Adaptive Filtering Algorithms for Set-Valued Observations--Symmetric Measurement Approach to Unlabeled and Anonymized Data

Suppose L simultaneous independent stochastic systems gen- erate observations, where the observations from each system depend on the underlying parameter of that system. The observations are unlabeled (anonymized), in the sense that an analyst does not know which observation came from which stochastic system. How can the analyst estimate the under- lying parameters of the L systems? Since the anonymized observations at each time are an unordered set of L measurements (rather than a vector), classical stochastic gradient algorithms cannot be directly used. By using symmetric polynomials, we formulate a symmetric measurement equation that maps the observation set to a unique vector. We then construct an adaptive filtering algorithm that yields a statistically consistent estimate of the underlying parameters.