Reconstruction and Higher Dimensional Geometry

In this paper, we give a new proof on a Theorem of Tutte which says that the determinants of the adjacency matrices of two hypomorphic graphs are the same. We also study the lowest eigenvectors.Comment: 9 pages, to appear in Journal of Combinatorial Theory Series

A Deep Network with Visual Text Composition Behavior

While natural languages are compositional, how state-of-the-art neural models achieve compositionality is still unclear. We propose a deep network, which not only achieves competitive accuracy for text classification, but also exhibits compositional behavior. That is, while creating hierarchical representations of a piece of text, such as a sentence, the lower layers of the network distribute their layer-specific attention weights to individual words. In contrast, the higher layers compose meaningful phrases and clauses, whose lengths increase as the networks get deeper until fully composing the sentence.Comment: accepted to ACL201

On Matrix-Valued Square Integrable Positive Definite Functions

In this paper, we study matrix valued positive definite functions on a unimodular group. We generalize two important results of Godement on square integrable positive definite functions to matrix valued square integrable positive definite functions. We show that a matrix-valued continuous $L^2$ positive definite function can always be written as a convolution of a $L^2$ positive definite function with itself. We also prove that, given two $L^2$ matrix valued positive definite functions $\Phi$ and $\Psi$, $\int_G Trace(\Phi(g) \bar{\Psi(g)}^t) d g \geq 0$. In addition this integral equals zero if and only if $\Phi * \Psi=0$. Our proofs are operator-theoretic and independent of the group.Comment: 11 page

Gan-Gross-Prasad Conjecture for U(p,q)

In this paper, we give a proof of the Gan-Gross-Prasad conjecture for the discrete series of U(p,q). Given a discrete series representation $D(\lambda)$ in terms of the Harish-Chandra parameter, the restriction of $D(\lambda)$ to U(p-1,q) contains $D(\mu)$ as a subrepresentation if and only if $\lambda$ and $\mu$ interlaces in a very special way.Comment: 34 pages, to appear in Inventiones Mathematica