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Title: Deep Learning of Determinantal Point Processes via Proper Spectral Sub-gradient

Paper content:
Determinantal point processes (DPPs) is an effective tool to deliver diversity on multiple machine learning and computer vision tasks.
Under deep learning framework, DPP is typically optimized via approximation, which is not straightforward and has some conflict with diversity requirement.
We note, however, there has been no deep learning paradigms to optimize DPP directly since it involves matrix inversion which may result in highly computational instability.
This fact greatly hinders the wide use of DPP on some specific objectives where DPP serves as a term to measure the feature diversity.
In this paper, we devise a simple but effective algorithm to address this issue to optimize DPP term directly expressed with L-ensemble in spectral domain over gram matrix, which is more flexible than learning on parametric kernels.
By further taking into account some geometric constraints, our algorithm seeks to generate valid sub-gradients of DPP term in case when 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.
Diversity is desired in multiple machine learning and computer vision tasks (e.g., image hashing (Chen et al., 2017; Carreira-Perpinán & Raziperchikolaei, 2016) , descriptor learning , metric learning (Mishchuk et al., 2017) and video summarization (Sharghi et al., 2018; Liu et al., 2017) ), in which sub-sampled points or learned features need to spread out through a specific bounded space.
Originated from quantum physics, determinantal point processes (DPP) have shown its power in delivering such properties Kulesza & Taskar, 2011b) .
Compared with other diversity-oriented techniques (e.g., entropy (Zadeh et al., 2017) and orthogonality ), DPP shows its superiority as it incorporates only one single metric and delivers genuine diversity on any bounded space Affandi et al., 2013; Gillenwater et al., 2012) .
Therefore, DPP has been utilized in a large body of diversity-oriented tasks.
In general, sample points from a DPP tend to distribute diversely within a bounded space A .
Given a positive semi-definite kernel function κ : A × A → R, the probability of a discrete point set X ⊂ A under a DPP with kernel function κ can be characterized as:
where L is a |X | × |X | matrix with entry L ij = κ(x i , x j ) and x i , x j ∈ X .
L is called L-ensemble.
Note that A is a continuous space, whereas X is finite.
In the Hilbert space associated with κ, larger determinant implies larger spanned volume, thus the mapped points tend not to be similar or linearly dependent.
DPP can be viewed from two perspectives: sampling and learning.
A comprehensive introduction to mathematical fundamentals of DPP for sampling from a discrete space can be found in .
Based on this, a line of works has been proposed (Kulesza & Taskar, 2011a; Kang, 2013; Hennig & Garnett, 2016) .
In this paper, we concentrate on learning DPPs.
In learning of DPP, the term det(L) is typically treated as a singleton diversity measurement and is extended to learning paradigms on continuous space (Chao et al., 2015; Kulesza & Taskar, 2010; Affandi et al., 2014) .
There are generally two lines of strategies to learn DPPs:
Approximation.
This type of methods is to convert DPP into a simpler format which can ease and stabilize the computation.
low-rank approximation proves powerful in easing the computational burden (Gartrell et al., 2017) , in which the gram matrix is factorized as L = BB where B ∈ n×m with m n.
This decomposition can also reduce the complexity which is originally a cubic time of |L|.
Kulesza & Taskar (2011b) explicitly expressed the kernel with κ(x,
y) = σ 1 σ 2 δ(x
) δ(y
), where σ measures the intrinsic quality of the feature and δ(·) is function mapping input x to a feature space. In
this sense, the pairwise similarity is calculated in Euclidean feature space with cosine distance. Elfeki
et al. (2019) suggest approximating a given distribution by approximating the eigenvalues of the corresponding DPP. As such
, the computation can be eased and become stable. Following
this, DPP is also applied on some visual tasks, such as video summarization (Sharghi et al., 2018) , ranking (Liu et al., 2017) and image classification (Xie et al., 2017) . It can be
noted that the approximation is not straightforward for DPP, thus cannot fully deliver the diversity property (e.g. resulting in rank-deficiency).
Direct optimization.
While the aforementioned methods optimize DPP with specific approximation, a series of efforts also seek to optimize the DPP term directly (Gillenwater et al., 2014; Mariet & Sra, 2015; Bardenet & Titsias, 2015) .
In this setting, the whole gram matrix L corresponding to the pairwise similarity among features is updated directly, which allows accommodating more flexible feature mapping functions rather than an approximation.
Gillenwater et al. (2014) proposed an Expectation-Maximization algorithm to update marginal kernel DPP K = L(L + I) −1 , together with a baseline K-Ascent derived from projected gradient ascent (Levitin & Polyak, 1966) .
Mariet & Sra (2015) extended DPP from a fixed-point perspective and Bardenet & Titsias (2015) proposed to optimize DPP upon a lower bound in variational inference fashion.
A key problem of such line of works is that the computation is not differentiable, making it difficult to be used in deep learning frameworks.
To the best of our knowledge, there is no previous method incorporating DPP as a feature-level diversity metric in deep learning.
A key difficulty in doing so is that the calculation of the gradient of det(L) involves matrix inversion, which can be unstable and inaccurate in GPUs.
Though KAscent seems to be a naive rule, it still needs explicit matrix inversion in the first step before the projection procedure.
This fact greatly hinders the tight integration of DPP with deep networks.
Some alternative methods seek to reach diversity under more constrained settings.
For example, resorted to a global pairwise orthogonality constraint in hyper-sphere and Zadeh et al. (2017) employed statistical moments to measure the diversity.
However, compared with DPP, such measurements are unable to fully characterize diversity in an arbitrary bounded space.
In this paper, rather than providing more efficient DPP solvers, we concentrate on delivering a feasible feature-level DPP integration under the deep learning framework.
To this end, we revisit the spectral decomposition of DPP and propose a sub-gradient generation method which can be tightly integrated with deep learning.
Our method differs from either approximation or direct optimization by introducing a "differentiable direct optimization" procedure, thus can produce genuinely diverse features in continuous bounded space.
Our method is stable and scalable to the relatively large dataset with a specific mini-batch sampling strategy, which is verified by several experiments on various tasks.
Notations: Bold lower case x and bold upper case K represent vector and matrix, respectively.
det(·) and Tr(·) calculate the determinant and trace of a matrix, respectively.
A ⊗ B is the element-wise product of matrices A and B. |X | and |x| measure the cardinality of a finite set X and the L 2 length of a vector x, respectively.
x, y calculates the inner product of the two vectors.
x = diag(X) transforms a diagonal matrix X into its vector form x, and vice versa.
We refer "positive semi-definite" and "positive definite" to PSD and PD, respectively.
Denote the real numbers.
In this paper, we investigated the problem of learning diverse features via a determinantal point process under deep learning framework.
To overcome the instability in computing the gradient which involves the matrix inverse, we developed an efficient and reliable procedure called proper spectral sub-gradient generation.
The generated proper sub-gradient can replace the true gradient and performs well in applications.
We also considered how to constrain the features into a bounded space, since in such a way one can ensure the behavior of the network more predictable.
To this end, we further incorporated Wasserstein GAN into our framework.
Together, DPP+WGAN showed significant performance on both some common criteria and feature space utility.
A APPENDIX

assistant: text: We proposed a specific back-propagation method via proper spectral sub-gradient to integrate determinantal point process to deep learning framework.