Vector cross product in n-dimensional vector space

The definition of vector cross product (VCP) introduced by Eckmann only exists in thethree- and the seven- dimensional vector space. In this paper, according to the orthogonal completeness, magnitude of basis vector cross product and all kinds of combinations of basis vector $\hat{e}_i$, the generalized definition of VCP in the odd n-dimensional vector space is given by introducing a cross term $X_{AB}$. In addition, the definition is validated by reducing the generalization definition to the fundamental three- and seven-dimensional vector space.Comment: 9 pages, 1 figure, comments are welcom

Dynamic Power Splitting Policies for AF Relay Networks with Wireless Energy Harvesting

Wireless energy harvesting (WEH) provides an exciting way to supply energy for relay nodes to forward information for the source-destination pairs. In this paper, we investigate the problem on how the relay node dynamically adjusts the power splitting ratio of information transmission (IT) and energy harvesting (EH) in order to achieve the optimal outage performance. According to the knowledge of channel state information (CSI) at the relay, optimal dynamic power splitting policy with full CSI and partial CSI are both provided. Finally, through simulations, the proposed power splitting policies can improve the outage performances and the policy with full CSI achieves the best performance. It is also shown that the policy with partial CSI can approach the policy with full CSI closely and incurs far less system overhead.Comment: accepted by IEEE ICC 2015 - Workshop on Green Communications and Networks with Energy Harvesting, Smart Grids, and Renewable Energie

Being Negative but Constructively: Lessons Learnt from Creating Better Visual Question Answering Datasets

Visual question answering (Visual QA) has attracted a lot of attention lately, seen essentially as a form of (visual) Turing test that artificial intelligence should strive to achieve. In this paper, we study a crucial component of this task: how can we design good datasets for the task? We focus on the design of multiple-choice based datasets where the learner has to select the right answer from a set of candidate ones including the target (\ie the correct one) and the decoys (\ie the incorrect ones). Through careful analysis of the results attained by state-of-the-art learning models and human annotators on existing datasets, we show that the design of the decoy answers has a significant impact on how and what the learning models learn from the datasets. In particular, the resulting learner can ignore the visual information, the question, or both while still doing well on the task. Inspired by this, we propose automatic procedures to remedy such design deficiencies. We apply the procedures to re-construct decoy answers for two popular Visual QA datasets as well as to create a new Visual QA dataset from the Visual Genome project, resulting in the largest dataset for this task. Extensive empirical studies show that the design deficiencies have been alleviated in the remedied datasets and the performance on them is likely a more faithful indicator of the difference among learning models. The datasets are released and publicly available via http://www.teds.usc.edu/website_vqa/.Comment: Accepted for Oral Presentation at NAACL-HLT 201