5,063 research outputs found
Convergence Rates for Learning Linear Operators from Noisy Data
This paper studies the learning of linear operators between
infinite-dimensional Hilbert spaces. The training data comprises pairs of
random input vectors in a Hilbert space and their noisy images under an unknown
self-adjoint linear operator. Assuming that the operator is diagonalizable in a
known basis, this work solves the equivalent inverse problem of estimating the
operator's eigenvalues given the data. Adopting a Bayesian approach, the
theoretical analysis establishes posterior contraction rates in the infinite
data limit with Gaussian priors that are not directly linked to the forward map
of the inverse problem. The main results also include learning-theoretic
generalization error guarantees for a wide range of distribution shifts. These
convergence rates quantify the effects of data smoothness and true eigenvalue
decay or growth, for compact or unbounded operators, respectively, on sample
complexity. Numerical evidence supports the theory in diagonal and non-diagonal
settings.Comment: To appear in SIAM/ASA Journal on Uncertainty Quantification (JUQ); 34
pages, 5 figures, 2 table
Nonparametric Uncertainty Quantification for Stochastic Gradient Flows
This paper presents a nonparametric statistical modeling method for
quantifying uncertainty in stochastic gradient systems with isotropic
diffusion. The central idea is to apply the diffusion maps algorithm to a
training data set to produce a stochastic matrix whose generator is a discrete
approximation to the backward Kolmogorov operator of the underlying dynamics.
The eigenvectors of this stochastic matrix, which we will refer to as the
diffusion coordinates, are discrete approximations to the eigenfunctions of the
Kolmogorov operator and form an orthonormal basis for functions defined on the
data set. Using this basis, we consider the projection of three uncertainty
quantification (UQ) problems (prediction, filtering, and response) into the
diffusion coordinates. In these coordinates, the nonlinear prediction and
response problems reduce to solving systems of infinite-dimensional linear
ordinary differential equations. Similarly, the continuous-time nonlinear
filtering problem reduces to solving a system of infinite-dimensional linear
stochastic differential equations. Solving the UQ problems then reduces to
solving the corresponding truncated linear systems in finitely many diffusion
coordinates. By solving these systems we give a model-free algorithm for UQ on
gradient flow systems with isotropic diffusion. We numerically verify these
algorithms on a 1-dimensional linear gradient flow system where the analytic
solutions of the UQ problems are known. We also apply the algorithm to a
chaotically forced nonlinear gradient flow system which is known to be well
approximated as a stochastically forced gradient flow.Comment: Find the associated videos at: http://personal.psu.edu/thb11
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