## 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.

### MSC:

 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 65D05 Numerical interpolation 60B20 Random matrices (probabilistic aspects)

### Keywords:

determinantal point process; kernel method; flat limit

### Citations:

Zbl 1278.68240; Zbl 1446.62351
Full Text:

### References:

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