The neural network autoencoder is a useful tool for the restoration of missing sensors when enough known sensors with some relation to those missing are available. Through the idea of a contraction mapping, this paper provides some insight into the convergence of several iterative methods of sensor restoration using the autoencoder to some unique answer given a specific operating point (i.e., the known sensnr values), regardless of how the missing sensor values are initialized Previous work has established the ability of the autoassociative neural network encoder (or simply "autoencoder") to aid in the restoration of sensor data which may be missing or corrupt, given some sort of correlation between the numeric outputs of the various sensors in a system. [I], [2] Narayanan et al. [I] describe a method by which. the missing 'sensor data may he reconstructed using an iterative approach; in this paper we show that, under a set of conditions relating to the specific parameters of the neural network, we can provide a sufficient condition for the convergence of the iterative approach to sensor restoration. We approach this through the idea of a contraction mapping. Moreover, we will show compelling evidence that there exists a unique point of convergence for a fully trained autoencoder given an "operating point" defined by the set of known sensors, and that this convergence point should be reached regardless of how the missing sensors are initialized. or, more clearly, a contraction mapping is one in which the output distance between two points is less than the input distance. Now let us look at this property in Y', where our metric is simply the Euclidean norm, and 0 is simply some functional mappingfix): Now, suppose we replace y with x+dr, yielding, with some minor rearrangement: the limit of which as dx-0 gives us, as's less strict requirement for contraction This idea is demonstrated clearly in Figure 1 . It would then be sufficient to show that, for some function f %' +!R' , the derivative offis less than unity for all x. Consider the Banach Fixed-point Theorem: If/ is a contractive mapping, then there exists a unique fixed point x,) for whichflxo) = xu. Moreover, the sequence (x.), for which any element xnii = fix"), converges, and that convergent point is x, With this theorem, it becomes much clearer how any contractive tendency of the autoencoder can help us show whether or not the sensor restoration process will converge to some unique value. A contractive mapping is defined  Missing Sensor Restoration with mapping 0%-X on a complete metric space (X. d) in Autoencoders which, for any x and y in that space: There are three methods that we will examine for the restoration of missing sensors. The first is a simple application of altemating projections onto convex sets (POCS). The second and the third both employ search d(Ox, o y ) _i k . d(x, y )
Thompson et al. REF used contractive autoencoder for the restoration of missing sensor values and showed it is generally a non-expensive procedure for most data types.
On the contractive nature of autoencoders: application to missing sensor restoration
Proceedings of the International Joint Conference on Neural Networks, 2003.
Proceedings of the International Joint Conference on Neural Networks, 2003.
Computer Science