# Observable dictionary learning for high-dimensional statistical inference

## TL;DR

This paper presents a novel observable dictionary learning method for high-dimensional inference, enabling efficient and accurate estimation of quantities from limited observations using a Bayesian framework and tailored dictionaries.

## Contribution

It introduces a new observable dictionary learning approach that aligns the basis with the observation chain for improved inference accuracy.

## Key findings

- Outperforms PCA and K-SVD in velocity field estimation
- Provides closed-form posterior distribution for Bayesian inference
- Demonstrates effectiveness on cavity flow velocity estimation

## Abstract

This paper introduces a method for efficiently inferring a high-dimensional distributed quantity from a few observations. The quantity of interest (QoI) is approximated in a basis (dictionary) learned from a training set. The coefficients associated with the approximation of the QoI in the basis are determined by minimizing the misfit with the observations. To obtain a probabilistic estimate of the quantity of interest, a Bayesian approach is employed. The QoI is treated as a random field endowed with a hierarchical prior distribution so that closed-form expressions can be obtained for the posterior distribution. The main contribution of the present work lies in the derivation of \emph{a representation basis consistent with the observation chain} used to infer the associated coefficients. The resulting dictionary is then tailored to be both observable by the sensors and accurate in approximating the posterior mean. An algorithm for deriving such an observable dictionary is presented. The method is illustrated with the estimation of the velocity field of an open cavity flow from a handful of wall-mounted point sensors. Comparison with standard estimation approaches relying on Principal Component Analysis and K-SVD dictionaries is provided and illustrates the superior performance of the present approach.