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**Determinantal point processes in the flat limit.**
*(English)*
Zbl 1535.60081

Determinantal point processes are by now perhaps the most famous example of repulsive point processes. Repulsive point processes may also be constructed with some application in mind: in machine learning, it may be used to improve or accelerate learning [A. Kulesza and B. Taskar, Found. Trends Mach. Learn. 5, No. 2–3, 123–286 (2012; Zbl 1278.68240)]. In machine learning applications of repulsive point processes, a subset \(X\) of size \(m\) needs to be extracted from a ground set \(\Omega\) of size \(n\). \(\Omega\) may represent for instance a training set, too large for practical computation, and \(X\) a subset that is in some sense representative of \(\Omega\) for the purposes of training a learning algorithm. If \(X\) includes too many elements that are similar, it fails to be representative of the whole of \(\Omega\). A solution to this problem is to induce repulsivity between the elements sampled, or in other words, to sample the elements not independently, but with negative correlation [N. Tremblay et al., J. Mach. Learn. Res. 20, Paper No. 168, 70 p. (2019; Zbl 1446.62351)]. Determinantal point processes are by now perhaps the most famous example of negatively correlated point processes. The notion of diversity in a determinantal point process is defined relative to a notion of similarity represented by a positive-definite kernel. For instance, if the items are vectors in \(\mathbb{R}^d\), similarity may be defined via the squared-exponential (Gaussian) kernel: \(k_{\varepsilon}(x; y) = \exp \left(-(\varepsilon \|x - y\|)^2\right)\), where \(x\) and \(y\) are two items, and similarity is a decreasing function of distance. The class of determinantal point processes can be separated into two subclasses: \(L\)-ensembles and the rest. The organization of the paper is as follows. The paper begins with some definitions and background in Section 1. Section 2 introduces extended \(L\)-ensembles and partial-projection determinantal point processes and gives some major properties. For clarity, flat limit results are given in increasing order of complexity. Sections 3 and 4 study fixed-size \(L\)-ensembles in the flat limit. The authors begin with univariate results (where the points are a subset of the real line), before giving the results for the multivariate case, which require some background on multivariate polynomials. Section 5 gives the results in complete generality, meaning that they cover the multivariate case in both fixed-size and varying-size determinantal point processes. Note that all results given in prior sections are corollaries of the two main theorems of Section 5. Section 6 details some practical consequences of the results, in terms of eliminating hyperparameters.

Reviewer: Viktor Ohanyan (Erevan)

### MSC:

60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |

65D05 | Numerical interpolation |

60B20 | Random matrices (probabilistic aspects) |

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\textit{S. Barthelmé} et al., Bernoulli 29, No. 2, 957--983 (2023; Zbl 1535.60081)

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