This is a small python program that implements logistic regression in order to learn a subset of the MNIST dataset. It learns all the one and zero digits. I followed the concepts layed out on this website: http://ufldl.stanford.edu/tutorial/supervised/LogisticRegression/.
I have added a good amount of comments, so the code should be understandable. The hyperparameters in train.py are a good place to start messing around with the model. In order to run the program, you will need to have numpy installed. Check out http://www.numpy.org/.
Finally, I downloaded the MNIST dataset from http://yann.lecun.com/exdb/mnist/. The format of the images is described in detail on that page.