Comparison Theorems for Gibbs Measures

Rebeschini, Patrick; Handel, Ramon van

doi:10.1007/s10955-014-1087-7

Cited by 11 publications

(19 citation statements)

References 27 publications

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“…Since one can obtain contraction estimates for Markov kernels using Wasserstein matrices, it is natural to ask whether Chatterjee's result can be derived as a special case of a more general method, which would let us freely choose an arbitrary Markov kernel K that leaves µ invariant and control the constants in the resulting concentration inequality by means of a judicious choice of a Wasserstein matrix for K. Such a method would most likely rely on general comparison theorems for Gibbs measures [31].…”

Section: Open Questionsmentioning

confidence: 99%

Concentration of Measure Without Independence: A Unified Approach Via the Martingale Method

Kontorovich

Raginsky

2017

Convexity and Concentration

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The concentration of measure phenomenon may be summarized as follows: a function of many weakly dependent random variables that is not too sensitive to any of its individual arguments will tend to take values very close to its expectation. This phenomenon is most completely understood when the arguments are mutually independent random variables, and there exist several powerful complementary methods for proving concentration inequalities, such as the martingale method, the entropy method, and the method of transportation inequalities. The setting of dependent arguments is much less well understood. This chapter focuses on the martingale method for deriving concentration inequalities without independence assumptions. In particular, we use the machinery of so-called Wasserstein matrices to show that the Azuma-Hoeffding concentration inequality for martingales with almost surely bounded differences, when applied in a sufficiently abstract setting, is powerful enough to recover and sharpen several known concentration results for nonproduct measures. Wasserstein matrices provide a natural formalism for capturing the interplay between the metric and the probabilistic structures, which is fundamental to the concentration phenomenon.

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Section: Open Questionsmentioning

confidence: 99%

Concentration of Measure Without Independence: A Unified Approach Via the Martingale Method

Kontorovich

Raginsky

2017

Convexity and Concentration

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“…The complexity of the BPF is (N |K| ∞ card K) per time step where |K| ∞ := max K∈K card K. Under strong mixing assumptions, [13], [25] show that local errors ofπ N t,BPF are bounded uniformly in t and V .…”

Section: Blocked Particle Filtersmentioning

confidence: 99%

“…1) Model ( m,p t ,g t ), defined in Subsection B, represents the asymptotic target distribution of the BPF as stated in [25]. It defines a joint smoothing distribution Q T and filtersπ t .…”

Section: A Outlinementioning

confidence: 99%

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Approximate Smoothing and Parameter Estimation in High-Dimensional State-Space Models

Finke

Singh

2017

IEEE Trans. Signal Process.

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We present approximate algorithms for performing smoothing in a class of high-dimensional state-space models via sequential Monte Carlo methods ('particle filters'). In high dimensions, a prohibitively large number of Monte Carlo samples ('particles'), growing exponentially in the dimension of the state space, is usually required to obtain a useful smoother. Employing blocking approximations, we exploit the spatial ergodicity properties of the model to circumvent this curse of dimensionality. We thus obtain approximate smoothers that can be computed recursively in time and parallel in space. First, we show that the bias of our blocked smoother is bounded uniformly in the time horizon and in the model dimension. We then approximate the blocked smoother with particles and derive the asymptotic variance of idealised versions of our blocked particle smoother to show that variance is no longer adversely effected by the dimension of the model. Finally, we employ our method to successfully perform maximum-likelihood estimation via stochastic gradient-ascent and stochastic expectationmaximisation algorithms in a 100-dimensional state-space model.

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“…The general characterization that we give to ∂x (b)i ∂ba (Theorem 2.1 below) can be interpreted as a first instance of comparison theorems for constrained optimization procedures, along the lines of the comparison theorems established in probability theory to capture stochastic decay of correlation and control the difference of high-dimensional distributions (see [8] and [3]). …”

Section: Introductionmentioning

confidence: 98%

Decay of correlation in network flow problems

Rebeschini

Tatikonda

2016

2016 Annual Conference on Information Science and Systems (CISS)

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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

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Comparison Theorems for Gibbs Measures

Cited by 11 publications

References 27 publications

Concentration of Measure Without Independence: A Unified Approach Via the Martingale Method

Concentration of Measure Without Independence: A Unified Approach Via the Martingale Method

Approximate Smoothing and Parameter Estimation in High-Dimensional State-Space Models

Decay of correlation in network flow problems

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