Hyperbolic Hierarchical Knowledge Graph Embeddings for Link Prediction in Low Dimensions

Knowledge graph embeddings (KGE) have been validated as powerful methods for inferring missing links in knowledge graphs (KGs) since they map entities into Euclidean space and treat relations as transformations of entities. Currently, some Euclidean KGE methods model semantic hierarchies prevalent in KGs and promote the performance of link prediction. For hierarchical data, instead of traditional Euclidean space, hyperbolic space as an embedding space has shown the promise of high fidelity and low memory consumption; however, existing hyperbolic KGE methods neglect to model them. To address this issue, we propose a novel KGE model -- hyperbolic hierarchical KGE (HypHKGE). To be specific, we first design the attention-based learnable curvatures for hyperbolic space to preserve rich semantic hierarchies. Moreover, we define the hyperbolic hierarchical transformations based on the theory of hyperbolic geometry, which utilize hierarchies that we preserved to infer the links. Experiments show that HypHKGE can effectively model semantic hierarchies in hyperbolic space and outperforms the state-of-the-art hyperbolic methods, especially in low dimensions.

Total Variation Regularization for Compartmental Epidemic Models with Time-varying Dynamics

Traditional methods to infer compartmental epidemic models with time-varying dynamics can only capture continuous changes in the dynamic. However, many changes are discontinuous due to sudden interventions, such as city lockdown and opening of field hospitals. To model the discontinuities, this study introduces the tool of total variation regularization, which regulates the temporal changes of the dynamic parameters, such as the transmission rate. To recover the ground truth dynamic, this study designs a novel yet straightforward optimization algorithm, dubbed iterative Nelder-Mead, which repeatedly applies the Nelder-Mead algorithm. Experiments on the simulated data show that the proposed approach can qualitatively reproduce the discontinuities of the underlying dynamics. To extend this research to real data as well as to help researchers worldwide to fight against COVID-19, the author releases his research platform as an open-source package.

$hv$-Block Cross Validation is not a BIBD: a Note on the Paper by Jeff Racine (2000)

This note corrects a mistake in the paper "consistent cross-validatory model-selection for dependent data: $hv$-block cross-validation" by Racine (2000). In his paper, he implied that the therein proposed $hv$-block cross-validation is consistent in the sense of Shao (1993). To get this intuition, he relied on the speculation that $hv$-block is a balanced incomplete block design (BIBD). This note demonstrates that this is not the case, and thus the theoretical consistency of $hv$-block remains an open question. In addition, I also provide a Python program counting the number of occurrences of each sample and each pair of samples.

A Distributed Frank-Wolfe Framework for Learning Low-Rank Matrices with the Trace Norm

We consider the problem of learning a high-dimensional but low-rank matrix from a large-scale dataset distributed over several machines, where low-rankness is enforced by a convex trace norm constraint. We propose DFW-Trace, a distributed Frank-Wolfe algorithm which leverages the low-rank structure of its updates to achieve efficiency in time, memory and communication usage. The step at the heart of DFW-Trace is solved approximately using a distributed version of the power method. We provide a theoretical analysis of the convergence of DFW-Trace, showing that we can ensure sublinear convergence in expectation to an optimal solution with few power iterations per epoch. We implement DFW-Trace in the Apache Spark distributed programming framework and validate the usefulness of our approach on synthetic and real data, including the ImageNet dataset with high-dimensional features extracted from a deep neural network.

Toward a Better Understanding of Leaderboard

The leaderboard in machine learning competitions is a tool to show the performance of various participants and to compare them. However, the leaderboard quickly becomes no longer accurate, due to hack or overfitting. This article gives two pieces of advice to prevent easy hack or overfitting. By following these advice, we reach the conclusion that something like the Ladder leaderboard introduced in [blum2015ladder] is inevitable. With this understanding, we naturally simplify Ladder by eliminating its redundant computation and explain how to choose the parameter and interpret it. We also prove that the sample complexity is cubic to the desired precision of the leaderboard.

Two Differentially Private Rating Collection Mechanisms for Recommender Systems

We design two mechanisms for the recommender system to collect user ratings. One is modified Laplace mechanism, and the other is randomized response mechanism. We prove that they are both differentially private and preserve the data utility.