Computation Coding

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The repeated computation of arbitrarily large-dimensional linear functions is simplified by means of multiplicative matrix decompositions. For this purpose, codebooks similar to those that are currently used in source and channel coding are utilized. For any matrix decomposition, one matrix factor can be interpreted as a codebook, the other matrix factor as a set of pointers into the codebook. If implemented using reconfigurable computer architectures these pointers just define a wiring.As in state-of-the-art algorithms, the computation error decays exponentially with the number of computations. In contrast to practically useful state-of-the-art algorithms, the decay exponent is not constant, but grows unboundedly with the dimension of the function. In preliminary tests, no multiplications, only 1.6 additions and register shifts per matrix entry were sufficient to reach the accuracy of 16-bit fixed-point arithmetic for 12×4096 matrices. For low-end applications, the required number of additions and register-shifts is way below the number of matrix entries. Although a preliminary implementation of the decomposition algorithm uses ideas from compressive sensing, it neither requires the linear function nor its argument to be sparse in any domain.The proposed project shall establish the research field of computation coding and explore it into various directions: Generalization to nonlinear functions, optimization of the codebook, new decomposition algorithms, theoretical performance analysis, and implementation on field programmable gate arrays. Nonlinear functions are addressed by their approximations via deep neural networks. Codebook constraints in computation coding are entirely different from source and channel coding and constitute a novel area of coding theory. Current decomposition algorithms have super-cubic complexity. Finding faster ones, will make computation coding attractive for an even wider range of applications. Theoretical performance analysis will utilize a recent breakthrough of the applicant in performance analysis of finite-length random coding as well as replica symmetry breaking in maximum-likelihood decoding. The implementation on field programmable gate arrays shall demonstrate the practical feasibility of the novel approach and give important insights for future improvements.