Compressive sensing is not the only place where multiplying a vector or a matrix with a Gaussian matrix is a big deal for large scale problems (see the recent Random Matrices Are Too Damn Large ! and Another Comment on " Random Matrices Are Too Damn Large !"). If you recall this is also a problem for Random Kitchen Sinks, a randomized version of AdaBoost (a connection with compressive sensing is mentioned here). There, the training set in Machine Learning is used as a dictionary in order to learn a function. Those dictionaries are, however too large and the authors of the paper resort to a fast random projections to learn the function faster.
We had the talk for over a year now, we now have the attendant paper. There is actually more in the paper than what was shown in the presentation earlier: Fastfood: Approximate Kernel Expansions in Loglinear Time by Quoc Viet Le, Tamas Sarlos, Alexander Johannes Smola
Despite their successes, what makes kernel methods difficult to use in many large scale problems is the fact that storing and computing the decision function is typically expensive, especially at prediction time. In this paper, we overcome this difficulty by proposing Fastfood, an approximation that accelerates such computation significantly. Key to Fastfood is the observation that Hadamard matrices, when combined with diagonal Gaussian matrices, exhibit properties similar to dense Gaussian random matrices. Yet unlike the latter, Hadamard and diagonal matrices are inexpensive to multiply and store. These two matrices can be used in lieu of Gaussian matrices in Random Kitchen Sinks proposed by Rahimi and Recht (2009) and thereby speeding up the computation for a large range of kernel functions. Specifically, Fastfood requires O(n log d) time and O(n) storage to compute n non-linear basis functions in d dimensions, a significant improvement from O(nd) computation and storage, without sacrificing accuracy.Our method applies to any translation invariant and any dot-product kernel, such as the popular RBF kernels and polynomial kernels. We prove that the approximation is unbiased and has low variance. Experiments show that we achieve similar accuracy to full kernel expansions and Random Kitchen Sinks while being 100x faster and using 1000x less memory. These improvements, especially in terms of memory usage, make kernel methods more practical for applications that have large training sets and/or require real-time prediction.
 Uniform Approximation of Functions with Random Bases, Ali Rahimi and Benjamin Recht
 Nystrom Method vs Random Fourier Features:: A Theoretical and Empirical Comparison Tianbao Yang, Yu-Feng Li, Mehrdad Mahdavi, Rong Jin, Zhi-Hua Zhou
[9 Pruning random features with correlated kitchen sinks -poster- Brian McWilliams and David Balduzzi
Liked this entry ? subscribe to Nuit Blanche's feed, there's more where that came from. You can also subscribe to Nuit Blanche by Email, explore the Big Picture in Compressive Sensing or the Matrix Factorization Jungle and join the conversations on compressive sensing, advanced matrix factorization and calibration issues on Linkedin.