Small Noise Regime Research Articles

The most likely transition path for a class of distribution-dependent stochastic systems

Distribution-dependent stochastic dynamical systems arise widely in engineering and science. We consider a class of such systems which model the limit behaviors of interacting particles moving in a vector field with random fluctuations. We aim to examine the most likely transition path between equilibrium stable states of the vector field. In the small noise regime, the action functional does not involve the solution of the skeleton equation which describes the unperturbed deterministic flow of the vector field shifted by the interaction at zero distance. As a result, we are led to study the most likely transition path for a stochastic differential equation without distribution dependency. This enables the computation of the most likely transition path for these distribution-dependent stochastic dynamical systems by the adaptive minimum action method and we illustrate our approach in two examples.

Large deviations for Lévy diffusions in the small noise regime

This paper concerns the large deviations regime and the consequent solution of the Kramers problem for a two-time scale stochastic system driven by a common jump noise signal perturbed in small intensity [Formula: see text] and with accelerated jumps by intensity [Formula: see text]. We establish Freidlin–Wentzell estimates for the slow process of the multiscale system in the small noise limit [Formula: see text] using the weak convergence approach to large deviations theory. The core of our proof is the reduction of the large deviations principle to the establishment of a stochastic averaging principle for auxiliary controlled processes. As consequence we solve the first exit time/exit locus problem from a bounded domain containing the stable state of the averaged dynamics for the family of the slow processes in the small noise limit.

Computing high-dimensional invariant distributions from noisy data

The invariant distribution is an important object in the study of randomly perturbed dynamical systems. The existing methods, including traditional finite difference or finite element methods as well as recently developed machine learning-based methods, all require the knowledge of the dynamical equations or adequate equilibrium data for estimating the invariant distribution. In this work, we propose a data-driven method for inferring the dynamics and simultaneously learning the invariant distribution from noisy trajectory data of the dynamical system. The data is not necessarily at equilibrium and may be collected from the transient period of the dynamics. The proposed method combines the idea of maximum likelihood estimation and a decomposition of the force field as suggested by the Fokker-Planck equation. The drift term (or force field), which is in the form of a decomposition with the constraint specified by the Fokker-Planck equation, and the diffusion term are learned from the data using an alternate updating algorithm. The generalized potential, which is the negative logarithm of the invariant distribution multiplied by the noise, is then obtained from the potential part of the decomposition. The proposed method is able to deal with high-dimensional dynamical systems and the small noise regime, without prior knowledge of the dynamical equations. The proposed method is demonstrated by four numerical examples, including a practical 10-dimensional biological network model.

Entropic optimal transport: convergence of potentials

We study the potential functions that determine the optimal density for \(\varepsilon \)-entropically regularized optimal transport, the so-called Schrödinger potentials, and their convergence to the counterparts in classical optimal transport, the Kantorovich potentials. In the limit \(\varepsilon \rightarrow 0\) of vanishing regularization, strong compactness holds in \(L^{1}\) and cluster points are Kantorovich potentials. In particular, the Schrödinger potentials converge in \(L^{1}\) to the Kantorovich potentials as soon as the latter are unique. These results are proved for all continuous, integrable cost functions on Polish spaces. In the language of Schrödinger bridges, the limit corresponds to the small-noise regime.

Resolving molecular contributions of ion channel noise to interspike interval variability through stochastic shielding.

Molecular fluctuations can lead to macroscopically observable effects. The random gating of ion channels in the membrane of a nerve cell provides an important example. The contributions of independent noise sources to the variability of action potential timing have not previously been studied at the level of molecular transitions within a conductance-based model ion-state graph. Here we study a stochastic Langevin model for the Hodgkin-Huxley (HH) system based on a detailed representation of the underlying channel state Markov process, the "[Formula: see text]D model" introduced in (Pu and Thomas in Neural Computation 32(10):1775-1835, 2020). We show how to resolve the individual contributions that each transition in the ion channel graph makes to the variance of the interspike interval (ISI). We extend the mean return time (MRT) phase reduction developed in (Cao et al. in SIAM JApplMath 80(1):422-447, 2020) to the second moment of the return time from an MRT isochron to itself. Because fixed-voltage spike detection triggers do not correspond to MRT isochrons, the inter-phase interval (IPI) variance only approximates the ISI variance. We find the IPI variance and ISI variance agree to within a few percent when both can be computed. Moreover, we prove rigorously, and show numerically, that our expression for the IPI variance is accurate in the small noise (large system size) regime; our theory is exact in the limit of small noise. By selectively including the noise associated with only those few transitions responsible for most of the ISI variance, our analysis extends the stochastic shielding (SS) paradigm (Schmandt and Galán in PhysRevLett 109(11):118101, 2012) from the stationary voltage clamp case to the current clamp case. We show numerically that the SS approximation has a high degree of accuracy even for larger, physiologically relevant noise levels. Finally, we demonstrate that the ISI variance is not an unambiguously defined quantity, but depends on the choice of voltage level set as the spike detection threshold. We find a small but significant increase in ISI variance, the higher the spike detection voltage, both for simulated stochastic HH data and for voltage traces recorded in in vitro experiments. In contrast, the IPI variance is invariant with respect to the choice of isochron used as a trigger for counting "spikes."

A formula for the time derivative of the entropic cost and applications

In the recent years the Schrödinger problem has gained a lot of attention because of the connection, in the small-noise regime, with the Monge-Kantorovich optimal transport problem. Its optimal value, the entropic costCT, is here deeply investigated. In this paper we study the regularity of CT with respect to the parameter T under a curvature condition and explicitly compute its first and second derivative. As applications:-we determine the large-time limit of CT and provide sharp exponential convergence rates; we obtain this result not only for the classical Schrödinger problem but also for the recently introduced Mean Field Schrödinger problem [3];-we improve the Taylor expansion of T↦TCT around T=0 from the first to the second order.

Super-Resolution Limit of the ESPRIT Algorithm

The problem of imaging point objects can be formulated as estimation of an unknown atomic measure from its $M+1$ consecutive noisy Fourier coefficients. The standard resolution of this inverse problem is $1/M$ and super-resolution refers to the capability of resolving atoms at a higher resolution. When any two atoms are less than $1/M$ apart, this recovery problem is highly challenging and many existing algorithms either cannot deal with this situation or require restrictive assumptions on the sign of the measure. ESPRIT is an efficient method that does not depend on the sign of the measure. This paper provides an explicit error bound on the support matching distance of ESPRIT in terms of the minimum singular value of Vandermonde matrices. When the support consists of multiple well-separated clumps and noise is sufficiently small, the support error by ESPRIT scales like ${\rm SRF}^{2\lambda+2} \times {\rm Noise}$, where the Super-Resolution Factor (${\rm SRF}$) governs the difficulty of the problem and $\lambda$ is the cardinality of the largest clump. {If the support contains one clump of closely spaced atoms, the min-max error is ${\rm SRF}^{2\lambda +2} \times {\rm Noise}/M$. Our error bound matches the min-max rate up to a factor of $M$ in the small noise regime. Our results therefore establishes the near-optimality of ESPRIT,} and our theory is validated by numerical experiments.

Large Deviation Principles for Stochastic Volatility Models with Reflection and Three Faces of the Stein and Stein Model

We introduce stochastic volatility models, in which the volatility is described by a time-dependent nonnegative function of a reflecting diffusion. The idea to use reflecting diffusions as building blocks of the volatility came into being because of a certain volatility misspecification in the classical Stein and Stein model. A version of this model that uses the reflecting Ornstein-Uhlenbeck process as the volatility process is a special example of a stochastic volatility model with reflection. The main results obtained in the present paper are sample path and small-noise large deviation principles for the log-price process in a stochastic volatility model with reflection under rather mild restrictions. We use these results to study the asymptotic behavior of binary barrier options and call prices in the small-noise regime.

Fisher information in Poissonian model neurons

Mutual information (MI) is being widely used to analyze the neural code in a variety of stochastic neuronal sensory systems. Unfortunately, MI is analytically tractable only for simple coding problems. One way to address this difficulty is to relate MI to Fisher information which is relatively easier to compute and interpret with regard to neurophysiological parameters. The relationship between the two measures is not always clear and often depends on the probability distribution function that best describes the distribution of the noise. Using Stam’s inequality we show here that deviations from Gaussianity in neuronal response distribution function can result in a large overestimation of MI, even in the small noise regime. This result is especially relevant when studying neural codes represented by Poissonian neurons.

Mean field games in the weak noise limit : A WKB approach to the Fokker–Planck equation

Motivated by the study of a Mean Field Game toy model called the “seminar problem”, we consider the Fokker–Planck equation in the small noise regime for a specific drift field. This gives us the opportunity to discuss the application to diffusion problem of the WKB approach “à la Maslov (Maslov and Fedoriuk, 1981)”, making it possible to solve directly the time dependent problem in an especially transparent way.

Sparse regularization on thin grids I: the Lasso

This article analyzes the recovery performance in the presence of noise of sparse regularization, which is often referred to as the Lasso or Basis-Pursuit. We study the behavior of the method for inverse problems regularization when the discretization step size tends to zero. We assume that the sought after sparse sum of Diracs is recovered when there is no noise (a condition which has been thoroughly studied in the literature) and we study what is the support (in particular the number of Dirac masses) estimated by the Lasso when noise is added to the observation. We identify a precise non-degeneracy condition that guarantees that the recovered support is close to the initial one. More precisely, we show that, in the small noise regime, when the non-degeneracy condition holds, this method estimates twice the number of spikes as the number of original spikes. Indeed, we prove that the Lasso detects two neighboring spikes around each location of an original spike. While this paper is focussed on cases where the observations vary smoothly with the spikes locations (e.g. the deconvolution problem with a smooth kernel), an interesting by-product is an abstract analysis of the support stability of discrete regularization, which is of an independent interest. We illustrate the usefulness of this abstract analysis to analyze for the first time the support instability of compressed sensing recovery.

Gaussian approximations of small noise diffusions in Kullback–Leibler divergence

We study Gaussian approximations to the distribution of a diffusion. The approximations are easy to compute: they are defined by two simple ordinary differential equations for the mean and the covariance. Time correlations can also be computed via solution of a linear stochastic differential equation. We show, using the Kullback-Leibler divergence, that the approximations are accurate in the small noise regime. An analogous discrete time setting is also studied. The results provide both theoretical support for the use of Gaussian processes in the approximation of diffusions, and methodological guidance in the construction of Gaussian approximations in applications.

Coinfection in a stochastic model for bacteriophage systems

A system modeling bacteriophage treatments with coinfections in a noisy context is analysed. We prove that in a small noise regime, the system converges in the long term to a bacteria-free equilibrium. Moreover, we compare the treatment with coinfection with the treatment without coinfection, showing how coinfection affects the convergence to the bacteria-free equilibrium.

Model reduction algorithms for optimal control and importance sampling of diffusions

We propose numerical algorithms for solving optimal control and importance sampling problems based on simplified models. The algorithms combine model reduction techniques for multiscale diffusions and stochastic optimization tools, with the aim of reducing the original, possibly high-dimensional problem to a lower dimensional representation of the dynamics, in which only a few relevant degrees of freedom are controlled or biased. Specifically, we study situations in which either a reaction coordinate onto which the dynamics can be projected is known, or situations in which the dynamics shows strongly localized behavior in the small noise regime. No explicit assumptions about small parameters or scale separation have to be made. We illustrate the approach with simple, but paradigmatic numerical examples.

Generalisation of the Eyring–Kramers Transition Rate Formula to Irreversible Diffusion Processes

In the small noise regime, the average transition time between metastable states of a reversible diffusion process is described at the logarithmic scale by Arrhenius' law. The Eyring-Kramers formula classically provides a subexponential prefactor to this large deviation estimate. For irreversible diffusion processes, the equivalent of Arrhenius' law is given by the Freidlin-Wentzell theory. In this paper, we compute the associated prefactor and thereby generalise the Eyring-Kramers formula to irreversible diffusion processes. In our formula, the role of the potential is played by Freidlin-Wentzell's quasipotential, and a correction depending on the non-Gibbsianness of the system along the instanton is highlighted. Our analysis relies on a WKB analysis of the quasistationary distribution of the process in metastable regions, and on a probabilistic study of the process in the neighbourhood of saddle-points of the quasipotential.

Multilevel Monte Carlo for Stochastic Differential Equations with Small Noise

We consider the problem of numerically estimating expectations of solutions to stochastic differential equations driven by Brownian motions in the commonly occurring small noise regime. We consider (i) standard Monte Carlo methods combined with numerical discretization algorithms tailored to the small noise setting, and (ii) a multilevel Monte Carlo method combined with a standard Euler-Maruyama implementation. Under the assumptions we make on the underlying model, the multilevel method combined with Euler-Maruyama is often found to be the most efficient option. Moreover, under a wide range of scalings the multilevel method is found to give the same asymptotic complexity that would arise in the idealized case where we have access to exact samples of the required distribution at a cost of $O(1)$ per sample. A key step in our analysis is to analyze the variance between two coupled paths directly, as opposed to their $L^2$ distance. Careful simulations are provided to illustrate the asymptotic results.

Mutual Information, Fisher Information, and Efficient Coding

Fisher information is generally believed to represent a lower bound on mutual information (Brunel & Nadal, 1998), a result that is frequently used in the assessment of neural coding efficiency. However, we demonstrate that the relation between these two quantities is more nuanced than previously thought. For example, we find that in the small noise regime, Fisher information actually provides an upper bound on mutual information. Generally our results show that it is more appropriate to consider Fisher information as an approximation rather than a bound on mutual information. We analytically derive the correspondence between the two quantities and the conditions under which the approximation is good. Our results have implications for neural coding theories and the link between neural population coding and psychophysically measurable behavior. Specifically, they allow us to formulate the efficient coding problem of maximizing mutual information between a stimulus variable and the response of a neural population in terms of Fisher information. We derive a signature of efficient coding expressed as the correspondence between the population Fisher information and the distribution of the stimulus variable. The signature is more general than previously proposed solutions that rely on specific assumptions about the neural tuning characteristics. We demonstrate that it can explain measured tuning characteristics of cortical neural populations that do not agree with previous models of efficient coding.

Small‐Noise Analysis and Symmetrization of Implicit Monte Carlo Samplers

AbstractImplicit samplers are algorithms for producing independent, weighted samples from multivariate probability distributions. These are often applied in Bayesian data assimilation algorithms. We use Laplace asymptotic expansions to analyze two implicit samplers in the small noise regime. Our analysis suggests a symmetrization of the algorithms that leads to improved implicit sampling schemes at a relatively small additional cost. Computational experiments confirm the theory and show that symmetrization is effective for small noise sampling problems.© 2016 Wiley Periodicals, Inc.

Low noise regimes for ℓ regularization : continuous and discrete settings

AbstractWe focus on support recovery for signal deconvolution with sparsity assumption. We adopt the continuous setting defined by several recent works and we try to reconstruct a sum of Dirac masses from its low frequencies (possibly perturbed by some noise), by using a total variation prior for Radon measures (i.e. the generalization to measures of the ℓ1 norm). We show that, under a non degenerate source condition, there exists a small noise regime in which the model recovers exactly the same number of spikes as the original signal, and the spikes converge to those of the original signal as the noise vanishes. This continuous setting, by allowing the spikes to “move”, provides robust support recovery for signals composed of well separated spikes.In a discrete setting, where the spikes are reconstructed on a grid, similar low noise regimes which guarantee the exact recovery of the support also exist (see [3]). Yet, this property only concerns a small class of signals. Considering the asymptotics of the discrete problems as the size of the grid tends to zero, we show that the support of the original signal cannot be stable on thin grids, and that the discrete models actually reconstruct pairs of spikes near each original spike. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Gregarious versus individualistic behavior in Vicsek swarms and the onset of first-order phase transitions

The standard Vicsek model (SVM) is a minimal non-equilibrium model of self-propelled particles that appears to capture the essential ingredients of critical flocking phenomena. In the SVM, particles tend to align with each other and form ordered flocks of collective motion; however, perturbations controlled by a noise term lead to a noise-driven continuous order–disorder phase transition. In this work, we extend the SVM by introducing a parameter α that allows particles to be individualistic instead of gregarious, i.e. to choose a direction of motion independently of their neighbors. By focusing on the small-noise regime, we show that a relatively small probability of individualistic motion (around 10%) is sufficient to drive the system from a Vicsek-like ordered phase to a disordered phase. Despite the fact that the α-extended model preserves the O(n) symmetry and the interaction range, as well as the dimensionality of the underlying SVM, this novel phase transition is found to be discontinuous (first order), an intriguing manifestation of the richness of the non-equilibrium flocking/swarming phenomenon.