We thank the authors for their thoughtful comments. Here are some responses.

* Comparisons with shallower networks, but using unsupervised pre-training:
e will add those results to the paper. Previous work in our group with
very similar data (the InfiniteMNIST dataset were published in JMLR in 20102
"Why Does Unsupervised Pre-training Help Deep Learning?"). The results indeed
show improvement when going from 1 to 2 and then 3 layers, even when using
unsupervised pre-training (RBM or Denoising Auto-Encoder).

* Comparisons with SVMs. We have tried several kinds of SVMs. The main limitation
of course is the size of the training set. One option is to use a non-linear SVM
with a reduced training set, and the other is to use an online linear SVM.
Another option we have considered is to project the input non-linearly in a
high-dimensional but sparse representation and then use an online linear SVM on that space.
For this experiment we have thresholded input pixel gray levels considered a
low-order polynomial expansion (e.g. only looking at pairs of non-zero pixels).
We have obtained the following results until now, all substantially worse than those
obtained with the MLP and deep nets. 

SVM type   training set   input               online    validation test set
            type / size   features            training  set error    error
                                              error
Linear SVM,  NIST,  651k,  original,           36.62%,  34.41%,     42.26%
Linear SVM,  NIST,  651k,  sparse quadratic,   30.96%,  28.00%,     41.28%
Linear SVM,  NISTP, 800k,  original,           88.50%,  85.24%,     87.36%
Linear SVM,  NISTP, 800k,  sparse quadratic,   81.76%,  83.69%,     85.56%
RBF SVM,     NISTP, 100k,  original,           74.73%,  56.57%,     64.22%

The best results were obtained with the sparse quadratic input features, and
training on the CLEAN data (NIST) rather than the perturbed data (NISTP).


* Using distorted characters as the corruption process of the Denoising
Auto-Encoder (DAE). We had already performed preliminary experiments with this idea
and it did not work very well (in fact it depends on the kind of distortion
considered), i.e., it did not improve on the simpler forms of noise we used
for the AISTATS submission.  We have several interpretations for this, which should
probably go (along with more extensive simulations) into another paper.
The main interpretation for those results is that the DAE learns good
features by being given as target (to reconstruct) a pattern of higher
density (according to the unknown, underlying generating distribution) than
the network input. This is how it gets to know where the density should
concentrate. Hence distortions that are *plausible* in the input distribution
(such as translation, rotation, scaling, etc.) are not very useful, whereas
corruption due to a form of noise are useful. In fact, the most useful 
is a very simple form of noise, that guarantees that the input is much
less likely than the target, such as Gaussian noise. Another way to think
about it is to consider the symmetries involved. A corruption process should
be such that swapping input for target should be very unlikely: this is
true for many kinds of noises, but not for geometric transformations
and deformations.

* Human labeling: 

* Size of labeled set: