Analog Error-Correcting Codes

• Published in: IEEE transactions on information theory Vol. 66; no. 7; pp. 4075 - 4088
• Main Author: Roth, Ron M.
• Format: Journal Article
• Language: English
• Published: New York IEEE 01-07-2020; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Analog arithmetic circuits; approximate computation; Codes; Coding; Computational modeling; Conductors; Decoding; Error correcting codes; Error correction; fault-tolerant computing; Linear codes; linear codes over the real field; Mathematical analysis; Matrix algebra; Matrix methods; Multiplication; Quantization (signal); Redundancy; Resistors; vector–matrix multiplication
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2020.2977918

Coding schemes are presented that provide the ability to locate computational errors above a prescribed threshold while using analog resistive devices for approximate real vector-matrix multiplication. In such devices, the matrix is programmed into the device by setting an array of resistors to have conductances proportional to the respective entries in the matrix. In the coding scheme that is considered in this work, redundancy columns are appended so that each row in the programmed matrix forms a codeword of a prescribed linear code <inline-formula> <tex-math notation="LaTeX">{\mathcal {C}} </tex-math></inline-formula> over the real field; the result of the multiplication of any input real row vector by the matrix is then also a codeword of <inline-formula> <tex-math notation="LaTeX">{\mathcal {C}} </tex-math></inline-formula>. While error values within <inline-formula> <tex-math notation="LaTeX">\pm \delta </tex-math></inline-formula> in the entries of the result are tolerable (for some prescribed <inline-formula> <tex-math notation="LaTeX">\delta > 0 </tex-math></inline-formula>), outlying errors, with values outside the range <inline-formula> <tex-math notation="LaTeX">\pm \Delta </tex-math></inline-formula> (for a prescribed <inline-formula> <tex-math notation="LaTeX">\Delta \ge \delta </tex-math></inline-formula>) should be located and corrected. As a design and analysis tool for such a setting, a certain functional is defined for the code <inline-formula> <tex-math notation="LaTeX">{\mathcal {C}} </tex-math></inline-formula>, through which a characterization is obtained for the number of outlying errors that can be handled, as a function of the ratio <inline-formula> <tex-math notation="LaTeX">\Delta /\delta </tex-math></inline-formula>. Several code constructions are then presented, primarily for the case of single outlying error handling. For this case, the coding problem is shown to be related to certain extremal problems on convex polygons.