Paper:
Detection of photo manipulation relies on subtle statistical traces, notoriously removed by aggressive lossy compression employed online. We demonstrate that end-to-end modeling of complex photo dissemination channels allows for codec optimization with explicit provenance objectives. We design a lightweight trainable lossy image codec, that delivers good rate-distortion performance, comparable with the popular hand-crafted BPG, and has low computational footprint on modern GPU-enabled platforms. Our results show that significant improvements in manipulation detection accuracy are possible at fractional costs in bandwidth/storage. Our codec improved the accuracy from 37% to 86% even at very low bit-rates, well below the practicality of JPEG (QF 20).

Review:
The paper describes a pipeline for image compression which allows to reliably detect specific manipulation patterns in compressed images.  The results show that it is possible to learn image compression that performs similarly to a modern image compression algorithm while in the same time is optimized to reveal specific kinds of manipulations. The authors build upon (Korus & Memon, 2019), but use a learnable codec instead of differentiable JPEG. 
The idea to regularize entropy of the latent representation of images is interesting. A method to train a well-performing image compression system which can also follow additional constraints (such as ability to reveal certain manipulations) is very valuable for practice. Unfortunately, there are already available trainable compression methods and the authors do not compare to these methods. However, in my opinion to detect manipulation in the image one should prove that visual content in some area of the image was significantly changed with respect to some original, while in the other parts of the image it was not changed. Otherwise it becomes impossible to distinguish in-camera filtering and secondary postprocessing. Basically, the authors present a method to detect whether a very particular configuration of some basic image processing filters (Gaussian blur, median filter, resampling)  was applied to the image. Therefore the particular problem formulation looks very artificial. 
With regards to the experiments in the paper, I was somewhat lost. Compare the Fig. 5 and the Fig. 8. In the Fig. 8, we see a big set of possible system configurations having different manipulation detection accuracy, image quality and compression performance.  In the Fig. 5, we see a compression efficiency-image quality dependency. However, it remains unclear how do the systems represented at these two graphs relate to each other, or, in the other words, what is he mapping between points of these graphs. Next, poor performance of JPEG manipulation detection by the proposed  network does not prove that JPEG manipulation cannot be detected, it just shows that the proposed architecture does not perform well in this problem. A comparative study which relates a new system to a current state of the art is required to claim that a proposed approach is better. Finally, SSIM is not a standard way to compute image quality. MS-SSIM and PSNR are also popular, and a user study is usually recommended to claim that some method generates images of better visual quality.
Summarizing, the authors do not provide a new best-performing image compression algorithm, and neither solve a problem of image manipulation detection, but show that it is possible to learn an image compression system with some additional constraints. I believe it is an interesting contribution, and I hope the authors can improve presentation of the experiments.