Erasure recovery matrices for encoder protection |
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| Institution: | 1. Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA;2. Department of Mathematics, Texas A&M University, College Station, TX, USA;3. Department of Mathematics, University of Wisconsin-Eau Claire, Eau Claire, WI, USA;4. School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China;1. Department of Mathematics, Vanderbilt University, Nashville, TN, 37212, USA;2. Department of Mathematical Science, Northern Illinois University, Dekalb, IL, 60115, USA;3. Department of Mathematics, Johns Hopkins University, Baltimore, MD, 21218, USA;1. Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, United Kingdom;2. Department of Mathematics, University of Oslo, 0316 Oslo, Norway;1. School of Mathematical Sciences, Tongji University, 200092 Shanghai, China;2. Beijing International Center for Mathematical Research, Peking University, 100871 Beijing, China;3. Institute of Natural Sciences and School of Mathematical Sciences; MOE-LSC, Shanghai Jiao Tong University, 200240 Shanghai, China |
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| Abstract: | In this article, we investigate the privacy issues that arise from a new frame-based kernel analysis approach to reconstruct from frame coefficient erasures. We show that while an erasure recovery matrix is needed in addition to a decoding frame for a receiver to recover the erasures, the erasure recovery matrix can be designed in such a way that it protects the encoding frame. The set of such erasure recovery matrices is shown to be an open and dense subset of a certain matrix space. We present algorithms to construct concrete examples of encoding frame and erasure recovery matrix pairs for which the erasure reconstruction process is robust to additive channel noise. Using the Restricted Isometry Property, we also provide quantitative bounds on the amplification of sparse additive channel noise. Numerical experiments are presented on the amplification of additive normally distributed random channel noise. In both cases, the amplification factors are demonstrated to be quite small. |
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| Keywords: | Frames Erasures Fourier series Erasure recovery matrices Encoding and decoding frames |
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