Counting Humps in Motzkin paths

In this paper we study the number of humps (peaks) in Dyck, Motzkin and Schr\"{o}der paths. Recently A. Regev noticed that the number of peaks in all Dyck paths of order $n$ is one half of the number of super Dyck paths of order $n$. He also computed the number of humps in Motzkin paths and found a similar relation, and asked for bijective proofs. We give a bijection and prove these results. Using this bijection we also give a new proof that the number of Dyck paths of order $n$ with $k$ peaks is the Narayana number. By double counting super Schr\"{o}der paths, we also get an identity involving products of binomial coefficients.Comment: 8 pages, 2 Figure

Data-Driven Rational Drug Design

Vast amount of experimental data in structural biology has been generated, collected and accumulated in the last few decades. This rich dataset is an invaluable mine of knowledge, from which deep insights can be obtained and practical applications can be developed. To achieve that goal, we must be able to manage such Big Data\u27\u27 in science and investigate them expertly. Molecular docking is a field that can prominently make use of the large structural biology dataset. As an important component of rational drug design, molecular docking is used to perform large-scale screening of putative associations between small organic molecules and their pharmacologically relevant protein targets. Given a small molecule (ligand), a molecular docking program simulates its interaction with the target protein, and reports the probable conformation of the protein-ligand complex, and the relative binding affinity compared against other candidate ligands. This dissertation collects my contributions in several aspects of molecular docking. My early contribution focused on developing a novel metric to quantify the structural similarity between two protein-ligand complexes. Benchmarks show that my metric addressed several issues associated with the conventional metric. Furthermore, I extended the functionality of this metric to cross different systems, effectively utilizing the data at the proteome level. After developing the novel metric, I formulated a scoring function that can extract the biological information of the complex, integrate it with the physics components, and finally enhance the performance. Through collaboration, I implemented my model into an ultra-fast, adaptive program, which can take advantage of a range of modern parallel architectures and handle the demanding data processing tasks in large scale molecular docking applications

Support Neighbor Loss for Person Re-Identification

Person re-identification (re-ID) has recently been tremendously boosted due to the advancement of deep convolutional neural networks (CNN). The majority of deep re-ID methods focus on designing new CNN architectures, while less attention is paid on investigating the loss functions. Verification loss and identification loss are two types of losses widely used to train various deep re-ID models, both of which however have limitations. Verification loss guides the networks to generate feature embeddings of which the intra-class variance is decreased while the inter-class ones is enlarged. However, training networks with verification loss tends to be of slow convergence and unstable performance when the number of training samples is large. On the other hand, identification loss has good separating and scalable property. But its neglect to explicitly reduce the intra-class variance limits its performance on re-ID, because the same person may have significant appearance disparity across different camera views. To avoid the limitations of the two types of losses, we propose a new loss, called support neighbor (SN) loss. Rather than being derived from data sample pairs or triplets, SN loss is calculated based on the positive and negative support neighbor sets of each anchor sample, which contain more valuable contextual information and neighborhood structure that are beneficial for more stable performance. To ensure scalability and separability, a softmax-like function is formulated to push apart the positive and negative support sets. To reduce intra-class variance, the distance between the anchor's nearest positive neighbor and furthest positive sample is penalized. Integrating SN loss on top of Resnet50, superior re-ID results to the state-of-the-art ones are obtained on several widely used datasets.Comment: Accepted by ACM Multimedia (ACM MM) 201