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.

Causal inference aims to draw a causal relationship between some treatment and target responses.

A central result in statistical theory is Pinsker's theorem, which characterizes the minimax rate in the normal means model of nonparametric estimation. In this paper, we present an extension to Pinsker's theorem where estimation is carried out under storage or communication constraints. In particular, we place limits on the number of bits used to encode an estimator, and analyze the excess risk in terms of this constraint, the signal size, and the noise level. We give sharp upper and lower bounds for the case of a Euclidean ball, which establishes the Pareto-optimal minimax tradeoff between storage and risk in this setting.

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.

Generalization theory has been established for sparse deep neural networks under high-dimensional regime. Beyond generalization, parameter estimation is also important since it is crucial for variable selection and interpretability of deep neural networks. Current theoretical studies concerning parameter estimation mainly focus on two-layer neural networks, which is due to the fact that the convergence of parameter estimation heavily relies on the regularity of the Hessian matrix, while the Hessian matrix of deep neural networks is highly singular. To avoid the unidentifiability of deep neural networks in parameter estimation, we propose to conduct nonparametric estimation of partial derivatives with respect to inputs. We first show that model convergence of sparse deep neural networks is guaranteed in that the sample complexity only grows with the logarithm of the number of parameters or the input dimension when the $\ell_{1}$-norm of parameters is well constrained. Then by bounding the norm and the divergence of partial derivatives, we establish that the convergence rate of nonparametric estimation of partial derivatives scales as $\mathcal{O}(n^{-1/4})$, a rate which is slower than the model convergence rate $\mathcal{O}(n^{-1/2})$. To the best of our knowledge, this study combines nonparametric estimation and parametric sparse deep neural networks for the first time. As nonparametric estimation of partial derivatives is of great significance for nonlinear variable selection, the current results show the promising future for the interpretability of deep neural networks.

A central result in statistical theory is Pinsker's theorem, which characterizes the minimax rate in the normal means model of nonparametric estimation. In this paper, we present an extension to Pinsker's theorem where estimation is carried out under storage or communication constraints. In particular, we place limits on the number of bits used to encode an estimator, and analyze the excess risk in terms of this constraint, the signal size, and the noise level. We give sharp upper and lower bounds for the case of a Euclidean ball, which establishes the Pareto-optimal minimax tradeoff between storage and risk in this setting.