However, the Achilles' heel of all these approaches is that

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In the vast majority of recent work on sparse estimation algorithms, performance has been evaluated using ideal or quasi-ideal dictionaries (e.g., random Gaussian or Fourier) characterized by unit l

In recent years, a rich variety of shrinkage priors have been proposed that have great promise in addressing massive regression problems. In general, these new priors can be expressed as scale mixtures of normals, but have more complex forms and better properties than traditional Cauchy and double exponential priors. We first propose a new class of normal scale mixtures through a novel generalized beta distribution that encompasses many interesting priors as special cases. This encompassing framework should prove useful in comparing competing priors, considering properties and revealing close connections. We then develop a class of variational Bayes approximations through the new hierarchy presented that will scale more efficiently to the types of truly massive data sets that are now encountered routinely.

Finding maximally sparse representations from overcomplete feature dictionaries frequently involves minimizing a cost function composed of a likelihood (or data fit) term and a prior (or penalty function) that favors sparsity. While typically the prior is factorial, here we examine non-factorial alternatives that have a number of desirable properties relevant to sparse estimation and are easily implemented using an efficient, globally-convergent reweighted \ell_1 minimization procedure. The first method under consideration arises from the sparse Bayesian learning (SBL) framework. Although based on a highly non-convex underlying cost function, in the context of canonical sparse estimation problems, we prove uniform superiority of this method over the Lasso in that, (i) it can never do worse, and (ii) for any dictionary and sparsity profile, there will always exist cases where it does better. These results challenge the prevailing reliance on strictly convex penalty functions for finding sparse solutions.

In the vast majority of recent work on sparse estimation algorithms, performance has been evaluated using ideal or quasi-ideal dictionaries (e.g., random Gaussian or Fourier) characterized by unit \ell_2 norm, incoherent columns or features. But in reality, these types of dictionaries represent only a subset of the dictionaries that are actually used in practice (largely restricted to idealized compressive sensing applications). In contrast, herein sparse estimation is considered in the context of structured dictionaries possibly exhibiting high coherence between arbitrary groups of columns and/or rows. Sparse penalized regression models are analyzed with the purpose of finding, to the extent possible, regimes of dictionary invariant performance. In particular, a Type II Bayesian estimator with a dictionary-dependent sparsity penalty is shown to have a number of desirable invariance properties leading to provable advantages over more conventional penalties such as the \ell_1 norm, especially in areas where existing theoretical recovery guarantees no longer hold. This can translate into improved performance in applications such as model selection with correlated features, source localization, and compressive sensing with constrained measurement directions.

We study the contextual bandits with knapsack (CBwK) problem under the high-dimensional setting where the dimension of the feature is large. The reward of pulling each arm equals the multiplication of a sparse high-dimensional weight vector and the feature of the current arrival, with additional random noise. In this paper, we investigate how to exploit this sparsity structure to achieve improved regret for the CBwK problem. To this end, we first develop an online variant of the hard thresholding algorithm that performs the sparse estimation in an online manner. We further combine our online estimator with a primal-dual framework, where we assign a dual variable to each knapsack constraint and utilize an online learning algorithm to update the dual variable, thereby controlling the consumption of the knapsack capacity. We show that this integrated approach allows us to achieve a sublinear regret that depends logarithmically on the feature dimension, thus improving the polynomial dependency established in the previous literature. We also apply our framework to the high-dimension contextual bandit problem without the knapsack constraint and achieve optimal regret in both the data-poor regime and the data-rich regime. We finally conduct numerical experiments to show the efficient empirical performance of our algorithms under the high dimensional setting.

We propose statistically robust and computationally efficient linear learning methods in the high-dimensional batch setting, where the number of features $d$ may exceed the sample size $n$. We employ, in a generic learning setting, two algorithms depending on whether the considered loss function is gradient-Lipschitz or not. Then, we instantiate our framework on several applications including vanilla sparse, group-sparse and low-rank matrix recovery. This leads, for each application, to efficient and robust learning algorithms, that reach near-optimal estimation rates under heavy-tailed distributions and the presence of outliers. For vanilla $s$-sparsity, we are able to reach the $s\log (d)/n$ rate under heavy-tails and $\eta$-corruption, at a computational cost comparable to that of non-robust analogs. We provide an efficient implementation of our algorithms in an open-source $\mathtt{Python}$ library called $\mathtt{linlearn}$, by means of which we carry out numerical experiments which confirm our theoretical findings together with a comparison to other recent approaches proposed in the literature.

This paper addresses the problem of fault diagnosis in multistation assembly systems. Fault diagnosis is to identify process faults that cause the excessive dimensional variation of the product using dimensional measurements. For such problems, the challenge is solving an underdetermined system caused by a common phenomenon in practice; namely, the number of measurements is less than that of the process errors. To address this challenge, this paper attempts to solve the following two problems: (1) how to utilize the temporal correlation in the time series data of each process error and (2) how to apply prior knowledge regarding which process errors are more likely to be process faults. A novel sparse Bayesian learning method is proposed to achieve the above objectives. The method consists of three hierarchical layers. The first layer has parameterized prior distribution that exploits the temporal correlation of each process error. Furthermore, the second and third layers achieve the prior distribution representing the prior knowledge of process faults. Then, these prior distributions are updated with the likelihood function of the measurement samples from the process, resulting in the accurate posterior distribution of process faults from an underdetermined system. Since posterior distributions of process faults are intractable, this paper derives approximate posterior distributions via Variational Bayes inference. Numerical and simulation case studies using an actual autobody assembly process are performed to demonstrate the effectiveness of the proposed method.

Robust and sparse estimation of linear regression coefficients is investigated. The situation addressed by the present paper is that covariates and noises are sampled from heavy-tailed distributions, and the covariates and noises are contaminated by malicious outliers. Our estimator can be computed efficiently. Further, the error bound of the estimator is nearly optimal.