric/spacial queries and extremely well on others, so your mileage may vary.