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

1 INTRODUCTION . 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 ( Zhang et al. , 2017 ) , 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 et al. , 2012 ; Kulesza & Taskar , 2011b ) . Compared with other diversity-oriented techniques ( e.g. , entropy ( Zadeh et al. , 2017 ) and orthogonality ( Zhang et al. , 2017 ) ) , DPP shows its superiority as it incorporates only one single metric and delivers genuine diversity on any bounded space ( Kulesza et al. , 2012 ; 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 ( Kulesza et al. , 2012 ) . 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 : Pκ ( X ) ∝ det ( LX ) ( 1 ) where L is a |X | × |X | matrix with entry Lij = κ ( xi , xj ) and xi , xj ∈ 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 Kulesza et al . ( 2012 ) . 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 ) = σ1σ2δ ( 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 can not 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 , Zhang et al . ( 2017 ) 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 L2 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 . 2 BACKGROUND . 2.1 DETERMINANTAL POINT PROCESS . L-ensemble expression of DPP requires L to be PSD , whereas kernel expression further constrains K < I ( each eigenvalue of K is less than 1 ) . A conversion from L to K can thus be written as K = L ( L + I ) −1 following the truth ∑ X det ( LX ) = det ( L + I ) , which is the marginal normalization constant given a specific L. While there is always conversion from L to K , the inverse may not exist ( Kulesza et al. , 2012 ) . In practice , one may construct L-ensemble first , then normalize it into a marginal kernel . This fact may give rise to the difficulty of deep networks . Since a conversion from K to L might not exist , the network needs carefully adjusting the gradients under specific constraints to ensure the updated L to be valid . As L and K share the same eigenvectors vi , a pair of L and K holds the relation : K = ∑ i λiviv > i ⇐⇒ L = ∑ i λi 1− λi viv > i ( 2 ) where λi is the ith eigenvalue . It is seen that such conversion is not straightforward to be directly integrated with deep learning framework . Therefore , we optimize ensemble L directly in this paper . 2.2 GAUSSIAN KERNEL . We briefly introduce Gaussian kernel in this section , which works on Hilbert space with infinite dimension . Mercer ’ s theorem Friedman et al . ( 2001 ) ensures the PSD properties when constructing new kernels with existing ones under a specific procedure . Such procedure is also employed in multiple kernel learning paradigms ( Affandi et al. , 2014 ; Kulesza & Taskar , 2011b ; Chao et al. , 2015 ) , which is out of the scope of this paper . A Gaussian kernel is defined as κ ( xi , xj ) = exp ( −|xi − xj |2/σ2 ) , where σ is a controlling parameter . Thus an L-ensemble matrix becomes Lij = κ ( xi , xj ) . According to the definition , Lii = 1 and for any element in the matrix we have Lij ∈ ( 0 , 1 ] . With Gaussian kernel , we have a nice property 0 ≤ det ( L ) ≤ 1 . This can be easily verified by applying geometric inequality to the eigenvalues of L. Although not tight , this property shows that the determinant value with Gaussian kernel is bounded . This fact inspires one version of our algorithm detailed in the next section . Throughout this paper , our discussion is based on the Gaussian kernel unless specified . 3 METHOD . Given vectorized inputs Ii ∈ Rh where i = 1 , ... , n , our goal is to learn a map f such that the features xi = f ( Ii ) can spread out within a bounded feature space xi ∈ S. Hereafter we refer space to an Euclidean bounded space ( e.g. , [ −1 , 1 ] d ) without loss of generality . Given any loss function J , the chain rule of gradient involving DPP is written as : ∆J = ∂J ∂ det ( L ) ∂ det ( L ) ∂L ∂L ∂X ( 3 ) where X refer to the features before DPP layer . While calculating ∂J/∂ det ( L ) and ∂L/∂X is straightforward , the main difficulty lies on the calculation of ∂ det ( L ) /∂L . We will discuss the calculation of this term under two case : 1 ) When the inversion L−1 can be stably obained , we will derive the gradient of DPP det ( L ) on Sec 3.1 ; When L is not invertible or L−1 is difficult to calculate , we give the procedure to handle the case by generating valid sub-gradient in Sec 3.2 . Since our objective is to diverse features , det ( L ) will serve as a ( partial ) objective term to be directly maximized . Algorithm 1 DPPSG Input : K , tolerance ∆ ; Output : K̄ UΛU > ← K ( σ1 , ... , σn ) ← diag ( K ) for i in { 1 , ... , n } do if σi < ∆ then σi ← ∆ end if end for Λ̂← diag ( σ1 , ... , σn ) K̂← UΛ̂U > K̄← K̂−K Algorithm 2 DPPSG * Input : K , tolerance θ ; Output : K̄ UΛU > ← K ( σ1 , ... , σn ) ← diag ( K ) b = ( 1− θ ) ( σ1 , ... , σn ) − θ ( 1 , ... , 1 ) Λ̂← diag ( b ) K̂← UΛ̂U > K̄← K̂−K

Summary: The authors present an approach to optimize determinantal point processes directly (by gradient descent, instead of . via approximations), so that diversity could be modeled in objective functions for deep learning systems. The approach taken is to express the DPP term as an L-ensemble in the spectral domain over the gram matrix. They also generate sub-gradients for cases where the gradient does not exist (when the gram matrix is not invertible).