Nonparametric estimation of continuous DPPs with kernel methods
Authors: Michaël Fanuel
Rémi Bardenet
Venue: NeurIPS 2021 Poster (2021)
Keywords: determinantal point processes
kernel methods

Abstract: Determinantal Point Process (DPPs) are statistical models for repulsive point patterns. Both sampling and inference are tractable for DPPs, a rare feature among models with negative dependence that explains their popularity in machine learning and spatial statistics. Parametric and nonparametric inference methods have been proposed in the finite case, i.e. when the point patterns live in a finite ground set. In the continuous case, only parametric methods have been investigated, while nonparametric maximum likelihood for DPPs -- an optimization problem over trace-class operators -- has remained an open question. In this paper, we show that a restricted version of this maximum likelihood (MLE) problem falls within the scope of a recent representer theorem for nonnegative functions in an RKHS. This leads to a finite-dimensional problem, with strong statistical ties to the original MLE. Moreover, we propose, analyze, and demonstrate a fixed point algorithm to solve this finite-dimensional problem. Finally, we also provide a controlled estimate of the correlation kernel of the DPP, thus providing more interpretability.

Nonparametric estimation of continuous DPPs with kernel methods
Michaël Fanuel and Rémi Bardenet Université de Lille, CNRS, Centrale Lille UMR 9189 CRIStAL, F-59000 Lille. France {mi remi paardenetSauniv-llee
Abstract
Determinantal Point Process (DPPs) statistical models for repulsive point patterns. Both sampling and inference are tractable for DPPs, a rare feature among models with negative dependence that explains their popularity in machine learning and spatial statistics. Parametric and nonparametric inference methods have been proposed in the finite case, i.c. when the point patterns live in finite ground set. In the continuous case, only parametric methods have been investigated, while nonparametric maximum likelihood for DPPs an optimization problem over trace-class operators- has remained an open question. In this paper, we show that restricted version of this maximum likelihood (MLE) problem falls within the scope of recent representer theorem for nonnegative functions in an RKHS. This leads to a finiteddimensional problem, with strong statistical ties to the original MIE. Moreover, we propose, analyze, and demonstrate fixed point algorithm solve this finite-dimensional problem. Finally, we also provide a controlled estimate of the correlation kernel of the DPP, thus providing more interpretability.
Introduction
Determinantal point processes (DPPs) are tractable family of models for repulsive point patterns, where interaction between points is parametrized by positive semi-definite kernel. They were introduced by Macchi [1975] in the context of fermionic optics, and have gained a lot interest since the 2000s, in particular in probability (Hough al., 2006], spatial statisties [Lavancier et al., 2014], and machine learning [Kulesza and Taskar, 2012]. machine learning at large, DPPs have been used essentially for two purposes: 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 al., 2020b], 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 large corpus of documents [Kulesza and Taskar, 2012]. Since Affandi et al. [2014], a lot 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 mathematically and algorithmically very similar to finite DPPs, have been less popular in ML than in spatial statisties. It is thus natural that work 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 handful of interpretable parameters are learned. Relatedly, spatial statisticians typically learn the correlation kernel of a DPP, which is more
infinite ground sets, say Rdd while
35th Conference on Neural Information Processing Systems (NcurIPS 2021).

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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 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 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 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 available'.
1.1 Notation and background
Sets. It is customary to define DPPs on a compact Polish space X' endowed with a Radon measure H. so that we can define the space of square integrable functions (2')) for this measure [Hough et al., 2009]. Outside of generalities in Section 2, we consider a compact X' C Rd and μe the uniform probability measure on X. Let (H, (s) be an RKHS of functions on X' with bounded continuous kernel KH (x, y), and let x² = 2u (H(II,) x). Denote by x(x) = KH (z, -) E H the canonical feature map. For Hilbert space F, denote by (F) the space of symmetric and positive semi-defining trace-class operators on F. By slight abuse of notation, we denote by S_ (RR) the space of 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 (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(I, y) and (x y) are distinct functions. Other operators are written in standard fonts (e.g. A, S), while we write matrices and dinieeedmeiiional vectors in bold (e.g. K, C, v). The identity operator is written commonly as I, whereas the n x identity matrix is denoted by I, When is a subset of (1...., n and K is an n x n matrix, the matrix Kee 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 (S))(z) = g(x). Its adjoint S : (x(xx) - H is the reconstruction operator hhh Jx (z))(()du((z)) The classical integral operator given by T _fx (. (r)(()))((()))(()))) seen as an endomorphism of (2x()), thus takes the simple expression TAN = SS*. Similarly, the so-called covariance operator C : H + H. defined by C= olz) @ (x))d(()), writes C' = S. In the tensor product notation defining C. ((x) is an element of the dual of H and ((xr) @ ((x)) is the endomorphism of H defined by (()))(x); see e.g. Sterge et al. [2020]. Finally, for convenience, given a finite set xn C X, we also define discrete restriction and reconstruction operators, respectively, as S, H + R" such and S = (I/V7) Σ viO(I;) for any v E R". In particular, we have S.S, = (1/n) //n)KKnn where K [K (Ii, Sn is kernel matrix, which is defined for given ordering of the set {zi..... En }. 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 S,, and Sp-
that S,g- = (1/vT) / ())(())),
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2 Determinantal point processes and inference
Determinantal point processes and L-ensembles. Consider simple point process y on X, that is, a random discrete subset of X. For D C X, we denote by J2(D) the number of points of this process that fall within D. Letting m be positive integer, we say that X' has m-point correlation function 2m w.r.t to the reference measure μ if, for any mutually disjoint subsets Di...., Dm C X,
E HIDD
-ffr I1 D.
2m Fm)))(m))))) ).d(()))
In most cases, point process characterized by its correlation functions (Pm)m In particular, determinantal point process (DPP) is defined as having correlation functions in the form of determinant of Gram matrix, i.e. 2m (21.....,. that k is the correlation kernel of the DPP. Not all kernels yield DPP: if k(x, )) is the integral kernel of an operator K E S+ 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 the counting measure leads to conditions on the kernel matrix; Kulesza and Taskar [2012].
= det[k( r,)] for all m 2 1. We then say
the Macchi-Soshniker
theorem [Macchi, 1975; Soshnikov,
particular class of DPPs is formed by the so-called L-ensembles, for which the correlation kernel writes
K A(A + TTy-
with the likelihood operator A E S4 (L)) taken to be of the form Af(z) Sx a(z, yf (y)d(y) The kernel a of A is sometimes called the likelihood kernel of the L-ensemble, to distinguish from its correlation kernel k. The interest of L-ensembles is that their Janossy densities be computed in closed form, Informally, the m-Janossy density describes the probability that the point process has cardinality m, and that the points located around a given set of distinet points T1, ..... Fim X. For the rest the paper, we assume that X' C Rd is compact, and that j 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)- det(a(z;, 2,)
where the normalization constant is a Fredholm deteminant that implicitly depends on X. Assume now that we are given i.i.d. samples of DPP, denoted by the sets C,....,C C, in the window X. The associated maximum log-likelinn lillliieeoodd estimation problem reads
max AES-(12(X)) log det [a(x;, 22), - log det(I A).
Solving (3) is a nontrivial problem. First, it is difficult to calculate the Fredholm determinant. Second, not straightforward to optimize over the space of operators S., (X)) 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 (ar, y) of the operator 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 ct al. [2015]; Biscio and Lavancier [2017]: Poinas and Lavancier [2021] for general parametric estimation through (3) so-called contrrast inference. Still for the parametric case, two-step estimation was recently proposed for non-stationary processes by Lavancier et al. [2021]. J. In more general context, non-asymptotic risk bounds for estimating 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;
3

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Gartrell et al., 2017], learning with negative sampling [Mariet et al., 2019], learning with moments and cycles [Urschel 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.
3 A sampling approximation to constrained MLE
Using the machinery of kernel methods, we develop a controlled approximation of the MLE problem (3). Let us outline the main landmarks of our approach. First, we restrict the domain of the MLE problem (3) to smooth operators. On the one hand, this restriction allows us to develop a sampling approximation of the Fredholm determinant, On the other hand, the new optimization problem now admits a finite rank solution that can be obtained by solving finite-dimensional problem. This procedure is described in Algorithm and yields estimator for the likelihood kernel. Finally, use another sampling approximation and solve lincar system to estimate the correlation kernel of the fitted DPP; see Algorithm 2.
Restricting to smooth operators. order to later apply the representer theorem Marteau- Ferey et al. [2020], restrict the original maximum likelihood problem (3) to "smooth" operators A = SAS*, with (HH and S the restriction operator introduced Section 1.1. Note that the kernel of A now writes
a(z, y) = Ao(y))
(4)
With this restriction on its domain, the optimization problem (3) now reads
log det [(b(z;), A + log (SA")) Σ
min S(A) (H)+
(5)
8=1
Approximating the Fredholm determinant. We use a sampling approach to approximate the normalization constant in (5). We sample set of points I {at < i n) i.i.d. from the ambient probability measure μ. For definiteness, we place ourselves on an event happening with probability one where all the points in I and C₁ for 1 < l
distinct. We define the sample version of f(A)
as
(A) Σ log det ( Ab(2,) + +lg det + S,,A)),,
where the Fredholm determinant of SAS" has been replaced by the determinant of a finite matrix involving AS, = [(0(z)), A
Theorem (Approximation of the Fredholm determinant). Let 5 10,1/2). With probability at least 1 25,
llog det (I, SAAS,) log det + SAS") s log det(I + c),
with
4k² log 66
2x²
2n² 2x log
On
+
3n
n
where E Amax (anax(Tki) (Toy)) and x = suprex kx X) oc.
The proof of Theorem is given in Supplementary Material in Section S3.2. Several remarks in order. First, the high probability² the statement of Theorem that of the event {1|S* 5 S, < )x). Importantly, all the results given in what follows for the approximation of the solution of (5) only depend on this event, so that we do not need any union bound. Second, we emphasize that Tayy defined in Section 1.1. should not be confused with the correlation kernel (1). Third, to interpret the bound in Theorem 1, it is useful to recall that log det + A, S c. Tr(A),
We write S b if there exists constant
0 such that S cb.

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since A E (+H).. Thus, by penalizing Tr(A), one also improves the upper bound on the Fredholm
determinant approximation enror. This remark motivates the following infinite dimensional problem
min (A(A) TTr(A),
(6)
AES_ (H)
for some À > O. The penalty (r(A) is also intuitively needed so that the optimization problem selects a smooth solution, i.c., such a trace regularizer promotes fast decay eigenvalues of A. Note that this problem depends both on the data CI....,Co Cn C X' and the subset I used for approximating the Fredholm determinant.
Finite-dimensional representatives. In an RHKS, there is natural mapping between finite rank operators and matrices. For the sake of completeness, let K = [ 2y]lusi.jgm be a kernel matrix and let K = R R be Cholesky factorization. Throughout the paper, kernel matrices are always assumed be invertible. This is not strong assumption: if lyz is the Laplace, Gaussian or Sobolev kernel, this is true almost surely if 2; for 1 S S m are sampled e.g. w.r.t. the Lehesgue measure; Bochner's classical theorem [Wendland, 2004, Theorem 6.6 and Corollary 6.9]. In this case, can define a partial isometry V : H - "R as V = vm((-1) 1 Sm It satisfies VV* = I, and V*V
the orthogonal projector onto the span of o(2) for all 1 S S m. This is helpful to define
c = Vo(2)) R,"
the finite-dimensional representative of d(z;) E H for all 1 S m. This construction yields useful mapping between operator in S+ and a finite matrix, which is instrumental for obtaining our results.
Lemma 2 (Finite dimensional representatives, extension of Lemma 3 in Rudi al. [2020]). Let A E S+ (H). Then, the matrix B = VAV* is such that o B = A6(z)) for all 1 s i S m, and log det + B) < log (Il + A), as well as Tr(B) < (r(A).
The proof of Lemma 2 is given in Section $3.1. Notice that the partial isometry V also helps to map a matrix in (IR")) to an operator in S+ (H), as B + V "V, in such way that we have the matrix element matching (0(2,), V'BV (2,))2)) = $ B for all S i.j I m.
Finite rank solution thanks to representer theorem. The sampling approximation of the Fred- holm determinant also yields finite rank solution for (6). For simplicity, we define C & U C. and recall Z (z/,... ..., x,x) Then, write the set of points - CUI as {21....; Zm,) with m = [C] n, and denote the corresponding restriction operator S,, : H. + R". Consider the kernel matrix K (2, 2, gi.jsmm In particular, since we used a trace regularizer, the representer theorem of Marteau- Ferey et al. [2020, Theorem 1] holds: the optimal operator is of the form
m
=
Σ C2j((2)) & 0(2y) with C (R
(8)
(==1
In this paper, we call C the representer matrix of the operator A. we do the change variables B = RCR we have the following identities: A = mmS, CS* = V*BV and Tr(A) Tr(KC) Tr(B), thanks to Lemma S1 in Supplementary Material. Therefore, the problem (6) boils down to the finite non-convex problem:
min (V + A TT(B), B=0
where fv = -i log det [ B syecc + log det [5.g + $ B / j,jer We assume that there is global minimizer of (9) that we denote by B,, The final estimator of the integral kernel of the likelihood depends on C. = R R-'B,R--TT and reads á(x, y) = Emj=1 C. C.ykk((z,, (2,, y). The numerical strategy is summarized in Algorithm 1.
5

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Algorithm 1 Estimation of the integral kernel a v) of the DPP likelihood kernel A.
procedure ESTIMATHA(>, ....... C.)
Sample I = {-/,
i.i.d. from 1
Sample n points for Fredhom det. approx.
Define Z U= UI
Collect all samples Cholesky of kernel matrix Regularized Picard iteration
Compute K = R R with K = (2 zi)liszz
Solve (9) with iteration (14) to obtain B,
Compute C, R B.R
Representer matrix of á v) > Likelihood kernel
return á(z 8) = m C.ykk((,, I))
end procedure
Estimation of the correlation kernel. The exact computation of the correlation kernel of the L-ensemble DPP
requires the exact diagonalization K(7) A(A + +nd)
(10)
SAS" For more flexibility, we introduced scale parameter > which often takes the value 1. It is instructive to approximate K in order to easily express the correlation functions of the estimated point process. We propose here an approximation scheme based once again on sampling. Recall the form of the solution A = m.5, CS,, of (6), and consider the factorization C A A with A = FR-11 where F F = B. the Cholesky factorization of B, Let {ay( ..... " X' be sampled i.i.d. from the probability measure and denote by Sp H + R2 the corresponding restriction operator. The following integral operator
K k= mSSS A SpS,S, + + S
(11)
gives an approximation of The numerical approach for solving (11) relies on the computation of K mm Sp = [kx(=;, =²"" with 1 < i S and 1 S S rectangular kernel matrix, associated to fixed ordering of Z {21 { ... 2m) and {x" ................................ zx). Our strategy is described in Algorithm 2.
Algorithm 2 Estimation of the integral kernel k(r,, y) of the DPP correlation kernel K = A(A + II-I.
procedure ESTIMATEK(Z, C.)
Compute C, =
> Factorization of representer matrix >> Sample p points
Sample (x' 2x1 X i..d. from μ
Compute Kmp = [kx(zz, r²']] E Rxx
Cross kernel matrix
Compute Ω = (AKK mp
K mp
A +
Representer matrix of k(x:, y)
return k(x, v) Σxj=1 Dighi Ju )
Correlation kernel
end procedure
4 Implementation
We propose algorithm for solving the discrete problem (9) associated to (6). To simplify the
discussion and relate it to Mariet and Sra [2015], we define the objective g(X) = (V=.(VVB))))) + TT)) with the change of variables B(X) R XR-1 Then we can rephrase (9)
1 X + XTIK(X--), n
min (X) x=
log det (etcc))))) + log det
[iz] +
(12)
where we recall that n = [ Define for convenience U, as the matrix obtained by selecting the columns of the identity matrix which are indexed by Ct, so that, have in particular Xc,c, U. XU,. Similarly, define sampling matrix Uz associated to the subset I. Recall the Cholesky decomposition K = R R. To minimize (12), we start at some X, > 0 and use the following iteration
R (IIm + 4AR-1T ((X,)R 1/2 R Im R. 21
X X
(13)

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where p(X) X XAX and A(X) - Ei U.Xeic, Ux(11r + n = We dub this sequence regularized Picard iteration, as it generalization of the Picard iteration which was introduced by Mariet and Sra [2015] in the context of learning discrete Leensembee DPPs. In Mariet and Sra [2015], the Picard iteration, defined as X s+11 = p(Xx), is shown to be appropriate for minimizing different objective given by: Ei log det (x,c,) + log det X). The following theorem indicates that the iteration (13) is a good candidate for minimizing g g(X).
Theorem 3. Let Xx for integer k be the sequence generated by (13) and initialized with Xo 0. Then, the sequence g(X; ) is monotonically decreasing.
For proof, we refer to Section S3.4. In practice, use the iteration (13) with the inverse change of variables X(B) = R. BR. and solve
2A (Inm (Bx() -II with ((B) = B BRA
1
B
(14)
where A (X) is given hereabove. For the stopping criterion, we monitor the objective values of (9) and stop if the relative variation of two consecutive objectives is less than a predefined precision threshold tol. Contrary to (12), the objective (9) does not include the inverse of K, which might be
ill-conditioned. The interplay between À and is best understood by considering (9) with the change of variables B' = B/n, yielding the equivalent problem
min 0
Σ 2=1
C.C, log det ( BB'e), ) + +og det (I+ + B'
An Tr(B'),
where R. a matrix whose i-th column is 4; for < S m as defined in (7). Notice that,
up 1/n factor, $ B' is the in-sample Gram matrix of å (x: y) evaluated the data 2; Algorithm 1. Thus, in the limit + O, the above expression corresponds to the MLE estimation of finite DPP if U,C₁ C I. This is the intuitive connection with finite DPP: the continuous DPP is well approximated by finite DPP if the ground set Z is dense enough sampling within X.
Theoretical guarantees
We now describe the guarantees coming with the approximations presented in the previous section.
Statistical guarantees for approximating the maximum likelihood problem Next, we give a statistical guarantee for the approximation of the ledeeikl by its sample version.
Theorem 4 (Discrete optimal objective approximates full MLE objective). Let B, be the solution of (9). Let A, be the solution of (5). Let ò 102) If 2 (6)( then with probability at least 1 - 25, it holds that
3 (S(A)) JV B,V) < TH.)..
with 0 Cn I 1//Tt given in Theorem 1.
The above result, proved in Section S3.3, shows that, with high probability, the optimal objective value of the discrete problem is not far from the optimal log-likelihood provided n is large enough. As simple consequence, the discrete solution also yields finite rank operator VB. V whose likelihood is not far from the optimal likelihood ff (AA.. ). as it can be shown by using triangle inequality.
Corollary 5 (Approximation of the full MLE optimizer by a finite rank operator). Under the assumptions Theorem x 2c, (6), with probability least 25, it holds
(S(A)) - (V"BBV) S 3X Tr(A))
with C. 2 1/ V given in Theorem 1.
The proof of Corollary is also provided in Section $3.3.
Approximation of the correlation kernel An important quantity for the control of the amount of points necessary to approximate well the correlation kernel the so-called effective dimension (7) = Tr (AAA + -))) which is the expected sample size under the DPP with correlation kernel K A(A -
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1.00
1,50
0.75
0.25
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0.50
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0
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Figure Intensity estimation with a 0.1 and A 0.1 from DPP sample with p 50 (top row) and p 100 (bottom row). On the LHS, a DPP sample (hexagons) and = 1000 uniform samples (crosses). the color the diagonal of + B (in-sample likelihood kernel). On the RHS, out-of-sample estimated intensity k(x,,r) a)) of the learned process on a 100 100 grid.
Theorem (Correlation kernel approximation). Let 5 E (0, 1) be a failure probability, let e E (0. be an acurracy parameter and let y be a scale factor. Let K(?) be the correlation kernel (10) defined with = SAS". Consider K(r) defined in (11) with i.i.d sampling of p points in X wrt μ.. If we take p
8x² ||Allaz log 4ddar(1)) then, with probability 6, the following multiplicative error [|[K[|u>
7
bound holds K(>) 3 K(1) I k-))))
The proof of Theorem 6. given in Section S3.5, mainly relies on a matrix Bernstein inequality. Let us make few comments. First, we can simply take of the correlation kernel (1). Second, the presence of d (7) in the logarithm is welcome since it is the expected subset size of the L-ensemble. Third, the quantity ||A| op directly influences the typical sample size to get accurate approximation. A worst case bound is || S Amax C)) (K) mS,, CS*, in the light of (8). Thus, the
recover the common definition in Theorem
with K = [ (2, 2j) im and where we used that p may be large in practice. Although probably more costly, approach inspired
lower bound from approximate ridge leverage score sampling [Rudi et al., 2018] 8] is likely to allow lower p s. We leave this to future work.
6 Empirical evaluation
We consider an L-ensemble with correlation kernel k(æ, y) = pexp(--1a2 vll3/a2) defined on Rd with a 0.05. Following Lavancier et al. [2014], this valid kernel if p < ( (/aa)-d Note that the intensity, defined recovers that property. We draw samples from this continuous DPP in the window = [0, 1 Two such samples are shown as hexagons in the first column of Figure 1, with respective intensity p 50 and p = 100. For the estimation, we use a Gaussian kernel (a, y) = exp (-1| / (2o
+ k(æ, .z), is constant equal to P; shall check that the fitted kernel
with 0. The computation of the correlation kernel always uses p 1000 uniform samples.
Iteration (14) is rum until the precision threshold tol = 10-5 is achieved. For stability, we add 10
We used the code of Poinas and Lavancier [2021], available at https: //g/th/ttt com/ hwtp//PPonnnaaaaaaee /MLEDPPP It relies the R package spatstar [Baddeley et al., 2015], available under the GPL-2 GPL-3 licence.

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Figure 2: Analysis of the solution corresponding the example of Figure with p 100. Left:
eigenvalues of + Bd. Middle: Gram matrix of k(x, y) on a regular 30 30 grid within [0, 1]2. Right: Gram matrix of k(2, y) on the same grid.
1.00
-0
1.00
45
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45
30
25
30
-00
0.25
25
0.0 0.25 0.50 0.75
1.00
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1.00
Figure 3: Intensity estimation with 0.1 and À 0.01 from DPP sample with p 100. On the LHS, a DPP sample (hexagons) and = 1000 uniform samples (crosses), the color is the diagonal of T B$ (in-sample likelihood kernel). On the RHS, out-of-sample estimated intensity k(z, 2) of the learned process on a 100 100 grid.
to the diagonal of the Gram matrix K. The remaining parameter values are given in captions. We empirically observe that the regularized Picard iteration returns a matrix B, such that + BB, 4 low rank; see Figure 2 (left). A lesson from Figure is that the sample size of the DPP has to be large enough to retrieve constant intensity k(x,x). In particular, the top row of this figure illustrates case where a is too small. Also, due to the large regularization À = 0.1 and the use of only one DPP sample, the scale of p clearly underestimated in this example. On the contrary, in Figure 3, for smaller regularization parameter À = 0.01, the intensity scale estimate larger. We also observe that large regularization parameter tends to smooth out the local variations of the intensity, which surprising. comparison between Gram matrix of k(x, y) and k(r, y) is given in Figure corresponding to the example of Figure 1. The short-range diagonal structure is recovered, while some long-range structures are smoothed out. More illustrative simulations are given in Section $4, with study of the influence of the hyperparameters, including the use of particular, the estimation of the intensity is improved if several DPP samples are used with smaller value of X.
DPP samples. In
7 Discussion
We leveraged recent progress on kernel methods to propose a nonparametric approach to learning continuous DPPs. We see three major limitations of our procedure. First, our final objective function is nonconvex, and our algorithm is only guaranteed to increase its objective function. Experimental evidence suggests that our approach recovers the synthetic kernel, but more work needed study the maximizers of the likelihood, in the spirit of Brunel et al. [2017a] for finite DPPs, and the properties of our fixed point algorithm. Second, the estimated integral kernel does not have any explicit structure, other than being implicitly forced to be low-rank because of the trace penalty. Adding structural assumptions might be desirable, either for modelling or learning purposes. For modelling, it is not uncommon to assume that the underlying continuous DPP is stationary, for example, which implies that the correlation kernel k(r, y/ depends only on y. For learning, structural assumptions the kernel may reduce the computational cost, or reduce the number of

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maximizers of the likelihood. The third limitation of our pipeline is that, like most nonparametric methods, still requires to tune a handful of hyperparameters, and, our experience, the final performance varies significantly with the lengthscale of the RKHS kernel or the coefficient of the trace penalty, automatic tuning procedure with guarantees would make the pipeline turn-key.
Maybe unexpectedly, future work could also deal with transferring our pipeline to the finite DPP setting. Indeed, we saw Section 4 that in some asymptotic regime, our regularized MLE objective is close to a regularized version of the MLE objective for finite DPP. Investigating this maximum a posteriori inference problem may shed some new light on nonparametric inference for finite DPPs. Intuitively, regularization should improve learning and prediction when data is scarce.
Acknowledgements
We thank Arnaud Poinas and Guillaume Gautier for useful discussions on the manuseript. We acknowledge support from ERC grant BLACKJACK SCCC-209--TTG-8188000 and ANR Al chair BACCARAT
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Checklist
1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? [Yes] We checked that the abstract and introduction faithfully
summarize the main contributions of this paper.
(b) Did you describe the limitations of your work? [Yes] We describe the limitations in Sec. 7.
(c) Did you discuss any potential negative societal impacts of your work? [N/A] To our knowledge, there no negative impact of our work in society.
(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
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2. If you are including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results? [Yes]
(b) Did you include complete proofs of all theoretical results? [Yes] All the proofs are in Supplementary Material.
3. If you experiments..
(a) Did you include the code, data, and instructions needed to reproduce the main exper- imental results (either in the supplemental material or as a URL)? [Yes] The code is available at https: / /tg/tttttbbb//// com/ mrf wamue//Larninnnooiinoooo
j1.
(b) Did you specify all the training details (e.g., data splits. hyperparameters, how they were chosen)? [Yes] This is described both in the paper and in Supplementary Material.
(c) Did you report error bars (e.g.. with respect to the random seed after running experi- ments multiple times)? [N/A] This does not apply to our simulation setting.
(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, cloud provider)? [Yes] This described in Supplementary Material.
4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
(a) your work uses existing assets, did you cite the creators? [Yes] Some code for sampling DPPs is used in Section 6 and we give the corresponding paper in a footnote. The majority of the code was written for this paper.
(b) Did you mention the license of the assets? [Yes] We used code based on the spatstat R package which is available under license GPL-2 / GPL-3.
(c) Did you include any
assets either in the supplemental material a URL? [Yes] The code is available at https: thub. com/cro///e//oo coommrranuell BeariingContimmuusPPP....
(d) Did you discuss whether and how consent was obtained from people whose data you're using/curating? [N/A] Only simulated data are used.
(e) Did you discuss whether the data you using/curating contains personally identifi- able information or offensive content? [N/A] Only simulated data are used and our simulations do not contain offensive content.
5. If you used crowdsourcing conducted research with human subjects...
(a) Did you include the full text of instructions given to participants and screenshots, applicable? [N/A]
(b) Did you describe any potential participant risks, with links Institutional Review Board (IRB) approvals, if applicable? [N/A]
(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]
13


Reviews:
[{"review_id": "Zy9P2weXaKm", "reviewer": "Reviewer_jwGL", "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. \n\nThe 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.", "questions": "", "limitations": "", "rating": 7, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Originality: The proposed method to learn a non-parameteric, continuous DPP is new; it is, to the extent of my knowledge, the first result on learning non-parametric, continuous DPPs.\n\nQuality: This paper is of high quality; the proofs are detailed, the theory meticulous (to the extent that I can judge it, being unfamiliar with results in Marteau-Ferey et al., 2020, and Rudi et al., 2020.). Furthermore, the theoretical analysis is regularly tied back to core concepts in DPPs (e.g., the effective sample size), and key arguments are regularly highlighted to support the provided analysis. \n\nClarity: With the caveat that I am not familiar with continuous DPPs (but familiar with their discrete counterparts), I found this paper very well written and accessible. The authors carefully frame their contributions in terms of previously known results within continuous and discrete DPP landscapes, draw the reader's attention to crucial arguments in their proofs, and describe in detail the assumptions and potential limitations of their work. Overall, this is a very well written paper.\n\nSignificance: The results are significant to the community, in particular given growing interest in the applications of continuous DPPs. This is the first result on learning non-parametric continuous DPPs, and also has interesting implications for potential extensions to learning algorithms for discrete DPPs. One potential limitation of this work is that the empirical verification focuses only on exponentiated quadratic kernels.\n\n\nQuestions & Comments:\n- You allude in the final section that the regularized objective might transfer to the finite DPP setting. Do you mean that by introducing a trace-based regularizer to the discrete Picard iteration, you may be able to recover a guarantee of the style of Theorem 4?\n- Have you tried learning L-ensembles with correlation kernels that are not of the same form as the RKHS kernel? Would you expect this to influence the learning process?", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "The authors clearly describe the limitations of their work; I do not see a societal impact that requires discussing.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "5 hours"}, "extra_text": {}}, {"review_id": "8MBLUfUGOeV", "reviewer": "Reviewer_dbvc", "summary": "This paper studies nonparametric MLE inference of continuous determinantal point processes (DPPs). The authors analyze various representations between discrete DPPs and continuous DPPs. In particular, they observe that the difference of normalization of DPP is bounded by the trace of a matrix. Then, they consider a new optimization problem penalizing the trace and propose a method of estimating likelihood/correlation DPP kernels. Moreover, they study an iterative method (a.k.a Picard iteration) for solving the MLE problem with the regularization term. Statistical guarantees of the MLE objective and correlation kernel are also studied. ", "questions": "", "limitations": "", "rating": 7, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "This work tries to bridge the gap between continuous DPPs and discrete DPPs and provides some rigorous results. However, some derivation is unnatural and various analysis methods are already known. Moreover, as DPPs are used in many ML applications and mostly they are given by discrete settings, one should justify importance of continuous DPPs rather than discrete DPPs.\n\nThe equation (6) comes from penalizing the difference of normalization term as shown in Theorem 1. However, this only considers the error of normalization term and one needs to consider the first term (i.e., numerator in probability of DPP) in equation (5). Does the Theorem 1 similarly validate without identity operator?\n\nIn Theorem 6, the authors provide a statistical guarantee of correlation kernel. Is it possible to provide similar result in terms of the likelihood kernel in Algorithm 1?\n\nOverall, it needs to be improved the justification of the contributions of this paper. Also, it would be better to provide more concrete results (e.g., error estimation) rather than visualization.\n\n=========================================================================\n\nI have read authors response and other reviews and noticed there was a misunderstanding regarding continuous DPPs. Authors clarified this in their response. I update my score.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "- In line 192, it seems that m <= |C| + n since samples in \\mathcal{I} can contain points in C_l.\n- In Theorem 6, what is the \\kappa?\n", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "5"}, "extra_text": {}}, {"review_id": "4s2_YhXeRa6", "reviewer": "Reviewer_UmsC", "summary": "This paper presents a new approach for maximum likelihood estimation (MLE) for continuous determinantal point processes (DPPs), which leverages recent work on kernel methods.  This approach involves nonparametric learning for a restricted class of the MLE problem, which is shown to fall within the scope of recent work on a representer theorem for nonnegative functions in an RKHS.  An optimization problem over matrices is characterized, and then a fixed point algorithm for solving the finite-dimensional optimization problem is proposed.  Statistical guarantees for the MLE approximation are presented.  Finally, the authors provide an empirical validation of their proposed method using experiments run on synthetic data sampled from a DPP.", "questions": "", "limitations": "", "rating": 7, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "The MLE learning approach for continuous DPPs proposed in this paper is novel, and appears to be a significant contribution.  As noted by the authors, substantial prior work exists on discrete or finite DPPs, but much less attention has been paid to continuous DPPs within the machine learning community.  This paper is reasonably well written, and appears to be one of the first contributions in the area of nonparametric learning for continuous DPPs.  The main theoretical contributions in Sections 3 - 5 of the paper appear to be sound, although I have not carefully checked the proofs.  Furthermore, the statistical guarantees in Section 5 establish approximation bounds on the MLE solution and correlation kernel that appear to be reasonably tight under realistic assumptions, with the exception of the lower bound on $p$ in Theorem 6.\n\nA few aspects of this paper could be improved.  First, the authors should discuss the computational complexity of the algorithms for estimating the DPP likelihood kernel (Alg. 1) and correlation kernel (Alg. 2).  What is the computational complexity of the fixed point algorithm in Sec. 4?  A very brief discussion of space and time complexity is given in Appendix S4.3, but this discussion should be significantly expanded upon.  Second, experiments that involve using the proposed approach to learn a continuous DPP from a real dataset should be performed.  Lastly, it would be helpful for the authors to provide a more detailed comparison of their nonparametric learning approach with prior work on parametric learning for continuous DPPs.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "The authors have included a short, but adequate, discussion of the limitations of their work in Section 7 of the paper.  Since this work involves very general algorithmic and theoretical contributions, a discussion of potential negative societal impact does not appear to be applicable.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "5 hours"}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper studies nonparametric maximum likelihood estimation for continuous DPPs. In contrast to discrete DPPs, for continuous ones there are many open questions and new challenges. The main results are based on recent representer theorems for continuous DPPs as nonnegative functions in an RKHS. The paper uses this machinery to develop a new fixed point algorithm which is guaranteed to increase the MLE objective after each step. Furthermore they give strong statistical guarantees – bounds on the error of their estimator in the appropriate metric, depending on the sample complexity. All the reviewers agreed that the results were interesting and original, and that the paper was well-written and accessible. 
Decision: Accept (Poster)