Abstract-We study the problem of corrupted sensing, a generalization of compressed sensing in which one aims to recover a signal from a collection of corrupted or unreliable measurements. While an arbitrary signal cannot be recovered in the face of arbitrary corruption, tractable recovery is possible when both signal and corruption are suitably structured. We quantify the relationship between signal recovery and two geometric measures of structure, the Gaussian complexity of a tangent cone and the Gaussian distance to a subdifferential. We take a convex programming approach to disentangling signal and corruption, analyzing both penalized programs that trade off between signal and corruption complexity, and constrained programs that bound the complexity of signal or corruption when prior information is available. In each case, we provide conditions for exact signal recovery from structured corruption and stable signal recovery from structured corruption with added unstructured noise. Our simulations demonstrate close agreement between our theoretical recovery bounds and the sharp phase transitions observed in practice. In addition, we provide new interpretable bounds for the Gaussian complexity of sparse vectors, block-sparse vectors, and low-rank matrices, which lead to sharper guarantees of recovery when combined with our results and those in the literature.

X 1 = λ 01 + ε 1 , X 2 = λ 02 + λ 12 X 1 + ε 2 , X 3 = λ 03 + λ 23 X 2 + ε 3 , with an error vector ε that has zero mean vector and covariance matrix Ω =   ω 11 0 0 0 ω 22 ω 23 0 ω 23 ω 33   . IDENTIFIABILITY OF LINEAR STRUCTURAL EQUATION MODELS 3The possibly nonzero entry ω 23 can absorb the effects that unobserved confounders (such as age, income, genetics, etc.) may have on both X 2 and X 3 ; compare Richardson and Spirtes (2002) and Wermuth (2011) for background on mixed graph representations of latent variable problems.Formally, a mixed graph is a triple G = (V, D, B), where V is a finite set of nodes and D, B ⊆ V × V are two sets of edges. In our context, the nodes correspond to the random variables X 1 , . . . , X m , and we simply let V = [m] := {1, . . . , m}. The pairs (v, w) in the set D represent directed edges and we will alwaysIf the directed part (V, D) does not contain directed cycles (i.e., no cycle v → · · · → v can be formed from the edges in D), then the mixed graph G is said to be acyclic.the remark after equation (2.3).] Similarly, let PD m be the cone of positive definite symmetric m × m-matrices Ω = (ω vw ) and define PD(B) ⊂ PD m to be the subcone of matrices with support B, that is, ω vw = 0 if v = w and v ↔ w / ∈ B.

A B S T R A C TThe current study examines the syntactic and prosodic characteristics of the maternal speech to two infants between six and ten months. Consistent with previous work, we find infant-directed speech to be characterized by generally short utterances, isolated words and phrases, and large numbers of questions, but longer utterances are also found. Prosodic information provides cues to grammatical units not only at utterance boundaries, but also at utterance-internal clause boundaries. Subject-verb phrase boundaries in questions also show reliable prosodic cues, although those of declaratives do not. Prosodic information may thus play an important role in providing preverbal infants with information about the grammatically relevant word groupings. Furthermore, questions may play an important role in infants' discovery of verb phrases in English.

Structural equation models are multivariate statistical models that are defined by specifying noisy functional relationships among random variables. We consider the classical case of linear relationships and additive Gaussian noise terms. We give a necessary and sufficient condition for global identifiability of the model in terms of a mixed graph encoding the linear structural equations and the correlation structure of the error terms. Global identifiability is understood to mean injectivity of the parametrization of the model and is fundamental in particular for applicability of standard statistical methodology.Comment: Published in at http://dx.doi.org/10.1214/10-AOS859 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Gaussian graphical models with sparsity in the inverse covariance matrix are of significant interest in many modern applications. For the problem of recovering the graphical structure, information criteria provide useful optimization objectives for algorithms searching through sets of graphs or for selection of tuning parameters of other methods such as the graphical lasso, which is a likelihood penalization technique. In this paper we establish the consistency of an extended Bayesian information criterion for Gaussian graphical models in a scenario where both the number of variables p and the sample size n grow. Compared to earlier work on the regression case, our treatment allows for growth in the number of non-zero parameters in the true model, which is necessary in order to cover connected graphs. We demonstrate the performance of this criterion on simulated data when used in conjunction with the graphical lasso, and verify that the criterion indeed performs better than either cross-validation or the ordinary Bayesian information criterion when p and the number of non-zero parameters q both scale with n.