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This paper is a corrected English translation of an original Japanese work introducing"power geometry,"a research programme centred on power transformations, deformations and dualities of functions in the sense of optimisation theory. Its two main motivations are the origin of probability distributions (via maximum-lik...
Helmke and Shayman showed that, for generic targets, least-squares distance to the manifold of rank-r matrices is a nondegenerate Morse function whose indices are governed by Bruhat-cell codimensions. We develop the analogous theory for Monarch matrices, products of a fixed permutation matrix with two block-diagonal fa...
Taking classical information geometry as its point of departure, this paper investigates, through gradient flows, how dually flat geometry extends beyond regular convexity, non-degeneracy, and smoothness. The regular theory is developed from the log-determinant potential on positive definite Gram matrices, establishing...
It is proved that the extracted trend is invariant across representations, and a closed-form Bregman-type divergence quantifying their disagreement under a shared coefficient vector is obtained, and this divergence vanishes as $\lambda\to\infty$.
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