The Fast Fourier Transform (FFT) is one of the most fundamental numerical algorithms. It computes the Discrete Fourier Transform (DFT) of an n-dimensional signal in O(n log n) time. The algorithm plays an important role in many areas. It is not known whether its running time can be improved. However, in many applications, most of the Fourier coefficients of a signal are “small” or equal to zero, i.e., the output of the transform is (approximately) sparse. In this case, it is known that one can compute the set of non-zero coefficients faster than in O(n log n) time.
In this talk, I will describe a set of efficient algorithms for sparse Fourier Transform. One of the algorithms has the running time of O(k log n), where k is the number of non-zero Fourier coefficients of the signal. This improves over the runtime of the FFT for any k = o(n). If time allows, I will also describe some of the applications, to spectrum sensing and GPS locking, as well as mention a few outstanding open problems. The talk will cover the material from the joint papers with Fadel Adib, Badih Ghazi, Haitham Hassanieh, Dina Katabi, Eric Price and Lixin Shi. The papers are available at http://groups.csail.mit.edu/netmit/sFFT/.
Piotr Indyk is a Professor of Electrical Engineering and Computer Science at MIT. He joined MIT in 2000, after earning PhD from Stanford University. Earlier, he received Magister degree from Uniwersytet Warszawski in 1995. Piotr’s research interests lie in the design and analysis of efficient algorithms. Specific interests include: high-dimensional computational geometry, sketching and streaming algorithms, sparse recovery and compressive sensing. He has received the Sloan Fellowship (2003), the Packard Fellowship (2003) and the Simons Investigator Award (2013). His work on sparse Fourier sampling has been named to Technology Review “TR10” in 2012, while his work on locality-sensitive hashing has received the 2012 ACM Kanellakis Theory and Practice Award.