Reconstruction of sequential data with density models

TL;DR

This paper presents a probabilistic approach to reconstruct missing data in sequences of multidimensional vectors, leveraging low-dimensional manifold assumptions and sequence continuity, with applications demonstrated in toy problems and robot inverse kinematics.

Contribution

It introduces a novel algorithm combining density models and dynamic programming to reconstruct sequences with missing data, addressing multivalued mappings and variable missing patterns.

Findings

Effective reconstruction in toy and robot kinematics problems.

Utilizes Gaussian mixture models for probabilistic inference.

Demonstrates robustness to complex missing data patterns.

Abstract

We introduce the problem of reconstructing a sequence of multidimensional real vectors where some of the data are missing. This problem contains regression and mapping inversion as particular cases where the pattern of missing data is independent of the sequence index. The problem is hard because it involves possibly multivalued mappings at each vector in the sequence, where the missing variables can take more than one value given the present variables; and the set of missing variables can vary from one vector to the next. To solve this problem, we propose an algorithm based on two redundancy assumptions: vector redundancy (the data live in a low-dimensional manifold), so that the present variables constrain the missing ones; and sequence redundancy (e.g. continuity), so that consecutive vectors constrain each other. We capture the low-dimensional nature of the data in a probabilistic way with a joint density model, here the generative topographic mapping, which results in a Gaussian mixture. Candidate reconstructions at each vector are obtained as all the modes of the conditional distribution of missing variables given present variables. The reconstructed sequence is obtained by minimising a global constraint, here the sequence length, by dynamic programming. We present experimental results for a toy problem and for inverse kinematics of a robot arm.