Quillen proved that repeated multiplication of the standard sesquilinear form to a positive Hermitian bihomogeneous polynomial eventually results in a sum of Hermitian squares,which was the first Hermitian analogue of...Quillen proved that repeated multiplication of the standard sesquilinear form to a positive Hermitian bihomogeneous polynomial eventually results in a sum of Hermitian squares,which was the first Hermitian analogue of Hilbert s seventeenth problem in the nondegenerate case.Later Catlin-D’Angelo generalized this positivstellensatz of Quillen to the case of Hermitian algebraic functions on holomorphic line bundles over compact complex manifolds by proving the eventual positivity of an associated integral operator.The arguments of Catlin-D’Angelo involve subtle asymptotic estimates of the Bergman kernel.In this article,the authors give an elementary and geometric proof of the eventual positivity of this integral operator,thereby yielding another proof of the corresponding positivstellensatz.展开更多
The feature selection characterized by relatively small sample size and extremely high-dimensional feature space is common in many areas of contemporary statistics. The high dimensionality of the feature space causes ...The feature selection characterized by relatively small sample size and extremely high-dimensional feature space is common in many areas of contemporary statistics. The high dimensionality of the feature space causes serious difficulties: (i) the sample correlations between features become high even if the features are stochastically independent; (ii) the computation becomes intractable. These difficulties make conventional approaches either inapplicable or inefficient. The reduction of dimensionality of the feature space followed by low dimensional approaches appears the only feasible way to tackle the problem. Along this line, we develop in this article a tournament screening cum EBIC approach for feature selection with high dimensional feature space. The procedure of tournament screening mimics that of a tournament. It is shown theoretically that the tournament screening has the sure screening property, a necessary property which should be satisfied by any valid screening procedure. It is demonstrated by numerical studies that the tournament screening cum EBIC approach enjoys desirable properties such as having higher positive selection rate and lower false discovery rate than other approaches.展开更多
基金partially supported by the Singapore Ministry of Education Academic Research Fund Tier 1 grant R-146-000-142-112。
文摘Quillen proved that repeated multiplication of the standard sesquilinear form to a positive Hermitian bihomogeneous polynomial eventually results in a sum of Hermitian squares,which was the first Hermitian analogue of Hilbert s seventeenth problem in the nondegenerate case.Later Catlin-D’Angelo generalized this positivstellensatz of Quillen to the case of Hermitian algebraic functions on holomorphic line bundles over compact complex manifolds by proving the eventual positivity of an associated integral operator.The arguments of Catlin-D’Angelo involve subtle asymptotic estimates of the Bergman kernel.In this article,the authors give an elementary and geometric proof of the eventual positivity of this integral operator,thereby yielding another proof of the corresponding positivstellensatz.
基金supported by Singapore Ministry of Educations ACRF Tier 1 (Grant No. R-155-000-065-112)supported by the National Science and Engineering Research Countil of Canada and MITACS,Canada
文摘The feature selection characterized by relatively small sample size and extremely high-dimensional feature space is common in many areas of contemporary statistics. The high dimensionality of the feature space causes serious difficulties: (i) the sample correlations between features become high even if the features are stochastically independent; (ii) the computation becomes intractable. These difficulties make conventional approaches either inapplicable or inefficient. The reduction of dimensionality of the feature space followed by low dimensional approaches appears the only feasible way to tackle the problem. Along this line, we develop in this article a tournament screening cum EBIC approach for feature selection with high dimensional feature space. The procedure of tournament screening mimics that of a tournament. It is shown theoretically that the tournament screening has the sure screening property, a necessary property which should be satisfied by any valid screening procedure. It is demonstrated by numerical studies that the tournament screening cum EBIC approach enjoys desirable properties such as having higher positive selection rate and lower false discovery rate than other approaches.
