Paper:
This paper proposes a novel approach for learning discriminative and sparse representations. It consists of utilizing two different models. A predefined number of non-linear transform models are used in the learning stage, and one sparsifying transform model is used at test time. The non-linear transform models have discriminative and minimum information loss priors. A novel measure related to the discriminative prior is proposed and defined on the support intersection for the transform representations. The minimum information loss prior is expressed as a constraint on the conditioning and the expected coherence of the transform matrix. An equivalence between the non-linear models and the sparsifying model is shown only when the measure that is used to define the discriminative prior goes to zero. An approximation of the measure used in the discriminative prior is addressed, connecting it to a similarity concentration. To quantify the discriminative properties of the transform representation, we introduce another measure and present its bounds. Reflecting the discriminative quality of the transform representation we name it as discrimination power. To support and validate the theoretical analysis a practical learning algorithm is presented. We evaluate the advantages and the potential of the proposed algorithm by a computer simulation. A favorable performance is shown considering the execution time, the quality of the representation, measured by the discrimination power and the recognition accuracy in comparison with the state-of-the-art methods of the same category.

Review:
Summary:
The paper proposes a model to estimate a non-linear transform of data with labels, trying to increase the discriminative power of the transformation while preserving information. 
Quality:
The quality is potentially good but I misunderstood too many things (see below) to emit a confident judgement.
Clarity:
Clarity is poor. The paper is overall difficult to follow (at least for me) for different reasons. First, with 17 pages + references + appendix, the length of the paper is way above the « strongly suggested limit of 8 pages ». Second, there are a number of small typos and the citations are not well formatted (use \citep instead of \citet). Third, and more importantly, several concepts are only vaguely formulated (or wrong), and I must say I certainly misunderstood some parts of the manuscript.
A few examples of things that should be clarified:
- p4: « per class c there exists an unknown nonlinear function […] that separates the data samples from different classes in the transformed domain ». What does « separate » mean here? There exists as many functions as there are classes, do you mean that each function separates all classes? Or that when you apply each class function to the elements of its own class, you get a separation between the classes? (which would not be very useful)
- p5: what do you mean exactly by « non-linear thresholding function »?
- p5: « The goal of learning a nonlinear transform (2) is to estimate… »: not sure what you mean by « accurate approximation » in this sentence.
- p5 equation 3: if I understand correctly, you not only want to estimate a single vector of thresholds for all classes, but also want it to be constant. Is there a reason for the second constraint?
- p5 After equation 4, you define different « priors ». The one on z_k seems to be Gaussian; however in equation 4, z_k seems to be the difference between the input and the output of the nonlinear threshold operator. If this is correct, then the nonlinear threshold operator is just the addition of a Gaussian random noise, which is really not a thresholding operator. I suppose I misunderstood something here, some clarification is probably needed.
- p5 before equation 5, you define a conditional distribution of \tau_c given y_{c,k} ; however if you define a prior on \tau_c given each point (i.e., each $k$), how do you define the law of \tau_c given all points?
- p5 Equation 6: I dont understand what the word « approximation » refers to, and how equation 6 is derived. 
-p6 equation 7 : missing exponent in the Gaussian distribution
- p6-7: to combine equations 7-8-9 and obtain 10, I suppose in equation 7 the distributions should be conditioned on A (at least the first one), and in equation 9 I suppose the second line should be removed and the third line should be conditioned on A; otherwise more explanations are needed.
- Lemma 1 is just unreadable. Please at least split equations in several lines.
Originality:
As far as I can tell, the approach is quite original and the results proved are new.
Significance:
The method provides a new way to learn discriminative features; as such it is a variant of several existing methods, which could have some impact if clarity is improved and a public code is provided.