(369i) Multiple Experts: Using the Mahalanobis Metric to Fuse Data from Different Partial Observations

Title: Multiple Experts: Using the Mahalanobis metric to recover a unified embedding of data from varied observables.

Authors: David Sroczynski, Felix Dietrich, Mahdi Kooshkbaghi, and Yannis Kevrekidis

Session: CAST, 10D02 Advances in Computational Methods and Numerical Analysis

Our understanding of physical processes can be greatly advanced by informative low-dimensional embeddings of high-dimensional data, such as those arising from molecular dynamics simulations or stochastic simulations of chemically reacting systems. Here we consider the problem of combining multiple data sets, each arising from different partial observations of the same system. We can imagine different “experts” observing the system in different ways, but each expert only sees part of the data. If we can estimate the Jacobian of our experts’ observations with respect to appropriately discovered intrinsic system states (e.g., from the covariance of short noisy simulation bursts at each data point), then the so-called Mahalanobis metric^[1,2], can filter out the effect of different observation functions, allowing us to estimate distances between data points in an intrinsic low-dimensional embedding of the system. We demonstrate our approach on stochastic simulation data for an enzyme reaction network with multiple time scales. This framework does not require that any expert sees all the data, nor that any data point be seen by all the experts. We also present results using alternating diffusion maps^[3], a recently developed manifold learning technique, to learn the evolution of a system from simultaneous measurements from two different sensors (different "experts").

[1] A. Singer and R.R. Coifman. Non-linear independent component analysis with diffusion maps. Applied and Computational Harmonic Analysis, 25(2):226-239, 2008.

[2] C. J. Dsilva, R. Talmon, N. Rabin, R.R. Coifman, and I.G. Kevrekidis. Nonlinear intrinsic variables and state reconstruction in multiscale simulations. J. Chem. Phys, 139, 184109, 2013.

[3] R.R. Lederman and R. Talmon, Learning the geometry of common latent variables using alternating-diffusion. Applied and Computational Harmonic Analysis, 44(3):509-536, 2018.