Determinantal point processes (DPPs) have emerged as a kernelized alternative to vanilla independent sampling for generating efficient minibatches, coresets and other parsimonious representations of large-scale datasets. While theoretical foundations and promising empirical performance have been demonstrated, there are two challenges for current proposals for DPP-based coresets or minibatches. The first is the need for families of DPPs with certain key variance reduction properties, usually constructed in a continuous setting, of which there are few known examples. The second is the need for an ad-hoc construction of a discrete DPP defined on a given dataset, that inherits such variance reduction. In this work, we contribute to the programme of establishing DPPs as a subsampling toolbox for ML by advancing on these two fronts. First, we propose new DPPs on the Euclidean space based on wavelets, with provably better accuracy guarantees than the best known rates. Second, we introduce a general method to convert such continuous DPPs, which are more amenable to proving analytical statements, into discrete kernels, which are pertinent for subsampling tasks such as minibatch and coreset constructions. This conversion mechanism simultaneously preserves the desired variance decay and reveals a low-rank decomposition of the discrete kernel, which makes sampling the corresponding DPP computationally inexpensive. En route, we enlarge the class of ML tasks amenable to improvements via DPP-based minibatches and coresets to include objective functions with arbitrarily low regularity, and rate guarantees that explicitly adapt to this regularity.

For the moment, it is worth noting that such sets of functions (e.g., Haar wavelets, Daubechies wavelets) are readily We are now in a position to present the main theorem of this subsection. To avoid repetition, we defer further discussions on the rates observed in Theorem A.1 to Remark 2.7 where a holistic In fact, by Proposition 1.1, there exists an optimal transport map Based on (B.2), the natural plug-in estimator of ρ Suppose that the same assumptions from Theorem 2.2 hold. B.2 Nonparametric independence testing: Optimal transport based Hilbert-Schmidt independence criterion Proposition B.2 shows that the test based on Further, when the sampling distribution is fixed, Proposition B.2 shows that In the following result (see Appendix C.2 for a proof), we show that if This section is devoted to proving our main results and is organized as follows: In Appendix C.1, we Further by Lemma D.2, we also have: ϕ Note that (C.10) immediately yields the following conclusions: S By (1.5) and some simple algebra, the following holds: null null null S Combining the above display with (C.9), we further have: null null null null 1 2 W Combining the above observation with Theorem 2.1, we have: lim sup For the next part, to simplify notation, let us begin with some notation. By using the exponential Markov's inequality coupled with the standard union Now by using [7, Theorem 2.10], we have P (B We are now in a position to complete the proof of Theorem 2.2 using steps I-III. Therefore, it is now enough to bound the right hand side of (C.17).

Whenconsidering networkdata,regularconvolutionsdonot hold since the data domain presents an irregular structure given by the network topology.

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The majority of work in robust statistics has focused on providing guarantees under the Huber -contamination model [Huber, 1965].