Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs

The chromatic polynomial of a graph G counts the number of proper colorings of G. We give an affirmative answer to the conjecture of Read and Rota-Heron-Welsh that the absolute values of the coefficients of the chromatic polynomial form a log-concave sequence. We define a sequence of numerical invariants of projective hypersurfaces analogous to the Milnor number of local analytic hypersurfaces. Then we give a characterization of correspondences between projective spaces up to a positive integer multiple which includes the conjecture on the chromatic polynomial as a special case. As a byproduct of our approach, we obtain an analogue of Kouchnirenko's theorem relating the Milnor number with the Newton polytope.Comment: Improved readability. Final version, to appear in J. Amer. Math. So

Positivity of Chern classes of Schubert cells and varieties

We show that the Chern-Schwartz-MacPherson class of a Schubert cell in a Grassmannian is represented by a reduced and irreducible subvariety in each degree. This gives an affirmative answer to a positivity conjecture of Aluffi and Mihalcea.Comment: Improved readability, 18 page

Correspondences between projective planes

We characterize integral homology classes of the product of two projective planes which are representable by a subvariety.Comment: Improved readability, 14 page

Cubic symmetroids and vector bundles on a quadric surface

We investigate the jumping conics of stable vector bundles \Ee of rank 2 on a smooth quadric surface $Q$ with the Chern classes c_1=\Oo_Q(-1,-1) and $c_2=4$ with respect to the ample line bundle \Oo_Q(1,1). We describe the set of jumping conics of \Ee, a cubic symmetroid in \PP_3, in terms of the cohomological properties of \Ee. As a consequence, we prove that the set of jumping conics, S(\Ee), uniquely determines \Ee. Moreover we prove that the moduli space of such vector bundles is rational.Comment: 6 pages; Comments welcom

Uncertainty-Aware Attention for Reliable Interpretation and Prediction

Department of Computer Science and EngineeringAttention mechanism is effective in both focusing the deep learning models on relevant features and interpreting them. However, attentions may be unreliable since the networks that generate them are often trained in a weakly-supervised manner. To overcome this limitation, we introduce the notion of input-dependent uncertainty to the attention mechanism, such that it generates attention for each feature with varying degrees of noise based on the given input, to learn larger variance on instances it is uncertain about. We learn this Uncertainty-aware Attention (UA) mechanism using variational inference, and validate it on various risk prediction tasks from electronic health records on which our model significantly outperforms existing attention models. The analysis of the learned attentions shows that our model generates attentions that comply with clinicians' interpretation, and provide richer interpretation via learned variance. Further evaluation of both the accuracy of the uncertainty calibration and the prediction performance with "I don't know'' decision show that UA yields networks with high reliability as well.ope