deep learning research papers always claim that deeper layers of CNN have good "semantic information" but poor "spatial information". What is the spatial information exactly. Is that some activations in deeper layers?
Spatial Information refers to information having location-based relation with other information.
This looks like the number "1".
If this was represented in a single line,
It wouldn't be recognizable.
Earlier layers of CNN are convolutional layers, which take into account the image as a 2D (spatial) information. Whereas, the deeper layers flatten that (convoluted) information.
I think the authors are referring to spatial invariance. I will explain it on image classification/recognition example.
Convolutional Neural Networks are designed to be spatially invariant, that is - they are not sensitive to the position of, for example, object in the picture. The deeper you go into layers, the originally not so (pixelwise) similar objects (or usually parts of objects) are becoming more similar (and this is achieved via convolution). At the deepest layers we have extracted features with no information on where they were positioned on the original image. We even lose the information on pixel-size of original objects because of another process in CNN called pooling.
Convolution is the key for why CNNs perform better than any other model in such "human-like" tasks like recognizing specific objects in the picture, words in a recorded speech,$\ldots$.
$\ $ - Convolution is nicely explained and visualized in this YouTube video.
$\ $ - A more lengthy and deeper video on this subject is Convolutional Neural Networks - The Math of Intelligence
Spatial refers to space.
So, what is space in images?
Space represents the 2D plane(x-y) in images. Coming back to the question, 'What is spatial information in cnn?', for example in first conv layer, it extracts spatial information like egdes, corners etc. and in other conv layer it extracts spatial information like eyes, nose etc.
This is spatial information in images. CNN's don't maintain spatial relationship among features. Advanced capsnet addresses this problem.