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.
We introduce a novel methodology for anomaly detection in time-series data. The method uses persistence diagrams and bottleneck distances to identify anomalies. Specifically, we generate multiple predictors by randomly bagging the data (reference bags), then for each data point replacing the data point for a randomly chosen point in each bag (modified bags). The predictors then are the set of bottleneck distances for the reference/modified bag pairs. We prove the stability of the predictors as the number of bags increases. We apply our methodology to traffic data and measure the performance for identifying known incidents.
The Kaczmarz algorithm is an iterative method for solving a system of linear equations. It can be extended so as to reconstruct a vector x in a (separable) Hilbert space from the inner-products { x, φn }. The Kaczmarz algorithms defines a sequence of approximations from the sequence { x, φn }; these approximations only converge to x when {φn} is effective. We dualize the Kaczmarz algorithm so that x can be obtained from { x, φn } by using a second sequence {ψn} in the reconstruction. This allows for the recovery of x even when the sequence {φn} is not effective; in particular, our dualization yields a reconstruction when the sequence {φn} is almost effective. We also obtain some partial results characterizing when the sequence of approximations from { x, φn } using {ψn} converges to x, in which case {(φn, ψn)} is called an effective pair.2010 Mathematics Subject Classification. Primary: 41A65, 65D15. Secondary: 42C15, 65F10.
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