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Regularization is a common tool in variational inverse problems to impose assumptions on the parameters of the problem. One such assumption is sparsity, which is commonly promoted using lasso and total variation-like regularization. Although the solutions to many such regularized inverse problems can be considered as points of maximum probability of well-chosen posterior distributions, samples from these distributions are generally not sparse. In this paper, we present a framework for implicitly defining a probability distribution that combines the effects of sparsity imposing regularization with Gaussian distributions. Unlike continuous distributions, these implicit distributions can assign positive probability to sparse vectors. We study these regularized distributions for various regularization functions including total variation regularization and piecewise linear convex functions. We apply the developed theory to uncertainty quantification for Bayesian linear inverse problems and derive a Gibbs sampler for a Bayesian hierarchical model. To illustrate the difference between our sparsity-inducing framework and continuous distributions, we apply our framework to small-scale deblurring and computed tomography examples.

In this paper, we study a spiked Wigner problem with an inhomogeneous noise profile. Our aim in this problem is to recover the signal passed through an inhomogeneous low-rank matrix channel. While the information-theoretic performances are well-known, we focus on the algorithmic problem. We derive an approximate message-passing algorithm (AMP) for the inhomogeneous problem and show that its rigorous state evolution coincides with the information-theoretic optimal Bayes fixed-point equations. We identify in particular the existence of a statistical-to-computational gap where known algorithms require a signal-to-noise ratio bigger than the information-theoretic threshold to perform better than random. Finally, from the adapted AMP iteration we deduce a simple and efficient spectral method that can be used to recover the transition for matrices with general variance profiles. This spectral method matches the conjectured optimal computational phase transition.

The Laplacian-constrained Gaussian Markov Random Field (LGMRF) is a common multivariate statistical model for learning a weighted sparse dependency graph from given data. This graph learning problem is formulated as a maximum likelihood estimation (MLE) of the precision matrix, subject to Laplacian structural constraints, with a sparsity-inducing penalty term. This paper aims to solve this learning problem accurately and efficiently. First, since the commonly-used $\ell_1$-norm penalty is less appropriate in this setting, we employ the nonconvex minimax concave penalty (MCP), which promotes sparse solutions with lower estimation bias. Second, as opposed to most existing first-order methods for this problem, we base our method on the second-order proximal Newton approach to obtain an efficient solver for large-scale networks. This approach is considered the most efficient for the related graphical LASSO problem and allows for several algorithmic features we exploit, such as using Conjugate Gradients, preconditioning, and splitting to active/free sets. Numerical experiments demonstrate the advantages of the proposed method in terms of \emph{both} computational complexity and graph learning accuracy compared to existing methods.

Spatial process models popular in geostatistics often represent the observed data as the sum of a smooth underlying process and white noise. The variation in the white noise is attributed to measurement error, or micro-scale variability, and is called the "nugget". We formally establish results on the identifiability and consistency of the nugget in spatial models based upon the Gaussian process within the framework of in-fill asymptotics, i.e. the sample size increases within a sampling domain that is bounded. Our work extends results in fixed domain asymptotics for spatial models without the nugget. More specifically, we establish the identifiability of parameters in the Mat\'ern covariance function and the consistency of their maximum likelihood estimators in the presence of discontinuities due to the nugget. We also present simulation studies to demonstrate the role of the identifiable quantities in spatial interpolation.

This paper explores reward mechanisms for a query incentive network in which agents seek information from social networks. In a query tree issued by the task owner, each agent is rewarded by the owner for contributing to the solution, for instance, solving the task or inviting others to solve it. The reward mechanism determines the reward for each agent and motivates all agents to propagate and report their information truthfully. In particular, the reward cannot exceed the budget set by the task owner. However, our impossibility results demonstrate that a reward mechanism cannot simultaneously achieve Sybil-proof (agents benefit from manipulating multiple fake identities), collusion-proof (multiple agents pretend as a single agent to improve the reward), and other essential properties. In order to address these issues, we propose two novel reward mechanisms. The first mechanism achieves Sybil-proof and collusion-proof, respectively; the second mechanism sacrifices Sybil-proof to achieve the approximate versions of Sybil-proof and collusion-proof. Additionally, we show experimentally that our second reward mechanism outperforms the existing ones.

Pervasive cross-section dependence is increasingly recognized as a characteristic of economic data and the approximate factor model provides a useful framework for analysis. Assuming a strong factor structure where $\Lop\Lo/N^\alpha$ is positive definite in the limit when $\alpha=1$, early work established convergence of the principal component estimates of the factors and loadings up to a rotation matrix. This paper shows that the estimates are still consistent and asymptotically normal when $\alpha\in(0,1]$ albeit at slower rates and under additional assumptions on the sample size. The results hold whether $\alpha$ is constant or varies across factor loadings. The framework developed for heterogeneous loadings and the simplified proofs that can be also used in strong factor analysis are of independent interest.

This paper develops and analyzes an accelerated proximal descent method for finding stationary points of nonconvex composite optimization problems. The objective function is of the form $f+h$ where $h$ is a proper closed convex function, $f$ is a differentiable function on the domain of $h$, and $\nabla f$ is Lipschitz continuous on the domain of $h$. The main advantage of this method is that it is "parameter-free" in the sense that it does not require knowledge of the Lipschitz constant of $\nabla f$ or of any global topological properties of $f$. It is shown that the proposed method can obtain an $\varepsilon$-approximate stationary point with iteration complexity bounds that are optimal, up to logarithmic terms over $\varepsilon$, in both the convex and nonconvex settings. Some discussion is also given about how the proposed method can be leveraged in other existing optimization frameworks, such as min-max smoothing and penalty frameworks for constrained programming, to create more specialized parameter-free methods. Finally, numerical experiments are presented to support the practical viability of the method.

Estimation of heterogeneous causal effects - i.e., how effects of policies and treatments vary across subjects - is a fundamental task in causal inference, playing a crucial role in optimal treatment allocation, generalizability, subgroup effects, and more. Many flexible methods for estimating conditional average treatment effects (CATEs) have been proposed in recent years, but questions surrounding optimality have remained largely unanswered. In particular, a minimax theory of optimality has yet to be developed, with the minimax rate of convergence and construction of rate-optimal estimators remaining open problems. In this paper we derive the minimax rate for CATE estimation, in a nonparametric model where distributional components are Holder-smooth, and present a new local polynomial estimator, giving high-level conditions under which it is minimax optimal. More specifically, our minimax lower bound is derived via a localized version of the method of fuzzy hypotheses, combining lower bound constructions for nonparametric regression and functional estimation. Our proposed estimator can be viewed as a local polynomial R-Learner, based on a localized modification of higher-order influence function methods; it is shown to be minimax optimal under a condition on how accurately the covariate distribution is estimated. The minimax rate we find exhibits several interesting features, including a non-standard elbow phenomenon and an unusual interpolation between nonparametric regression and functional estimation rates. The latter quantifies how the CATE, as an estimand, can be viewed as a regression/functional hybrid. We conclude with some discussion of a few remaining open problems.

The vanishing ideal of a set of points $X = \{\mathbf{x}_1, \ldots, \mathbf{x}_m\}\subseteq \mathbb{R}^n$ is the set of polynomials that evaluate to $0$ over all points $\mathbf{x} \in X$ and admits an efficient representation by a finite subset of generators. In practice, to accommodate noise in the data, algorithms that construct generators of the approximate vanishing ideal are widely studied but their computational complexities remain expensive. In this paper, we scale up the oracle approximate vanishing ideal algorithm (OAVI), the only generator-constructing algorithm with known learning guarantees. We prove that the computational complexity of OAVI is not superlinear, as previously claimed, but linear in the number of samples $m$. In addition, we propose two modifications that accelerate OAVI's training time: Our analysis reveals that replacing the pairwise conditional gradients algorithm, one of the solvers used in OAVI, with the faster blended pairwise conditional gradients algorithm leads to an exponential speed-up in the number of features $n$. Finally, using a new inverse Hessian boosting approach, intermediate convex optimization problems can be solved almost instantly, improving OAVI's training time by multiple orders of magnitude in a variety of numerical experiments.

Out-of-sample prediction is the acid test of predictive models, yet an independent test dataset is often not available for assessment of the prediction error. For this reason, out-of-sample performance is commonly estimated using data splitting algorithms such as cross-validation or the bootstrap. For quantitative outcomes, the ratio of variance explained to total variance can be summarized by the coefficient of determination or in-sample $R^2$, which is easy to interpret and to compare across different outcome variables. As opposed to the in-sample $R^2$, the out-of-sample $R^2$ has not been well defined and the variability on the out-of-sample $\hat{R}^2$ has been largely ignored. Usually only its point estimate is reported, hampering formal comparison of predictability of different outcome variables. Here we explicitly define the out-of-sample $R^2$ as a comparison of two predictive models, provide an unbiased estimator and exploit recent theoretical advances on uncertainty of data splitting estimates to provide a standard error for the $\hat{R}^2$. The performance of the estimators for the $R^2$ and its standard error are investigated in a simulation study. We demonstrate our new method by constructing confidence intervals and comparing models for prediction of quantitative $\text{Brassica napus}$ and $\text{Zea mays}$ phenotypes based on gene expression data.