Linear Computation Coding: A Framework for Joint Quantization and Computing

• Published in: Algorithms 2022-07, Vol.15 (7), p.253
• Main Authors: Müller, Ralf Reiner, Gäde, Bernhard Martin Wilhelm, Bereyhi, Ali
• Format: Article
• Language: English
• Subjects: Accuracy; Algorithms; approximate computing; Approximation; Artificial neural networks; Coding; computational complexity; Computing costs; Cost control; Data compression; estimation error; Field programmable gate arrays; fixed-point arithmetic; Integers; Linear functions; linear systems; Mathematical analysis; Matrices (mathematics); Matrix algebra; Neural networks; Partial differential equations; Random access memory; Random variables; rate-distortion theory; Sparsity; Wiring
• ISSN: 1999-4893; 1999-4893
• DOI: 10.3390/a15070253

Here we introduce the new concept of computation coding. Similar to how rate-distortion theory is concerned with the lossy compression of data, computation coding deals with the lossy computation of functions. Particularizing to linear functions, we present an algorithmic approach to reduce the computational cost of multiplying a constant matrix with a variable vector, which requires neither a matrix nor vector having any particular structure or statistical properties. The algorithm decomposes the constant matrix into the product of codebook and wiring matrices whose entries are either zero or signed integer powers of two. For a typical application like the implementation of a deep neural network, the proposed algorithm reduces the number of required addition units several times. To achieve the accuracy of 16-bit signed integer arithmetic for 4k-vectors, no multipliers and only 1.5 adders per matrix entry are needed.