Nonparametric estimation of continuous DPPs with kernel methods

1 Introduction . Determinantal point processes ( DPPs ) are a tractable family of models for repulsive point patterns , where interaction between points is parametrized by a positive semi-definite kernel . They were introduced by Macchi [ 1975 ] in the context of fermionic optics , and have gained a lot of interest since the 2000s , in particular in probability [ Hough et al. , 2006 ] , spatial statistics [ Lavancier et al. , 2014 ] , and machine learning [ Kulesza and Taskar , 2012 ] . In machine learning at large , DPPs have been used essentially for two purposes : as statistical models for diverse subsets of items , like in recommendation systems [ Gartrell et al. , 2019 ] , and as subsampling tools , like in experimental design [ Derezinski et al. , 2020 ] , column subset selection [ Belhadji et al. , 2020b ] , or Monte Carlo integration [ Gautier et al. , 2019 ; Belhadji et al. , 2019 ] . In this paper , we are concerned with DPPs used as statistical models for repulsion , and more specifically with inference for continuous DPPs . DPP models in Machine Learning ( ML ) have so far mostly been finite DPPs : they are distributions over subsets of a ( large ) finite ground set , like subsets of sentences from a large corpus of documents [ Kulesza and Taskar , 2012 ] . Since Affandi et al . [ 2014 ] , a lot of effort has been put into designing efficient inference procedures for finite DPPs . In particular , the fixed point algorithm of Mariet and Sra [ 2015 ] allows for nonparametric inference of a finite DPP kernel , thus learning the features used for modelling diversity from the data . DPP models on infinite ground sets , say Rd , while mathematically and algorithmically very similar to finite DPPs , have been less popular in ML than in spatial statistics . It is thus natural that work on inference for continuous DPPs has happened mostly in the latter community ; see e.g . the seminal paper [ Lavancier et al. , 2014 ] . Inference for continuous DPPs has however focused on the parametric setting , where a handful of interpretable parameters are learned . Relatedly , spatial statisticians typically learn the correlation kernel of a DPP , which is more 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . interpretable , while machine learners focus on the likelihood kernel , with structural assumptions to make learning scale to large ground sets . In this paper , we tackle nonparametric inference for continuous DPPs using recent results on kernel methods . More precisely , maximum likelihood estimation ( MLE ) for continuous DPPs is an optimization problem over trace-class operators . Our first contribution is to show that a suitable modification of this problem is amenable to the representer theorem of Marteau-Ferey , Bach , and Rudi [ 2020 ] . Further drawing inspiration from the follow-up work [ Rudi , Marteau-Ferey , and Bach , 2020 ] , we derive an optimization problem over matrices , and we prove that its solution has a near optimal objective in the original MLE problem . We then propose , analyze , and demonstrate a fixed point algorithm for the resulting finite problem , in the spirit [ Mariet and Sra , 2015 ] of nonparametric inference for finite DPPs . While our optimization pipeline focuses on the so-called likelihood kernel of a DPP , we also provide a controlled sampling approximation to its correlation kernel , thus providing more interpretability of our estimated kernel operator . A by-product contribution of independent interest is an analysis of a sampling approximation for Fredholm determinants . The rest of the paper is organized as follows . Since the paper is notation-heavy , we first summarize our notation and give standard definitions in Section 1.1 . In Section 2 , we introduce DPPs and prior work on inference . In Section 3 , we introduce our constrained maximum likelihood problem , and study its empirical counterpart . We analyze an algorithm to solve the latter in Section 4 . Statistical guarantees are stated in Section 5 , while Section 6 is devoted to numerically validating the whole pipeline . Our code is freely available1 . 1.1 Notation and background . Sets . It is customary to define DPPs on a compact Polish space X endowed with a Radon measure µ , so that we can define the space of square integrable functions L2 ( X ) for this measure [ Hough et al. , 2009 ] . Outside of generalities in Section 2 , we consider a compact X ⊂ Rd and µ the uniform probability measure on X . Let ( H , 〈· , ·〉 ) be an RKHS of functions on X with a bounded continuous kernel kH ( x , y ) , and let κ2 = supx∈X kH ( x , x ) . Denote by φ ( x ) = kH ( x , · ) ∈ H the canonical feature map . For a Hilbert space F , denote by S+ ( F ) the space of symmetric and positive semi-definite trace-class operators on F . By a slight abuse of notation , we denote by S+ ( Rn ) the space of n× n real positive semi-definite matrices . Finally , all sets are denoted by calligraphic letters ( e.g . C , I ) . Operators and matrices . Trace-class endomorphisms of L2 ( X ) , seen as integral operators , are typeset as uppercase sans-serif ( e.g . A , K ) , and the corresponding integral kernels as lowercase sans-serif ( e.g . a , k ) . Notice that k ( x , y ) and kH ( x , y ) are distinct functions . Other operators are written in standard fonts ( e.g . A , S ) , while we write matrices and finite-dimensional vectors in bold ( e.g . K , C , v ) . The identity operator is written commonly as I , whereas the n× n identity matrix is denoted by In . When C is a subset of { 1 , . . . , n } and K is an n× n matrix , the matrix KCC is the square submatrix obtained by selecting the rows and columns of K indexed by C. Restriction and reconstruction operators . Following Rosasco et al . [ 2010 , Section 3 ] , we define the restriction operator S : H → L2 ( X ) as ( Sg ) ( x ) = g ( x ) . Its adjoint S∗ : L2 ( X ) → H is the reconstruction operator S∗h = ∫ X h ( x ) φ ( x ) dµ ( x ) . The classical integral operator given by TkHh = ∫ X kH ( · , x ) h ( x ) dµ ( x ) , seen as an endomorphism of L 2 ( X ) , thus takes the simple expression TkH = SS ∗ . Similarly , the so-called covariance operator C : H → H , defined by C = ∫ X φ ( x ) ⊗ φ ( x ) dµ ( x ) , writes C = S ∗S . In the tensor product notation defining C , φ ( x ) is an element of the dual ofH and φ ( x ) ⊗φ ( x ) is the endomorphism ofH defined by ( ( φ ( x ) ⊗φ ( x ) ) ( g ) = g ( x ) φ ( x ) ; see e.g . Sterge et al . [ 2020 ] . Finally , for convenience , given a finite set { x1 , . . . , xn } ⊂ X , we also define discrete restriction and reconstruction operators , respectively , as Sn : H → Rn such that Sng = ( 1/ √ n ) [ g ( x1 ) , . . . , g ( xn ) ] > , and S∗nv = ( 1/ √ n ) ∑n i=1 viφ ( xi ) for any v ∈ Rn . In particular , we have SnS∗n = ( 1/n ) K where K = [ kH ( xi , xj ) ] 1≤i , j≤n is a kernel matrix , which is defined for a given ordering of the set { x1 , . . . , xn } . To avoid cumbersome expressions , when several discrete sets of different cardinalities , say n and p , are used , we simply write the respective sampling operators as Sn and Sp . 1https : //github.com/mrfanuel/LearningContinuousDPPs.jl 2 Determinantal point processes and inference . Determinantal point processes and L-ensembles . Consider a simple point process Y on X , that is , a random discrete subset of X . For D ⊂ X , we denote by Y ( D ) the number of points of this process that fall within D. Letting m be a positive integer , we say that X has m-point correlation function % m w.r.t . to the reference measure µ if , for any mutually disjoint subsets D1 , . . . , Dm ⊂ X , E [ m∏ i=1 Y ( Di ) ] = ∫ ∏m i=1Di % m ( x1 , . . . , xm ) dµ ( x1 ) . . . dµ ( xm ) . In most cases , a point process is characterized by its correlation functions ( ρm ) m≥1 . In particular , a determinantal point process ( DPP ) is defined as having correlation functions in the form of a determinant of a Gram matrix , i.e . % m ( x1 , . . . , xm ) = det [ k ( xi , xj ) ] for all m ≥ 1 . We then say that k is the correlation kernel of the DPP . Not all kernels yield a DPP : if k ( x , y ) is the integral kernel of an operator K ∈ S+ ( L2 ( X ) ) , the Macchi-Soshnikov theorem [ Macchi , 1975 ; Soshnikov , 2000 ] states that the corresponding DPP exists if and only if the eigenvalues of K are within [ 0 , 1 ] . In particular , for a finite ground set X , taking the reference measure to be the counting measure leads to conditions on the kernel matrix ; see Kulesza and Taskar [ 2012 ] . A particular class of DPPs is formed by the so-called L-ensembles , for which the correlation kernel writes K = A ( A + I ) −1 , ( 1 ) with the likelihood operator A ∈ S+ ( L2 ( X ) ) taken to be of the form Af ( x ) = ∫ X a ( x , y ) f ( y ) dµ ( y ) . The kernel a of A is sometimes called the likelihood kernel of the L-ensemble , to distinguish it from its correlation kernel k. The interest of L-ensembles is that their Janossy densities can be computed in closed form . Informally , the m-Janossy density describes the probability that the point process has cardinality m , and that the points are located around a given set of distinct points x1 , . . . , xm ∈ X . For the rest of the paper , we assume that X ⊂ Rd is compact , and that µ is the uniform probability measure on X ; our results straightforwardly extend to other densities w.r.t . Lebesgue . With these assumptions , the m-Janossy density is proportional to det ( I + A ) −1 · det [ a ( xi , xj ) ] 1≤i , j≤m , ( 2 ) where the normalization constant is a Fredholm deteminant that implicitly depends on X . Assume now that we are given s i.i.d . samples of a DPP , denoted by the sets C1 , . . . , Cs in the window X . The associated maximum log-likelihood estimation problem reads max A∈S+ ( L2 ( X ) ) 1 s s∑ ` =1 log det [ a ( xi , xj ) ] i , j∈C ` − log det ( I + A ) . ( 3 ) Solving ( 3 ) is a nontrivial problem . First , it is difficult to calculate the Fredholm determinant . Second , it is not straightforward to optimize over the space of operators S+ ( L2 ( X ) ) in a nonparametric setting . However , we shall see that the problem becomes tractable if we restrict the domain of ( 3 ) and impose regularity assumptions on the integral kernel a ( x , y ) of the operator A . For more details on DPPs , we refer the interested reader to Hough et al . [ 2006 ] ; Kulesza and Taskar [ 2012 ] ; Lavancier et al . [ 2014 ] . Previous work on learning DPPs . While continuous DPPs have been used in ML as sampling tools [ Belhadji et al. , 2019 , 2020a ] or models [ Bardenet and Titsias , 2015 ; Ghosh and Rigollet , 2020 ] , their systematic parametric estimation has been the work of spatial statisticians ; see Lavancier et al . [ 2015 ] ; Biscio and Lavancier [ 2017 ] ; Poinas and Lavancier [ 2021 ] for general parametric estimation through ( 3 ) or so-called minimum-contrast inference . Still for the parametric case , a two-step estimation was recently proposed for non-stationary processes by Lavancier et al . [ 2021 ] . In a more general context , non-asymptotic risk bounds for estimating a DPP density are given in Baraud [ 2013 ] . Discrete DPPs have been more common in ML , and the study of their estimation has started some years ago [ Affandi et al. , 2014 ] . Unlike continuous DPPs , nonparametric estimation procedures have been investigated for finite DPPs by Mariet and Sra [ 2015 ] , who proposed a fixed point algorithm . Moreover , the statistical properties of maximum likelihood estimation of discrete L-ensembles were studied by Brunel et al . [ 2017b ] . We can also cite low-rank approaches [ Dupuy and Bach , 2018 ; Gartrell et al. , 2017 ] , learning with negative sampling [ Mariet et al. , 2019 ] , learning with moments and cycles [ Urschel et al. , 2017 ] , or learning with Wasserstein training [ Anquetil et al. , 2020 ] . Learning non-symmetric finite DPPs [ Gartrell et al. , 2019 , 2021 ] has also been proposed , motivated by recommender systems . At a high level , our paper is a continuous counterpart to the nonparametric learning of finite DPPs with symmetric kernels in Mariet and Sra [ 2015 ] . Our treatment of the continuous case is made possible by recent advances in kernel methods .

Summary: This paper introduces an algorithm to learn non-parametric, continuous determinantal point processes (DPPs). This algorithm relies on recent representer results for nonnegative functions in a RKHS and on a previous, related algorithm to learn discrete DPP parameterizations.   The authors prove that the proposed algorithm is guaranteed to increase the MLE objective function at each iteration, as well as prove statistical guarantees for the resulting solution: distance to an optimal parameterization, sample complexity.