Smooth transforms of Gaussian noise

Taylor expansion

Not complicated but it can be a little subtle. For a general exposition which handles first and second-order transforms in context of all the other “easy“ transforms, I recommend Gustafsson and Hendeby (2012), which as a bonus proves some things which look obvious are not, in fact, obvious to prove, and disproves some things which looked obvious to me. Arras (1998) is also good.

Consider a general nonlinear differentiable transformation \(g\) and its second order Taylor expansion. We apply \(g:\mathbb{R}^{n_{x}}\to\mathbb{R}^{n_{z}}\) to a variable \(x,\) defining \(z:=g(x).\) Let \(\mathrm{E}(x)=\mu_{x}\) and \(\operatorname{Var}(x)=P_{x}.\) The Hessian of the \(i^{\text {th }}\) component of \(g\) is denoted \(g_{i}^{\prime \prime}.\) \([x_i]_i\) is a vector where the \(i\)th element is \(x_i\). We will approximate \(z\) using the Taylor expansion, \[z=g\left(\mu_{x}\right)+g^{\prime}\left(\mu_{x}\right)\left(x-\mu_{x}\right)+\left[\frac{1}{2}\left(x-\mu_{x}\right)^{T} g_{i}^{\prime \prime}\left(\mu_{x}\right)\left(x-\mu_{x}\right)\right]_{i}.\] Leaving aside questions of when this is convergent for now, and assume it is. Then we assert \(z\sim\mathcal{N}(\mu_z,P_z)\). The first moment of \(z\) is given by \[ \mu_{z}=g\left(\mu_{x}\right)+\frac{1}{2}\left[\operatorname{tr}\left(g_{i}^{\prime \prime}\left(\mu_{x}\right) P_{x}\right)\right]_{i} \] Further, let \(x \sim \mathcal{N}\left(\mu_{x}, P_{x}\right)\), then the second moment of \(z\) is given by \[ P_{z}=g^{\prime}\left(\mu_{x}\right) P_{x}\left(g^{\prime}\left(\mu_{x}\right)\right)^{T}+\frac{1}{2}\left[\operatorname{tr}\left(g_{i}^{\prime \prime}\left(\mu_{x}\right) P_{x} g_{j}^{\prime \prime}\left(\mu_{x}\right) P_{x}\right)\right]_{i j} \] with \(i, j=1, \ldots, n_{z}.\)

This approach is finite dimensional, but it also generalises to Gaussian processes, in that we can, at any finite number of test locations, once again find a first order approximation. See the [non-parametric case][#nonparametric].

Note that here I have assumed that we have the luxury of expanding the distribution about the mean, in which case I would probably usually only take a first order Taylor transform. Since I have a second-order expansion here, I should give the expansion about an arbitrary point which is not necessarily the mean, for the sake of making the generality worth it.

Unscented transform

The great invention of Uhlmann and Julier is the unscented transform, which uses a cunningly-chosen non-random empirical sample at so-called \(\sigma\)-points to approximate the transformed distribution via its moments. I think that anything using sigma points is an unscented transform? Otherwise it is just garden-variety moment-matching.

Mostly seen in the context of Kalman filtering.

What the Unscented Transform does is to replace the mean vector and its associated error covariance matrix with a special set of points with the same mean and covariance. In the case of the mean and covariance representing the current position estimate for a target, the UT is applied to obtain a set of points, referred to as sigma points, to which the full nonlinear equations of motion can be applied directly. In other words, instead of having to derive a linearized approximation, the equations could simply be applied to each of the points as if it were the true state of the target. The result is a transformed set of points, and the mean and covariance of that set represents the estimate of the predicted state of the target.

See, e.g., Roth, Hendeby, and Gustafsson (2016) and a comparison with the Taylor expansion in Gustafsson and Hendeby (2012).

Question: What would we need to do to apply the unscented transform to non-Gaussian distributions? See Ebeigbe et al. (2021).

• Unscented transform - Wikipedia
• The Unscented Transform
• Unscented Kalman Filter - fjp.github.io

Gaussian processes

Propagating error through Gaussian process inputs. See Emmanuel Johnson’s Linearized GPsite (mildly idiosyncratic notation and very idiosyncratic website navigation) and Arras (1998). The following references from Emmanuel Johnson’s lit review look promising: Deisenroth and Mohamed (2012); Girard and Murray-Smith (2003); Ko and Fox (2009) and McHutchon and Rasmussen (2011).

Incoming

• Byron Boots’ lecture on the extended Kalman filter.
• Robert Grosse’s lectures on CSC2541 Topics in Machine Learning: Neural Net Training Dynamics includes IMO the best introduction to applied Taylor expansions in arbitrary dimension here.

References