Low rank subspace clustering (LRSC)

We consider the problem of fitting a union of subspaces to a collection of data points drawn from one or more subspaces and corrupted by noise and/or gross errors. We pose this problem as a non-convex optimization problem, where the goal is to decompose the corrupted data matrix as the sum of a clean and self-expressive dictionary plus a matrix of noise and/or gross errors. By self-expressive we mean a dictionary whose atoms can be expressed as linear combinations of themselves with low-rank coefficients. In the case of noisy data, our key contribution is to show that this non-convex matrix decomposition problem can be solved in closed form from the SVD of the noisy data matrix. The solution involves a novel polynomial thresholding operator on the singular values of the data matrix, which requires minimal shrinkage. For one subspace, a particular case of our framework leads to classical PCA, which requires no shrinkage. For multiple subspaces, the low-rank coefficients obtained by our framework can be used to construct a data affinity matrix from which the clustering of the data according to the subspaces can be obtained by spectral clustering. In the case of data corrupted by gross errors, we solve the problem using an alternating minimization approach, which combines our polynomial thresholding operator with the more traditional shrinkage-thresholding operator. Experiments on motion segmentation and face clustering show that our framework performs on par with state-of-the-art techniques at a reduced computational cost.

A Global Climatology of Baroclinically Influenced Tropical Cyclogenesis*

Tropical cyclogenesis is generally considered to occur in regions devoid of baroclinic structures; however, an appreciable number of tropical cyclones (TCs) form in baroclinic environments each year. A global climatology of these baroclinically influenced TC developments is presented in this study. An objective classification strategy is developed that focuses on the characteristics of the environmental state rather than on properties of the vortex, thus allowing for a pointwise “development pathway” classification of reanalysis data. The resulting climatology shows that variability within basins arises primarily as a result of local surface thermal contrasts and the positions of time-mean features on the subtropical tropopause. The pathway analyses are sampled to generate a global climatology of 1948–2010 TC developments classified by baroclinic influence: nonbaroclinic (70%), low-level baroclinic (9%), trough induced (5%), weak tropical transition (11%), and strong tropical transition (5%). All basins other than the North Atlantic are dominated by nonbaroclinic events; however, there is extensive interbasin variability in secondary development pathways. Within each basin, subregions and time periods are identified in which the relative importance of the development pathways also differs. The efficiency of tropical cyclogenesis is found to be highly dependent on development pathway. The peak efficiency defined in the classification subspace straddles the nonbaroclinic/trough-induced boundary, suggesting that the optimal environment for TC development includes a baroclinic contribution from an upper-level disturbance. By assessing the global distribution of baroclinically influenced TC formations, this study identifies regions and pathways whose further study could yield improvements in our understanding of this important subset of TC developments.

ANOVA kernels and RKHS of zero mean functions for model-based sensitivity analysis

Given a reproducing kernel Hilbert space (H,〈.,.〉)(H,〈.,.〉) of real-valued functions and a suitable measure μμ over the source space D⊂RD⊂R, we decompose HH as the sum of a subspace of centered functions for μμ and its orthogonal in HH. This decomposition leads to a special case of ANOVA kernels, for which the functional ANOVA representation of the best predictor can be elegantly derived, either in an interpolation or regularization framework. The proposed kernels appear to be particularly convenient for analyzing the effect of each (group of) variable(s) and computing sensitivity indices without recursivity.

Frequency of Sobolev and quasiconformal dimension distortion

We study Hausdorff and Minkowski dimension distortion for images of generic affine subspaces of Euclidean space under Sobolev and quasiconformal maps. For a supercritical Sobolev map f defined on a domain in RnRn, we estimate from above the Hausdorff dimension of the set of affine subspaces parallel to a fixed m-dimensional linear subspace, whose image under f has positive HαHα measure for some fixed α>mα>m. As a consequence, we obtain new dimension distortion and absolute continuity statements valid for almost every affine subspace. Our results hold for mappings taking values in arbitrary metric spaces, yet are new even for quasiconformal maps of the plane. We illustrate our results with numerous examples.