Determinantal point processes (DPPs) is an effective tool to deliver diversity in multiple machine learning and computer vision tasks. Under the deep learning framework, DPP is typically optimized via approximation, which is not straightforward and has some conflicts with the diversity requirement. We note, however, there have been no deep learning paradigms to optimize DPP directly since it involves matrix inversion that may result in computational instability. This fact greatly hinders the use of DPP on some specific objective functions where DPP would otherwise serve as a term to measure the feature diversity. In this paper, we devise a simple but effective algorithm to optimize the DPP term directly through expressing with L-ensemble in the spectral domain over the gram matrix, which is more flexible than learning on parametric kernels. By further taking into account additional geometric constraints, our algorithm seeks to generate valid sub-gradients of the DPP term in cases where the DPP gram matrix is not invertible (no gradients exist in this case). In this sense, our algorithm can be easily incorporated with multiple deep learning tasks. Experiments show the effectiveness of our algorithm, indicating promising performance for practical learning problems.
Summary: the authors introduce a method to learn a deep-learning model whose loss function is augmented with a DPP-like regularization term to enforce diversity within the feature embeddings. 
Decision: I recommend that this paper be rejected. At a high level, this paper is experimentally focused, but I am not convinced that the experiments are sufficient for acceptance.
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My main concerns are as follows:
- Many mathematical claims should be more carefully stated. For example, the authors extend the discrete DPP formulation to continuous space. It is not clear to me, based on the choice of the kernel function embedding, that the resulting P_k(X) is a probability (Eq. 1). If it is not (using a DPP-based formulation as a regularizer does not require a distribution), the authors should clarify that fact; more generally, the authors should be more careful throughout the paper (for example, det=0 if features are proportional, not necessarily equal; the authors inconsistently switch between DPP kernel L and marginal kernel K throughout computations.)
- The authors do not describe their baselines for several experiments. In tables 1, 2, 3, the baseline is never described (I assume it's the same setup without regularization); I did not find a description of DCH (Tab 4) in the paper (Deep Cauchy Hashing?). The mAP-k metric should also be defined. Furthermore, the authors do not report standard deviations for their experiments.
- A key consideration when using DPPs is their compulational cost: most operations involving them require SVD (which seems to be used in this work), matrix inversion, and often both. This, unsurprisingly, limits the applications of DPPs, and has driven a lot of research focused on improving DPP overhead. I would like to see more discussion in this paper focused on to which extent the DPP's computational overhead remains tractable, and which methods were used (if any) to alleviate the computational burden.
- Finally, the paper itself appears to be somewhat incomplete: sentences are missing or incomplete (Section 4), and numbers are missing in some tables (Table 5).
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Questions and comments for the authors:
- When computing the proper sub-gradient, are you computing the subgradient as inv(L + \hat L)?
- You state that by avoiding matrix inversion, your method is more feasible. However, it seems like your method requires SVD, which is also O(N^3); could you please provide more detail for this?
- Could you report number of trials and standard deviations for your experiments?
- Do you have any insight into why DPPs do more poorly than the DCH baseline in Table 4 for mAP-k metrics?
- You might be able to save space by decreasing the space allocated to the exponentiated quadratic kernel.