I'll first try to share some intuition behind CNN and then comment the particular topics you listed.
The convolution and sub-sampling layers in a CNN are not different from the hidden layers in a common MLP, i. e. their function is to extract features from their input. These features are then given to the next hidden layer to extract still more complex features, or are directly given to a standard classifier to output the final prediction (usually a Softmax, but also SVM or any other can be used). In the context of image recognition, these features are images treats, like stroke patterns in the lower layers and object parts in the upper layers.
In natural images these features tend to be the same at all locations. Recognizing a certain stroke pattern in the middle of the images will be as useful as recognizing it close to the borders. So why don't we replicate the hidden layers and connect multiple copies of it in all regions of the input image, so the same features can be detected anywhere? It's exactly what a CNN does, but in a efficient way. After the replication (the "convolution" step) we add a sub-sample step, which can be implemented in many ways, but is nothing more than a sub-sample. In theory this step could be even removed, but in practice it's essential in order to allow the problem remain tractable.
Thus:
- Correct.
- As explained above, hidden layers of a CNN are feature extractors as in a regular MLP. The alternated convolution and sub-sampling steps are done during the training and classification, so they are not something done "before" the actual processing. I wouldn't call them "pre-processing", the same way the hidden layers of a MLP is not called so.
- Correct.
A good image which helps to understand the convolution is CNN page in the ULFDL tutorial. Think of a hidden layer with a single neuron which is trained to extract features from $3 \times 3$ patches. If we convolve this single learned feature over a $5 \times 5$ image, this process can be represented by the following gif:
In this example we were using a single neuron in our feature extraction layer, and we generated $9$ convolved features. If we had a larger number of units in the hidden layer, it would be clear why the sub-sampling step after this is required.
The subsequent convolution and sub-sampling steps are based in the same principle, but computed over features extracted in the previous layer, instead of the raw pixels of the original image.
Sparse representations are expected in hierarchical models. Possibly, what you are discovering is a problem intrinsic to the hierarchical structure of deep learning models. You will find quite a few scientific papers on "sparse representations", especially in memory research.
I think you would benefit from reading about "receptor fields" in visual cortex. Not only are there ON and OFF cells in the mammal brain, but also RF cells that fire both during ON and OFF. Perhaps the edge/sparsity problem could be circumvented by updating the model to reflect current neuroscience on vision, especially in animal models.
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(Source: Must Know Tips/Tricks in Deep Neural Networks (by Xiu-Shen Wei))