Reduced-rank Least Squares Parameter Estimation in the Presence of Byzantine Sensors

In this paper, we study the impact of the presence of byzantine sensors on the reduced-rank linear least squares (LS) estimator. A sensor network with N sensors makes observations of the physical phenomenon and transmits them to a fusion center which computes the LS estimate of the parameter of interest. It is well-known that rank reduction exploits the bias-variance tradeoff in the full-rank estimator by putting higher priority on highly informative content of the data. The low-rank LS estimator is constructed using this highly informative content, while the remaining data can be discarded without affecting the overall performance of the estimator. We consider the scenario where a fraction 0 <= alpha <= 1 of the N sensors are subject to data falsification attack from byzantine sensors, wherein an intruder injects a higher noise power (compared to the unattacked sensors) to the measurements of the attacked sensors. Our main contribution is an analytical characterization of the impact of data falsification attack of the above type on the performance of reduced-rank LS estimator. In particular, we show how optimally prioritizing the highly informative content of the data gets affected in the presence of attacks. A surprising result is that, under sensor attacks, when the elements of the data matrix are all positive the error performance of the low-rank estimator experiences a phenomenon wherein the estimate of the mean-squared error comprises negative components. A complex nonlinear programming-based recipe is known to exist that resolves this undesirable effect; however, the phenomenon is oftentimes considered very objectionable in the statistical literature. On the other hand, to our advantage this effect can serve to detect cyber attacks on sensor systems. Numerical results are presented to complement the theoretical findings of the paper.