Large Deviation Methods Research Articles

The concentration of zero-noise limits of invariant measures for stochastic dynamical systems

We study the zero-noise concentration phenomena of invariant measures for SDE. The models are with locally Lipschitz coefficients and have more than one ergodic state. By the large deviations method and an expression of invariant measures, we estimate the invariant measures in neighborhoods of stable sets, unstable sets and their complements. Our results illustrate that the zero-noise limiting invariant measures will concentrate on the stable sets, where a cost functional W(Ki) is minimized. This implies that the long time behaviors of ODE and SDE must have different judging criteria. Furthermore, we prove the large deviations principle of invariant measures.

Reconstructing volatility: Pricing of index options under rough volatility

AbstractAvellaneda et al. (2002, 2003) pioneered the pricing and hedging of index options – products highly sensitive to implied volatility and correlation assumptions – with large deviations methods, assuming local volatility dynamics for all components of the index. We present an extension applicable to non‐Markovian dynamics and in particular the case of rough volatility dynamics.

Large deviations approach to a one-dimensional, time-periodic stochastic model of pattern formation

In this work we consider the problem of pattern formation modeled by a one dimensional stochastic reaction–diffusion equation with time periodic coefficients. In particular, we apply Large Deviations methods to obtain lower bounds on the probability that certain evenly spaced patterns will develop. Our estimates are optimized when the number of interfaces scales as (ϵT)−1, where ϵ is the length-scale and T is the time-scale. For large times T=ρ|lnϵ| our lower bound is of order exp(−ϵ2ρ), suggesting high likelihood for evenly spaced patterns whose number of interfaces is of order (ϵρ|lnϵ|)−1. Numerical simulations provide support to the idea that the more likely number of interfaces, even among unevenly spaced patterns, follows this law.

Weighted approximate Bayesian computation via Sanov\u2019s theorem

We consider the problem of sample degeneracy in Approximate Bayesian Computation. It arises when proposed values of the parameters, once given as input to the generative model, rarely lead to simulations resembling the observed data and are hence discarded. Such “poor” parameter proposals do not contribute at all to the representation of the parameter’s posterior distribution. This leads to a very large number of required simulations and/or a waste of computational resources, as well as to distortions in the computed posterior distribution. To mitigate this problem, we propose an algorithm, referred to as the Large Deviations Weighted Approximate Bayesian Computation algorithm, where, via Sanov’s Theorem, strictly positive weights are computed for all proposed parameters, thus avoiding the rejection step altogether. In order to derive a computable asymptotic approximation from Sanov’s result, we adopt the information theoretic “method of types” formulation of the method of Large Deviations, thus restricting our attention to models for i.i.d. discrete random variables. Finally, we experimentally evaluate our method through a proof-of-concept implementation.

Leveraging large-deviation statistics to decipher the stochastic properties of measured trajectories

Extensive time-series encoding the position of particles such as viruses, vesicles, or individual proteins are routinely garnered in single-particle tracking experiments or supercomputing studies. They contain vital clues on how viruses spread or drugs may be delivered in biological cells. Similar time-series are being recorded of stock values in financial markets and of climate data. Such time-series are most typically evaluated in terms of time-averaged mean-squared displacements (TAMSDs), which remain random variables for finite measurement times. Their statistical properties are different for different physical stochastic processes, thus allowing us to extract valuable information on the stochastic process itself. To exploit the full potential of the statistical information encoded in measured time-series we here propose an easy-to-implement and computationally inexpensive new methodology, based on deviations of the TAMSD from its ensemble average counterpart. Specifically, we use the upper bound of these deviations for Brownian motion (BM) to check the applicability of this approach to simulated and real data sets. By comparing the probability of deviations for different data sets, we demonstrate how the theoretical bound for BM reveals additional information about observed stochastic processes. We apply the large-deviation method to data sets of tracer beads tracked in aqueous solution, tracer beads measured in mucin hydrogels, and of geographic surface temperature anomalies. Our analysis shows how the large-deviation properties can be efficiently used as a simple yet effective routine test to reject the BM hypothesis and unveil relevant information on statistical properties such as ergodicity breaking and short-time correlations.

Taming the diffusion approximation through a controlling-factor WKB method.

The diffusion approximation (DA) is widely used in the analysis of stochastic population dynamics, from population genetics to ecology and evolution. The DA is an uncontrolled approximation that assumes the smoothness of the calculated quantity over the relevant state space and fails when this property is not satisfied. This failure becomes severe in situations where the direction of selection switches sign. Here we employ the WKB (Wentzel-Kramers-Brillouin) large-deviations method, which requires only the logarithm of a given quantity to be smooth over its state space. Combining the WKB scheme with asymptotic matching techniques, we show how to derive the diffusion approximation in a controlled manner and how to produce better approximations, applicable for much wider regimes of parameters. We also introduce a scalable (independent of population size) WKB-based numerical technique. The method is applied to a central problem in population genetics and evolution, finding the chance of ultimate fixation in a zero-sum, two-types competition.

On Relations Between the Relative Entropy and χ2-Divergence, Generalizations and Applications.

This paper is focused on a study of integral relations between the relative entropy and the chi-squared divergence, which are two fundamental divergence measures in information theory and statistics, a study of the implications of these relations, their information-theoretic applications, and some generalizations pertaining to the rich class of f-divergences. Applications that are studied in this paper refer to lossless compression, the method of types and large deviations, strong data–processing inequalities, bounds on contraction coefficients and maximal correlation, and the convergence rate to stationarity of a type of discrete-time Markov chains.

Negative differential response in chemical reactions

Reaction currents in chemical networks usually increase when increasing their driving affinities. But far from equilibrium the opposite can also happen. We find that such negative differential response (NDR) occurs in reaction schemes of major biological relevance, namely, substrate inhibition and autocatalysis. We do so by deriving the full counting statistics of two minimal representative models using large deviation methods. We argue that NDR implies the existence of optimal affinities that maximize the robustness against environmental and intrinsic noise at intermediate values of dissipation. An analogous behavior is found in dissipative self-assembly, for which we identify the optimal working conditions set by NDR.

Large-Deviation Properties of Sequence Alignment of Correlated Sequences.

The significance of alignment scores of optimally aligned DNA sequences can be estimated through the score distribution of pairs of random sequences. It is necessary to obtain statistics for the relevant high-scoring tail of the distribution. For local alignments of iid drawn sequences it has already been shown that the often assumed Gumbel distribution does not hold in the distribution tail, but has to be corrected by a Gaussian factor. Real DNA sequences were observed to show long-range correlations within sequences, which are not correctly modeled by iid random sequences. In this publication the large deviation method that was used in previous studies is applied to local and global alignment of such sequences with long-range correlations. We study the distributions over the full range of the support and obtained probabilities as low as [Formula: see text]. We show that again a correction to the Gumbel distribution is necessary to study the dependence of the parameters on the correlation strength. For global alignments the Gamma distribution, which was found heuristically to be a good fit in earlier simple sampling studies, is found to be a poor fit.

Dynamical selection of Nash equilibria using reinforcement learning: Emergence of heterogeneous mixed equilibria.

We study the distribution of strategies in a large game that models how agents choose among different double auction markets. We classify the possible mean field Nash equilibria, which include potentially segregated states where an agent population can split into subpopulations adopting different strategies. As the game is aggregative, the actual equilibrium strategy distributions remain undetermined, however. We therefore compare with the results of a reinforcement learning dynamics inspired by Experience-Weighted Attraction (EWA) learning, which at long times leads to Nash equilibria in the appropriate limits of large intensity of choice, low noise (long agent memory) and perfect imputation of missing scores (fictitious play). The learning dynamics breaks the indeterminacy of the Nash equilibria. Non-trivially, depending on how the relevant limits are taken, more than one type of equilibrium can be selected. These include the standard homogeneous mixed and heterogeneous pure states, but also heterogeneous mixed states where different agents play different strategies that are not all pure. The analysis of the reinforcement learning involves Fokker-Planck modeling combined with large deviation methods. The theoretical results are confirmed by multi-agent simulations.

Orientational alignment in cavity quantum electrodynamics

We consider the orientational alignment of dipoles due to strong matter light coupling, for a non-vanishing density of excitations. We compare various approaches to this problem in the limit of large numbers of emitters, and show that direct Monte Carlo integration, mean-field theory, and large deviation methods match exactly in this limit. All three results show that orientational alignment develops in the presence of a macroscopically occupied polariton mode, and that the dipoles asymptotically approach perfect alignment in the limit of high density or low temperature.

Aspects of non-equilibrium in classical and quantum systems: Slow relaxation and glasses, dynamical large deviations, quantum non-ergodicity, and open quantum dynamics

In these four lectures I describe basic ideas and methods applicable to both classical and quantum systems displaying slow relaxation and non-equilibrium dynamics. The first half of these notes considers classical systems, and the second half, quantum systems. In Lecture 1, I briefly review the glass transition problem as a paradigm of slow relaxation and dynamical arrest in classical many-body systems. I discuss theoretical perspectives on how to think about glasses, and in particular how to model them in terms of kinetically constrained dynamics. In Lecture 2, I describe how via large deviation methods it is possible to define a statistical mechanics of trajectories which reveals the dynamical phase structure of systems with complex relaxation such as glasses. Lecture 3 is about closed (i.e. isolated) many-body quantum systems. I review thermalisation and many-body localisation, and consider the possibility of slow thermalisation and quantum non-ergodicity in the absence of disorder, thus connecting with some of the ideas of the first lecture. Lecture 4 is about open quantum systems, that is, quantum systems interacting with an environment. I review the description of open quantum dynamics within the Markovian approximation in terms of quantum master equations and stochastic quantum trajectories, and explain how to extend the dynamical large deviation method to study the statistical properties of ensembles of quantum jump trajectories. My overall aim is to draw analogies between classical and quantum non-equilibrium and find connections in the way we think about problems in these areas.

Design and Analysis of Transmit Beamforming for Millimeter Wave Base Station Discovery

In this paper, we develop an analytical framework for the initial access (also known as base station (BS) discovery) in a millimeter-wave communication system and propose an effective strategy for transmitting the reference signals (RSs) used for BS discovery. Specifically, by formulating the problem of BS discovery at user equipments (UEs) as hypothesis tests, we derive a detector based on the generalized likelihood ratio test and characterize the statistical behavior of the detector. The theoretical results obtained allow analysis of the impact of key system parameters on the performance of BS discovery, and show that RS transmission with narrow beams may not be helpful in improving the overall BS discovery performance due to the cost of spatial scanning. Using the method of large deviations, we identify the desirable beam pattern that minimizes the average miss-discovery probability of UEs within a targeted detectable region. We then propose to transmit the RS with sequential scanning, using a pre-designed codebook with narrow and/or wide beams to approximate the desirable patterns. The proposed design allows flexible choices of the codebook sizes and the associated beam widths to better approximate the desirable patterns. Numerical results demonstrate the effectiveness of the proposed method.

Millimeter Wave Beam Alignment: Large Deviations Analysis and Design Insights

In millimeter wave cellular communication, fast and reliable beam alignment via beam training is crucial to harvest sufficient beamforming gain for the subsequent data transmission. In this paper, we establish fundamental limits in beam-alignment performance under both the exhaustive search and the hierarchical search that adopts multi-resolution beamforming codebooks, accounting for time-domain training overhead. Specifically, we derive lower and upper bounds on the probability of misalignment for an arbitrary level in the hierarchical search, based on a single-path channel model. Using the method of large deviations, we characterize the decay rate functions of both bounds and show that the bounds coincide as the training sequence length goes large. We go on to characterize the asymptotic misalignment probability of both the hierarchical and exhaustive search, and show that the latter asymptotically outperforms the former, subject to the same training overhead and codebook resolution. We show via numerical results that this relative performance behavior holds in the non-asymptotic regime. Moreover, the exhaustive search is shown to achieve significantly higher worst-case spectrum efficiency than the hierarchical search, when the pre-beamforming signal-to-noise ratio (SNR) is relatively low. This study hence implies that the exhaustive search is more effective for users situated further from base stations, as they tend to have low SNR.

Rapid Mixing of Glauber Dynamics of Gibbs Ensembles via Aggregate Path Coupling and Large Deviations Methods

In this paper, we present a novel extension to the classical path coupling method to statistical mechanical models which we refer to as aggregate path coupling. In conjunction with large deviations estimates, we use this aggregate path coupling method to prove rapid mixing of Glauber dynamics for a large class of statistical mechanical models, including models that exhibit discontinuous phase transitions which have traditionally been more difficult to analyze rigorously. The parameter region for rapid mixing for the generalized Curie-Weiss-Potts model is derived as a new application of the aggregate path coupling method.

Rare-event trajectory ensemble approach to study dynamical phase transitions in the zero temperature Glauber model

The dynamics of a one-dimensional stochastic system of classical particles consisting of asymmetric death and branching processes is studied. The dynamical activity, defined as the number of configuration changes in a dynamical trajectory, is considered as a proper dynamical order parameter. By considering an ensemble of dynamical trajectories and applying the large deviation method, we have found that the system might undergo both continuous and discontinuous dynamical phase transitions at critical values of the counting field. Exact analytical results are obtained for an infinite system. Numerical investigations confirm our analytical calculations.

Moderate deviations for interacting processes

This article is concerned with moderate deviation principles of a class of interacting empirical processes. We derive an explicit description of the rate function, and we illustrate these results with Feynman-Kac particle models arising in nonlinear filtering, statistical machine learning, rare event analysis, and computational physics. We discuss functional moderate deviations of the occupation measures for both the strong τ -topology on the space of finite and bounded measures as well as for the corresponding stochastic processes on some class of functions equipped with the uniform topology, yielding the first results of this type for mean field interacting processes. Our approach is based on an original semigroup analysis combined with Orlicz norm inequalities, stochastic perturbation techniques, and projective limit large deviation methods.

Meta-work and the analogous Jarzynski relation in ensembles of dynamical trajectories

Recently there has been growing interest in extending the thermodynamic method from static configurations to dynamical trajectories. In this approach, ensembles of trajectories are treated in an analogous manner to ensembles of configurations in equilibrium statistical mechanics: generating functions of dynamical observables are interpreted as partition sums, and the statistical properties of trajectory ensembles are encoded in free-energy functions that can be obtained through large-deviation methods in a suitable large time limit. This establishes what one can call a ‘thermodynamics of trajectories’. In this paper we go a step further, and make a first connection to fluctuation theorems by generalising them to this dynamical context. We show that an effective ‘meta-dynamics’ in the space of trajectories gives rise to the celebrated Jarzynski relation connecting an appropriately defined ‘meta-work’ with changes in dynamical generating functions. We demonstrate the potential applicability of this result to computer simulations for two open quantum systems, a two-level system and the micromaser. We finally discuss the behavior of the Jarzynski relation across a first-order trajectory phase transition.

Elements of a unified framework for response formulae

We provide a physical interpretation of the first and second order terms occurring in Ruelle’s response formalism. We show that entropy fluxes play a major role in determining the response of the system to perturbations. Along this line, we show that our framework allows one to recover a wealth of previous results of response theory in both deterministic and stochastic contexts. In particular, we are able to shed light on the crosslinks between the dynamical systems approach á la Ruelle and large deviations methods.

SINR Statistics of Correlated MIMO Linear Receivers

Linear receivers offer a low complexity option for multiantenna communication systems. Therefore, understanding the outage behavior of the corresponding SINR is important in a fading mobile environment. In this paper, we introduce a large deviation method, valid nominally for a large number M of antennas, which provides the probability density of the SINR of Gaussian channel MIMO minimum mean square error (MMSE) and zero-forcing (ZF) receivers, with arbitrary transmission power profiles and in the presence of receiver antenna correlations. This approach extends the Gaussian approximation of the SINR, valid for large M asymptotically close to the center of the distribution, to obtain the non-Gaussian tails of the distribution. Our methodology allows us to calculate the SINR distribution to next-to-leading order ( O(1/M)) and showcase the deviations from approximations that have appeared in the literature (e.g., the Gaussian or the generalized Gamma distribution). We also analytically evaluate the outage probability, as well as the uncoded bit-error-rate. We find that our approximation is quite accurate even for the smallest antenna arrays (2 × 2).