On Computability of Pattern Recognition Problems

In statistical setting of the pattern recognition problem the number of examples required to approximate an unknown labelling function is linear in the VC dimension of the target learning class. In this work we consider the question whether such bounds exist if consider only computable pattern recognition methods, assuming that the unknown labelling function is also computable. We find that in this case the number of examples required for a computable method to approximate the labelling function not only is not linear, but grows faster (in the VC dimension of the class) than any computable function. No time or space constraints are put on the predictors or target functions; the only resource we consider is the training examples. The task of pattern recognition is considered in conjunction with another learning problem — data compression. An impossibility result for the task of data compression allows us to estimate the sample complexity for pattern recognition.