Compressed convolution

Abstract: We introduce the concept of compressed convolution, a technique to convolve a given data set with a large number of non-orthogonal kernels. In typical applications our technique drastically reduces the effective number of computations. The new method is applicable to convolutions with symmetric and asymmetric kernels and can be easily controlled for an optimal trade-off between speed and accuracy. It is based on linear compression of the collection of kernels into a small number of coefficients in an optimal eigenbasis. The final result can then be decompressed in constant time for each desired convolved output. The method is fully general and suitable for a wide variety of problems. We give explicit examples in the context of simulation challenges for upcoming multi-kilo-detector cosmic microwave background (CMB) missions. For a CMB experiment with O(10,000) detectors with similar beam properties, we demonstrate that the algorithm can decrease the costs of beam convolution by two to three orders of magnitude with negligible loss of accuracy. Likewise, it has the potential to allow the reduction of disk space required to store signal simulations by a similar amount. Applications in other areas of astrophysics and beyond are optimal searches for a large number of templates in noisy data, e.g. from a parametrized family of gravitational wave templates; or calculating convolutions with highly overcomplete wavelet dictionaries, e.g. in methods designed to uncover sparse signal representations.