We analyze an estimator based on the Bregman divergence for recovery of structured models from additive noise. The estimator can be seen as a regularized maximum likelihood estimator for an exponential family where the natural parameter is assumed to be structured. For all such Bregman denoising estimators, we provide an error bound for a natural associated error measure. Our error bound makes it possible to analyze a wide range of estimators, such as those in proximal denoising and inverse covariance matrix estimation, in a unified manner. In the case of proximal denoising, we exactly recover the existing tight normalized mean squared error bounds. In sparse precision matrix estimation, our bounds provide optimal scaling with interpretable constants in terms of the associated error measure.

Given samples lying on any of a number of subspaces, subspace clustering is the task of grouping the samples based on the their corresponding subspaces. Many subspace clustering methods operate by assigning a measure of affinity to each pair of points and feeding these affinities into a graph clustering algorithm. This paper proposes a new paradigm for subspace clustering that computes affinities based on the corresponding conic geometry. The proposed conic subspace clustering (CSC) approach considers the convex hull of a collection of normalized data points and the corresponding tangent cones. The union of subspaces underlying the data imposes a strong association between the tangent cone at a sample $x$ and the original subspace containing $x$. In addition to describing this novel geometric perspective, this paper provides a practical algorithm for subspace clustering that leverages this perspective, where a tangent cone membership test is used to estimate the affinities. This algorithm is accompanied with deterministic and stochastic guarantees on the properties of the learned affinity matrix, on the true and false positive rates and spread, which directly translate into the overall clustering accuracy.

The Stochastic Block Model (SBM) is a widely used random graph model for networks with communities. Despite the recent burst of interest in community detection under the SBM from statistical and computational points of view, there are still gaps in understanding the fundamental limits of recovery. In this paper, we consider the SBM in its full generality, where there is no restriction on the number and sizes of communities or how they grow with the number of nodes, as well as on the connectivity probabilities inside or across communities. For such stochastic block models, we provide guarantees for exact recovery via a semidefinite program as well as upper and lower bounds on SBM parameters for exact recoverability. Our results exploit the tradeoffs among the various parameters of heterogenous SBM and provide recovery guarantees for many new interesting SBM configurations.

Recovering structured models (e.g., sparse or group-sparse vectors, low-rank matrices) given a few linear observations have been well-studied recently. In various applications in signal processing and machine learning, the model of interest is structured in several ways, for example, a matrix that is simultaneously sparse and low rank. Often norms that promote the individual structures are known, and allow for recovery using an order-wise optimal number of measurements (e.g., 11 norm for sparsity, nuclear norm for matrix rank). Hence, it is reasonable to minimize a combination of such norms. We show that, surprisingly, using multiobjective optimization with these norms can do no better, orderwise, than exploiting only one of the structures, thus revealing a fundamental limitation in sample complexity. This result suggests that to fully exploit the multiple structures, we need an entirely new convex relaxation. Further, specializing our results to the case of sparse and low-rank matrices, we show that a nonconvex formulation recovers the model from very few measurements (on the order of the degrees of freedom), whereas the convex problem combining the 11 and nuclear norms requires many more measurements, illustrating a gap between the performance of the convex and nonconvex recovery problems. Our framework applies to arbitrary structure-inducing norms as well as to a wide range of measurement ensembles. This allows us to give sample complexity bounds for problems such as sparse phase retrieval and low-rank tensor completion.