This paper is concerned with pseudo almost automorphic functions, which are more general and complicated than pseudo almost periodic functions and asymptotically almost automorphic functions. New results, concerning the composition of pseudo almost automorphic functions, are established.
Divergence functions are the non-symmetric "distance" on the manifold, M θ , of parametric probability density functions over a measure space, (X, µ). Classical information geometry prescribes, on M θ : (i) a Riemannian metric given by the Fisher information; (ii) a pair of dual connections (giving rise to the family of α-connections) that preserve the metric under parallel transport by their joint actions; and (iii) a family of divergence functions (α-divergence) defined on M θ × M θ , which induce the metric and the dual connections. Here, we construct an extension of this differential geometric structure from M θ (that of parametric probability density functions) to the manifold, M, of non-parametric functions on X, removing the positivity and normalization constraints. The generalized Fisher information and α-connections on M are induced by an α-parameterized family of divergence functions, reflecting the fundamental convex inequality associated with any smooth and strictly convex function. The infinite-dimensional manifold, M, has zero curvature for all these α-connections; hence, the generally non-zero curvature of M θ can be interpreted as arising from an embedding of M θ into M. Furthermore, when a parametric model (after a monotonic scaling) forms an affine submanifold, its natural and expectation parameters form biorthogonal coordinates, and such a submanifold is dually flat for α = ±1, generalizing the results of Amari's α-embedding. The present analysis illuminates two different types of duality in information geometry, one concerning the referential status of a point (measurable function) expressed in the divergence function ("referential duality") and the other concerning its representation under an arbitrary monotone scaling ("representational duality").
When drifting bars or gratings are used as visual stimuli, information about orientation specificity (which has a period of 180 degrees) and direction specificity (which has a period of 360 degrees) is inherently confounded in the response of visual cortical neurons, which have long been known to be selective for both the orientation of the stimulus and the direction of its movement. It is essential to "unconfound" or separate these two components of the response as they may respectively contribute to form and motion perception, two of the main streams of information processing in the mammalian brain. Wörgötter and Eysel (1987) recently proposed the Fourier transform technique as a method of unconfounding the two components, but their analysis was incomplete. Here we formally develop the mathematical tools for this method to calculate the peak angles, bandwidths, and relative strengths, the three most important elements of a tuning curve, of both the orientational and the directional components, based on the experimentally-recorded neuron's response polar-plot. It will be shown that, in the 1-D Fourier decomposition of the polar-plot along its angular dimension, 1) the odd harmonics contain only the directional component, while the even harmonics are contributed to by both the orientational and the directional components; 2) the phases and the amplitudes of all the harmonics are related, respectively, to the peak angle and the bandwidth of the individual component. The basic assumption used here is that the two components are linearly additive; this in turn is immediately testable by the method itself.
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