Paper:
A KERNEL RANDOM MATRIX-BASED APPROACH FOR SPARSE PCA
In this paper, we present a random matrix approach to recover sparse principal components from n p-dimensional vectors. Specifically, considering the large dimensional setting where n, p → ∞ with p/n → c ∈ (0, ∞) and under Gaussian vector observations, we study kernel random matrices of the type f (Ĉ), where f is a three-times continuously differentiable function applied entry-wise to the sample covariance matrixĈ of the data. Then, assuming that the principal components are sparse, we show that taking f in such a way that f (0) = f (0) = 0 allows for powerful recovery of the principal components, thereby generalizing previous ideas involving more specific f functions such as the soft-thresholding function.

Review:
The paper aims at generalizing a result by Deshpande and Montanari.
Deshpande and Montanari prove a result for the covariance thresholding algorithm which consists of (1) removing noise from an empirical covariance matrix of a (single) spiked sparse PCA model using soft thresholding, then (2) computing the leading eigenvector of the denoised matrix and finally (3) picking the leading coordinates of the leading eigenvector.
The current paper focuses on step (1) in the above mentioned process. They claim to generalize the use of soft thesholding to more general functions applied element-wise to the empirical covariance matrix.
There is a questionable term in the paper's title: the word "kernel" is mistakenly used. In fact a symmetric matrix is the Gram matrix of a kernel if it is positive semi-definite. The empirical covariance is PSD, but when you apply a function to it elementwise it has no guarantee of conserving the PSD property.
Regarding the motivation of the paper (1) the paper claims to study the rank K case with arbitrary K>=1. Deshpande and Montanari studied K=1 and the analysis is already an interesting result. Generalizing it to higher ranks poses many questions, including the sparse vector supports' overlaps. 
It is not clear to me why authors insist on trying to generalize the result to a function f that has broader properties. The main motivation is to recover the support of the (leading) sparse eigenvector / PC. It should not be to try denoising the empirical covariance with a complicated function f. If the focus is on generalizing the soft-thresholding part of the approach, then the real question can be formulated as what is the optimal f given that we have this or that property in the data? This often leads to Bayesian analysis of the problem. 
The numerical experiments do not show any substantial improvement obtained using the prescribed method over using the baseline method (covariance thresholding). I suspect that authors can emphasis the benefit of their method by picking f to hold certain properties that reflect the noise process and beat covariance thresholding in those regimes.
Figure 1 right hand side. I do not see why authors refer to phase transition. I don't see a phase transition happening there.