259 research outputs found
Contour regression: A general approach to dimension reduction
We propose a novel approach to sufficient dimension reduction in regression,
based on estimating contour directions of small variation in the response.
These directions span the orthogonal complement of the minimal space relevant
for the regression and can be extracted according to two measures of variation
in the response, leading to simple and general contour regression (SCR and GCR)
methodology. In comparison with existing sufficient dimension reduction
techniques, this contour-based methodology guarantees exhaustive estimation of
the central subspace under ellipticity of the predictor distribution and mild
additional assumptions, while maintaining \sqrtn-consistency and computational
ease. Moreover, it proves robust to departures from ellipticity. We establish
population properties for both SCR and GCR, and asymptotic properties for SCR.
Simulations to compare performance with that of standard techniques such as
ordinary least squares, sliced inverse regression, principal Hessian directions
and sliced average variance estimation confirm the advantages anticipated by
the theoretical analyses. We demonstrate the use of contour-based methods on a
data set concerning soil evaporation.Comment: Published at http://dx.doi.org/10.1214/009053605000000192 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Consistent Computation of First- and Second-Order Differential Quantities for Surface Meshes
Differential quantities, including normals, curvatures, principal directions,
and associated matrices, play a fundamental role in geometric processing and
physics-based modeling. Computing these differential quantities consistently on
surface meshes is important and challenging, and some existing methods often
produce inconsistent results and require ad hoc fixes. In this paper, we show
that the computation of the gradient and Hessian of a height function provides
the foundation for consistently computing the differential quantities. We
derive simple, explicit formulas for the transformations between the first- and
second-order differential quantities (i.e., normal vector and principal
curvature tensor) of a smooth surface and the first- and second-order
derivatives (i.e., gradient and Hessian) of its corresponding height function.
We then investigate a general, flexible numerical framework to estimate the
derivatives of the height function based on local polynomial fittings
formulated as weighted least squares approximations. We also propose an
iterative fitting scheme to improve accuracy. This framework generalizes
polynomial fitting and addresses some of its accuracy and stability issues, as
demonstrated by our theoretical analysis as well as experimental results.Comment: 12 pages, 12 figures, ACM Solid and Physical Modeling Symposium, June
200
Deep Extreme Multi-label Learning
Extreme multi-label learning (XML) or classification has been a practical and
important problem since the boom of big data. The main challenge lies in the
exponential label space which involves possible label sets especially
when the label dimension is huge, e.g., in millions for Wikipedia labels.
This paper is motivated to better explore the label space by originally
establishing an explicit label graph. In the meanwhile, deep learning has been
widely studied and used in various classification problems including
multi-label classification, however it has not been properly introduced to XML,
where the label space can be as large as in millions. In this paper, we propose
a practical deep embedding method for extreme multi-label classification, which
harvests the ideas of non-linear embedding and graph priors-based label space
modeling simultaneously. Extensive experiments on public datasets for XML show
that our method performs competitive against state-of-the-art result
- …