This paper presents a general method for deriving approximate dynamical equations that satisfy a prescribed constraint typically chosen to filter out unwanted highfrequency motions. The approximate equations take a simple general form in arbitrary phase-space coordinates. The family of semigeostrophic equations for rapidly rotating flow derived by Salmon (1983, 1985) fits this general form when...

Tissot indicatrices have provided visual measures of local area and isotropy distortions. Here we show how large scale distortions of flexion (bending) and skewness (lopsidedness) can be measured. Area and isotropy distortions depend on the map projection metric, flexion and skewness, which manifest themselves on continental scales, depend on the first derivatives of the metric. We introduce ne...

We apply the L2 based Fisher-Rao vector-field projection by Brigo, Hanzon and LeGland (1999) to finite dimensional approximations of the Fokker Planck equation on exponential families. We show that if the sufficient statistics are chosen among the diffusion eigenfunctions the finite dimensional projection or the equivalent assumed density approximation provide the exact maximum likelihood densi...

Metric learning has been shown to significantly improve the accuracy of k-nearest neighbor (kNN) classification. In problems involving thousands of features, distance learning algorithms cannot be used due to overfitting and high computational complexity. In such cases, previous work has relied on a two-step solution: first apply dimensionality reduction methods to the data, and then learn a me...

SUMMARY We propose a novel adaptive filtering scheme named metric-combining normalized least mean square (MC-NLMS). The proposed scheme is based on iterative metric projections with a metric designed by combining multiple metric-matrices convexly in an adaptive manner, thereby taking advantages of the metrics which rely on multiple pieces of information. We compare the improved PNLMS (IPNLMS) a...

We consider the self-calibration problem for the generic imaging model that assigns projection rays to pixels without a parametric mapping. In this paper, we consider the central variant of this model, which encompasses all camera models with a single effective viewpoint. Self-calibration refers to calibrating a camera’s projection rays, purely from matches between images, i.e. without knowledg...

For a Banach space V we define its Lipschitz extension constant, LE(V ), to be the infimum of the constants c such that for every metric space (Z, ρ), every X ⊂ Z, and every f : X → V , there is an extension, g, of f to Z such that L(g) ≤ cL(f), where L denotes the Lipschitz constant. The basic theorem is that when V is finite-dimensional we have LE(V ) = PC(V ) where PC(V ) is the well-known p...