Time Independent ICA through a Fisher Game – Dr. Ravi C. Venkatesan (Systems Research Corp)
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Extreme Physical Information (EPI)  is a self contained theory to elicit physical laws from a system/process (Nature) based on a measurement-response framework. A specific form of the Fisher information measure (FIM) known as the Fisher channel capacity (FCC) is employed as a measure of uncertainty. The FCC is the trace of the FIM. EPI may be construed as being a zero-sum-game between a gedanken observer and a system under observation (characterized by a demon, reminiscent of the Maxwell demon, residing in a conjugate space). The payoff of the competitive game results in a variational principle that defines the physical law that generates the observations made by the gedanken observer, as a consequence of the response of the system to the measurements. A principled formulation for reconstructing pdfs from arbitrary discrete time independent random sequences based on an invariance preserving extension of the Extreme Physical Information (EPI) theory, is presented [2, 3]. Invariances are incorporated into the invariant EPI (IEPI) model through a Discrete Variational Complex inspired by the seminal work of T. D. Lee . A quantum mechanical connotation is provided to the Fisher game. This is accomplished through the IEPI Euler-Lagrange equation that acquires the form of a time independent Schrodinger-like equation, and, the quantum mechanical virial theorem . The concomitant constraints of the IEPI variational principle are consistent with the Heisenberg uncertainty principle. The ansatz describing the state estimators are obtained so as to selfconsistently satisfy an analog of the Fisher game corollary [1, 3]. The game corollary permits the demon to make the closing move in the Fisher game, by minimizing the FCC. This corresponds to a state of maximum uncertainty, and, is in keeping with the demon’s strategy of minimizing the information made available to the observer. A fundamental tenet of the EPI/IEPI model is the collection of statistically independent data by the observer. A principled IEPI Fisher game formulation guaranteeing the statistical independence of the quantum mechanical observables is presented, utilizing statistical analyses commonly employed in Independent Component Analysis (ICA) . Specifically, correlations are first eliminated using a whitening process (facilitated by a linear filter or PCA), in conjunction with Givens rotation (a unitary transform). Next, the IEPI Fisher game is played between the gedanken observer and the process inhabiting the conjugate system space. Finally, an inverse whitening filter is applied to the observables corresponding to the reconstructed state vectors obtained from the Fisher game. This yields a novel form of ICA based on minimizing the FCC. The prospect of obtaining an optimal whitening filter based on the Fisher game corollary is investigated into. Qualitative analogies and distinctions between the Fisher game ICA model and other prominent ICA theories are briefly discussed. Reconstruction of time independent random sequences generated from Gaussian mixture models demonstrates the efficacy of the Fisher game/ICA formulation. The utility of the Fisher game ICA formulation to achieve quantum clustering of data where a-priori knowledge of the number of clusters is unknown, is briefly discussed.