Is it ok to publish a picture from a dataset without permission? I am writing up some research I did using a Kaggle dataset:
https://www.kaggle.com/c/dogs-vs-cats

I want to include one or two images from the dataset in my paper to show the types of images I'm using. Will this run afoul of copyright issues?

The rules for the dataset say:


  These images have been published by Microsoft Research for the express purpose of furthering academic research. They may be used for non-commercial research purposes, but they may not be re-published without the express permission of Microsoft Research.


Including an image in my paper is use for non-commercial research purposes, but it is also re-publishing them without the express permission of Microsoft Research. I asked this question on the Kaggle forum, with no response in several weeks, and there is no contact information for anyone at Microsoft Research. 

More generally, is it acceptable to publish examples of images from publicly available datasets even if there is not an explicit copyright license? What about ImageNet, where they took images from all over the internet? Is any paper that publishes examples from ImageNet (there are plenty!) breaking copyright law?
If you are publishing (like in a journal) then it is for commercial use, just not for your commercial use - the publishing company makes money off of your publication.  Therefore you need to get permission as stated in the agreement.

It is almost always a requirement that you get permission under these circumstances.  When it is not is when the licence explicitly says so and you are following the rules