Density estimation is a building block for many other statistical methods, such as classification, nonparametric testing, and data compression. In this paper, we focus on a non-parametric approach to multivariate density estimation, and study its asymptotic properties under both frequentist and Bayesian settings. The estimated density function is obtained by considering a sequence of approximating spaces to the space of densities. These spaces consist of piecewise constant density functions supported by binary partitions with increasing complexity. To obtain an estimate, the partition is learned by maximizing either the likelihood of the corresponding histogram on that partition, or the marginal posterior probability of the partition under a suitable prior. We analyze the convergence rate of the maximum likelihood estimator and the posterior concentration rate of the Bayesian estimator, and conclude that for a relatively rich class of density functions the rate does not directly depend on the dimension. We also show that the Bayesian method can adapt to the unknown smoothness of the density function. The method is applied to several specific function classes and explicit rates are obtained. These include spatially sparse functions, functions of bounded variation, and Holder continuous functions. We also introduce an ensemble approach, obtained by aggregating multiple density estimates fit under carefully designed perturbations, and show that for density functions lying in a Holder space (H^(1,β),0<β≤1), the ensemble method can achieve minimax convergence rate up to a logarithmic term, while the corresponding rate of the density estimator based on a single partition is suboptimal for this function class.
more »
« less
Higher-Order Total Variation Classes on Grids: Minimax Theory and Trend Filtering Methods.
We consider the problem of estimating the values of a function over n nodes of a d-dimensional grid graph (having equal side lengths) from noisy observations. The function is assumed to be smooth, but is allowed to exhibit different amounts of smoothness at different regions in the grid. Such heterogeneity eludes classical measures of smoothness from nonparametric statistics, such as Holder smoothness. Meanwhile, total variation (TV) smoothness classes allow for heterogeneity, but are restrictive in another sense: only constant functions count as perfectly smooth (achieve zero TV). To move past this, we define two new higher-order TV classes, based on two ways of compiling the discrete derivatives of a parameter across the nodes. We relate these two new classes to Holder classes, and derive lower bounds on their minimax errors. We also analyze two naturally associated trend filtering methods; when d=2, each is seen to be rate optimal over the appropriate class.
more »
« less
- Award ID(s):
- 1712996
- NSF-PAR ID:
- 10061393
- Date Published:
- Journal Name:
- Advances in neural information processing systems
- Volume:
- 30
- ISSN:
- 1049-5258
- Page Range / eLocation ID:
- 5800--5810
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Density estimation is a building block for many other statistical methods, such as classification, nonparametric testing, and data compression. In this paper, we focus on a nonparametric approach to multivariate density estimation, and study its asymptotic properties under both frequentist and Bayesian settings. The estimated density function is obtained by considering a sequence of approximating spaces to the space of densities. These spaces consist of piecewise constant density functions supported by binary partitions with increasing complexity. To obtain an estimate, the partition is learned by maximizing either the likelihood of the corresponding histogram on that partition, or the marginal posterior probability of the partition under a suitable prior. We analyze the convergence rate of the maximum likelihood estimator and the posterior concentration rate of the Bayesian estimator, and conclude that for a relatively rich class of density functions the rate does not directly depend on the dimension. We also show that the Bayesian method can adapt to the unknown smoothness of the density function. The method is applied to several specific function classes and explicit rates are obtained. These include spatially sparse functions, functions of bounded variation, and Holder continuous functions. We also introduce an ensemble approach, obtained by aggregating multiple density estimates fit under carefully designed perturbations, and show that for density functions lying in a Holder space (H^(1,β), 0 < β ≤ 1), the ensemble method can achieve minimax convergence rate up to a logarithmic term, while the corresponding rate of the density estimator based on a single partition is suboptimal for this function class.more » « less
-
We study the problem of finding the maximum of a function defined on the nodes of a connected graph. The goal is to identify a node where the function obtains its maximum. We focus on local iterative algorithms, which traverse the nodes of the graph along a path, and the next iterate is chosen from the neighbors of the current iterate with probability distribution determined by the function values at the current iterate and its neighbors. We study two algorithms corresponding to a Metropolis-Hastings random walk with different transition kernels: (i) The first algorithm is an exponentially weighted random walk governed by a parameter gamma. (ii) The second algorithm is defined with respect to the graph Laplacian and a smoothness parameter k. We derive convergence rates for the two algorithms in terms of total variation distance and hitting times. We also provide simulations showing the relative convergence rates of our algorithms in comparison to an unbiased random walk, as a function of the smoothness of the graph function. Our algorithms may be categorized as a new class of “descent-based” methods for function maximization on the nodes of a graph.more » « less
-
more » « less
Abstract Finite volume, weighted essentially non-oscillatory (WENO) schemes require the computation of a smoothness indicator. This can be expensive, especially in multiple space dimensions. We consider the use of the simple smoothness indicator
$$\sigma ^{\textrm{S}}= \frac{1}{N_{\textrm{S}}-1}\sum _{j} ({\bar{u}}_{j} - {\bar{u}}_{m})^2$$ $$N_{\textrm{S}}$$ $${\bar{u}}_j$$ j , and indexm gives the target element. Reconstructions utilizing standard WENO weighting fail with this smoothness indicator. We develop a modification of WENO-Z weighting that gives a reliable and accurate reconstruction of adaptive order, which we denote as SWENOZ-AO. We prove that it attains the order of accuracy of the large stencil polynomial approximation when the solution is smooth, and drops to the order of the small stencil polynomial approximations when there is a jump discontinuity in the solution. Numerical examples in one and two space dimensions on general meshes verify the approximation properties of the reconstruction. They also show it to be about 10 times faster in two space dimensions than reconstructions using the classic smoothness indicator. The new reconstruction is applied to define finite volume schemes to approximate the solution of hyperbolic conservation laws. Numerical tests show results of the same quality as standard WENO schemes using the classic smoothness indicator, but with an overall speedup in the computation time of about 3.5–5 times in 2D tests. Moreover, the computational efficiency (CPU time versus error) is noticeably improved. -
Confident identification of pericytes (PCs) remains an obstacle in the field, as a single molecular marker for these unique perivascular cells remains elusive. Adding to this challenge is the recent appreciation that PC populations may be heterogeneous, displaying a range of morphologies within capillary networks. We found additional support on the ultrastructural level for the classification of these PC subtypes—“thin-strand” (TSP), mesh (MP), and ensheathing (EP)—based on distinct morphological characteristics. Interestingly, we also found several examples of another cell type, likely a vascular smooth muscle cell, in a medial layer between endothelial cells (ECs) and pericytes (PCs) harboring characteristics of the ensheathing type. A conserved feature across the different PC subtypes was the presence of extracellular matrix (ECM) surrounding the vascular unit and distributed in between neighboring cells. The thickness of this vascular basement membrane was remarkably consistent depending on its location, but never strayed beyond a range of 150–300 nm unless thinned to facilitate closer proximity of neighboring cells (suggesting direct contact). The density of PC-EC contact points (“peg-and-socket” structures) was another distinguishing feature across the different PC subtypes, as were the apparent contact locations between vascular cells and brain parenchymal cells. In addition to this thinning, the extracellular matrix (ECM) surrounding EPs displayed another unique configuration in the form of extensions that emitted out radially into the surrounding parenchyma. Knowledge of the origin and function of these structures is still emerging, but their appearance suggests the potential for being mechanical elements and/or perhaps signaling nodes via embedded molecular cues. Overall, this unique ultrastructural perspective provides new insights into PC heterogeneity and the presence of medial cells within the microvessel wall, the consideration of extracellular matrix (ECM) coverage as another PC identification criteria, and unique extracellular matrix (ECM) configurations (i.e., radial extensions) that may reveal additional aspects of PC heterogeneity.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- The DOI is not currently available.
