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Abstract Cantor sets are constructed from iteratively removing sections of intervals. This process yields a cumulative distribution function (CDF), constructed from the invariant Borel probability measure associated with their iterated function systems. Under appropriate assumptions, we identify sampling schemes of such CDFs, meaning that the underlying Cantor set can be reconstructed from sufficiently many samples of its CDF. To this end, we prove that two Cantor sets have almost-nowhere intersection with respect to their corresponding invariant measures. 

Our brains are, “prediction machines”, where we are continuously comparing our surroundings with predictions from internal models generated by our brains. This is demonstrated by observing our basic low level sensory systems and how they predict environmental changes as we move through space and time. Indeed, even at higher cognitive levels, we are able to do prediction. We can predict how the laws of physics affect people, places, and things and even predict the end of someone’s sentence. In our work, we sought to create an artificial model that is able to mimic early, low level biological predictive behavior in a computer vision system. Our predictive vision model uses spatiotemporal sequence memories learned from deep sparse coding. This model is implemented using a biologically inspired architecture: one that utilizes sequence memories, lateral inhibition, and top-down feed- back in a generative framework. Our model learns the causes of the data in a completely unsupervised manner, by simply observing and learning about the world. Spatiotemporal features are learned by minimizing a reconstruction error convolved over space and time, and can subsequently be used for recognition, classification, and future video prediction. Our experiments show that we are able to accurately predict what will happen in the future; furthermore, we can use our predictions to detect anomalous, unexpected events in both synthetic and real video sequences.