# Telescoping Recursive Representations and Estimation of Gauss-Markov Random Fields

## TL;DR

This paper introduces telescoping recursive representations for noncausal Gauss-Markov random fields, enabling efficient estimation algorithms that extend classical filters to higher-dimensional, noncausal settings.

## Contribution

It develops a novel telescoping recursive framework for noncausal Gauss-Markov random fields, simplifying multi-dimensional recursions to one dimension and deriving extended estimation algorithms.

## Key findings

- Recursions reduce from multi-dimensional to single dimension.
- Derived recursive estimation algorithms extend Kalman-Bucy and RTS smoothers.
- Applicable to both continuous and discrete noncausal Gaussian Markov fields.

## Abstract

We present \emph{telescoping} recursive representations for both continuous and discrete indexed noncausal Gauss-Markov random fields. Our recursions start at the boundary (a hypersurface in $\R^d$, $d \ge 1$) and telescope inwards. For example, for images, the telescoping representation reduce recursions from $d = 2$ to $d = 1$, i.e., to recursions on a single dimension. Under appropriate conditions, the recursions for the random field are linear stochastic differential/difference equations driven by white noise, for which we derive recursive estimation algorithms, that extend standard algorithms, like the Kalman-Bucy filter and the Rauch-Tung-Striebel smoother, to noncausal Markov random fields.