The discovery of particle filtering methods has enabled the use of nonlinear filtering in a wide array of applications. Unfortunately, the approximation error of particle filters typically grows exponentially in the dimension of the underlying model. This phenomenon has rendered particle filters of limited use in complex data assimilation problems. In this paper, we argue that it is often possible, at least in principle, to develop local particle filtering algorithms whose approximation error is dimension-free. The key to such developments is the decay of correlations property, which is a spatial counterpart of the much better understood stability property of nonlinear filters. For the simplest possible algorithm of this type, our results provide under suitable assumptions an approximation error bound that is uniform both in time and in the model dimension. More broadly, our results provide a framework for the investigation of filtering problems and algorithms in high dimension.Comment: Published at http://dx.doi.org/10.1214/14-AAP1061 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

We consider the problem of reconstructing an n-dimensional k-sparse signal from a set of magnitude-only measurements. Formulating the problem as an unregularized empirical risk minimization task, we study the sample complexity performance of gradient descent with Hadamard parametrization, which we call Hadamard Wirtinger flow (HWF). Provided knowledge of the signal sparsity k, we prove that a single step of HWF is able to recover the support from O(k(x * max ) −2 log n) samples, where x * max is the largest component of the signal in magnitude. This support recovery procedure can be used to initialize existing reconstruction methods and yields algorithms with total runtime proportional to the cost of reading the data and improved sample complexity, which is linear in k when the signal contains at least one large component. We numerically investigate the performance of HWF at convergence and show that, while not requiring any explicit form of regularization nor knowledge of k, HWF adapts to the signal sparsity and reconstructs sparse signals with fewer measurements than existing state-of-the-art methods.

The Dobrushin comparison theorem is a powerful tool to bound the difference between the marginals of high-dimensional probability distributions in terms of their local specifications. Originally introduced to prove uniqueness and decay of correlations of Gibbs measures, it has been widely used in statistical mechanics as well as in the analysis of algorithms on random fields and interacting Markov chains. However, the classical comparison theorem requires validity of the Dobrushin uniqueness criterion, essentially restricting its applicability in most models to a small subset of the natural parameter space. In this paper we develop generalized Dobrushin comparison theorems in terms of influences between blocks of sites, in the spirit of Dobrushin-Shlosman and Weitz, that substantially extend the range of applicability of the classical comparison theorem. Our proofs are based on the analysis of an associated family of Markov chains. We develop in detail an application of our main results to the analysis of sequential Monte Carlo algorithms for filtering in high dimension.Comment: 55 page

It has been established under very general conditions that the ergodic properties of Markov processes are inherited by their conditional distributions given partial information. While the existing theory provides a rather complete picture of classical filtering models, many infinite-dimensional problems are outside its scope. Far from being a technical issue, the infinite-dimensional setting gives rise to surprising phenomena and new questions in filtering theory. The aim of this paper is to discuss some elementary examples, conjectures, and general theory that arise in this setting, and to highlight connections with problems in statistical mechanics and ergodic theory. In particular, we exhibit a simple example of a uniformly ergodic model in which ergodicity of the filter undergoes a phase transition, and we develop some qualitative understanding as to when such phenomena can and cannot occur. We also discuss closely related problems in the setting of conditional Markov random fields.

This paper studies early-stopped mirror descent applied to noisy sparse phase retrieval, which is the problem of recovering a $k$-sparse signal $\textbf{x}^\star \in{\mathbb{R}}^n$ from a set of quadratic Gaussian measurements corrupted by sub-exponential noise. We consider the (non-convex) unregularized empirical risk minimization problem and show that early-stopped mirror descent, when equipped with the hypentropy mirror map and proper initialization, achieves a nearly minimax-optimal rate of convergence, provided the sample size is at least of order $k^2$ (modulo logarithmic term) and the minimum (in modulus) non-zero entry of the signal is on the order of $\|\textbf{x}^\star \|_2/\sqrt{k}$. Our theory leads to a simple algorithm that does not rely on explicit regularization or thresholding steps to promote sparsity. More generally, our results establish a connection between mirror descent and sparsity in the non-convex problem of noisy sparse phase retrieval, adding to the literature on early stopping that has mostly focused on non-sparse, Euclidean and convex settings via gradient descent. Our proof combines a potential-based analysis of mirror descent with a quantitative control on a variational coherence property that we establish along the path of mirror descent, up to a prescribed stopping time.

We develop a general theory for the local sensitivity of optimal points of constrained network optimization problems under perturbations of the constraints. For the network flow problem, we show that local perturbations on the constraints have an impact on the components of the optimal point that decreases exponentially with the graph-theoretical distance. The exponential rate is controlled by the spectral radius of a substochastic transition matrix of a killed random walk associated to the network. For graphs where this spectral radius is wellbehaved (bounded, for instance) as a function of the dimension of the network, our theory yields the first-known incarnation of the decay of correlation principle in constrained optimization.Index Terms-sensitivity of optimal point, decay of correlation, network flow, killed random walk

The local Rademacher complexity framework is one of the most successful general-purpose toolboxes for establishing sharp excess risk bounds for statistical estimators based on the framework of empirical risk minimization. Applying this toolbox typically requires using the Bernstein condition, which often restricts applicability to convex and proper settings. Recent years have witnessed several examples of problems where optimal statistical performance is only achievable via non-convex and improper estimators originating from aggregation theory, including the fundamental problem of model selection. These examples are currently outside of the reach of the classical localization theory.In this work, we build upon the recent approach to localization via offset Rademacher complexities, for which a general high-probability theory has yet to be established. Our main result is an exponential-tail excess risk bound expressed in terms of the offset Rademacher complexity that yields results at least as sharp as those obtainable via the classical theory. However, our bound applies under an estimator-dependent geometric condition (the "offset condition") instead of the estimator-independent (but, in general, distribution-dependent) Bernstein condition on which the classical theory relies. Our results apply to improper prediction regimes not directly covered by the classical theory.

We investigate implicit regularization schemes for gradient descent methods applied to unpenalized least squares regression to solve the problem of reconstructing a sparse signal from an underdetermined system of linear measurements under the restricted isometry assumption. For a given parametrization yielding a non-convex optimization problem, we show that prescribed choices of initialization, step size and stopping time yield a statistically and computationally optimal algorithm that achieves the minimax rate with the same cost required to read the data up to poly-logarithmic factors. Beyond minimax optimality, we show that our algorithm adapts to instance difficulty and yields a dimension-independent rate when the signal-to-noise ratio is high enough. Key to the computational efficiency of our method is an increasing step size scheme that adapts to refined estimates of the true solution. We validate our findings with numerical experiments and compare our algorithm against explicit 1 penalization. Going from hard instances to easy ones, our algorithm is seen to undergo a phase transition, eventually matching least squares with an oracle knowledge of the true support.