7 research outputs found
System Identification for Continuous-time Linear Dynamical Systems
The problem of system identification for the Kalman filter, relying on the
expectation-maximization (EM) procedure to learn the underlying parameters of a
dynamical system, has largely been studied assuming that observations are
sampled at equally-spaced time points. However, in many applications this is a
restrictive and unrealistic assumption. This paper addresses system
identification for the continuous-discrete filter, with the aim of generalizing
learning for the Kalman filter by relying on a solution to a continuous-time
It\^o stochastic differential equation (SDE) for the latent state and
covariance dynamics. We introduce a novel two-filter, analytical form for the
posterior with a Bayesian derivation, which yields analytical updates which do
not require the forward-pass to be pre-computed. Using this analytical and
efficient computation of the posterior, we provide an EM procedure which
estimates the parameters of the SDE, naturally incorporating irregularly
sampled measurements. Generalizing the learning of latent linear dynamical
systems (LDS) to continuous-time may extend the use of the hybrid Kalman filter
to data which is not regularly sampled or has intermittent missing values, and
can extend the power of non-linear system identification methods such as
switching LDS (SLDS), which rely on EM for the linear discrete-time Kalman
filter as a sub-unit for learning locally linearized behavior of a non-linear
system. We apply the method by learning the parameters of a latent,
multivariate Fokker-Planck SDE representing a toggle-switch genetic circuit
using biologically realistic parameters, and compare the efficacy of learning
relative to the discrete-time Kalman filter as the step-size irregularity and
spectral-radius of the dynamics-matrix increases.Comment: 31 pages, 3 figures. Only light changes and restructuring to previous
version mad
Support Recovery for the Drift Coefficient of High-Dimensional Diffusions
Abstract-Consider the problem of learning the drift coefficient of a p-dimensional stochastic differential equation from a sample path of length T . We assume that the drift is parametrized by a high-dimensional vector, and study the support recovery problem when both p and T can tend to infinity. In particular, we prove a general lower bound on the sample-complexity T by using a characterization of mutual information as a time integral of conditional variance, due to Kadota, Zakai, and Ziv. For linear stochastic differential equations, the drift coefficient is parametrized by a p × p matrix which describes which degrees of freedom interact under the dynamics. In this case, we analyze a 1-regularized least squares estimator and prove an upper bound on T that nearly matches the lower bound on specific classes of sparse matrices
Information Theoretic Limits on Learning Stochastic Differential Equations
Consider the problem of learning the drift coefficient of a stochastic differential equation from a sample path. In this paper, we assume that the drift is parametrized by a high dimensional vector. We address the question of how long the system needs to be observed in order to learn this vector of parameters. We prove a general lower bound on this time complexity by using a characterization of mutual information as time integral of conditional variance, due to Kadota, Zakai, and Ziv. This general lower bound is applied to specific classes of linear and non-linear stochastic differential equations. In the linear case, the problem under consideration is the one of learning a matrix of interaction coefficients. We evaluate our lower bound for ensembles of sparse and dense random matrices. The resulting estimates match the qualitative behavior of upper bounds achieved by computationally efficient procedures.