Min, Max and Plus nodes in RxInfer

[apiedrafTNO]

Hi,
I would like to use RxInfer to model a dataflow graph with stochastic timings. The basic blocks that I would need would be nodes that can take any number of variables $$X_i \sim N(\mu_i ,\sigma_i) for i=1\dots N$$ and approximate the distributions of $$Z_1=Max(X_1,\dots,X_N)$$
$$Z_2=Sum(X_1,\dots,X_N)$$ and $$Z_3=Min(X_1,\dots,X_N)$$. The sum node I have seen for sure, but not the other ones.
I've seen that max-plus algorithm for computing deterministic timings likened to MAP estimation here where all the messages are point mass distributions, but simple MAP doesn't work when the variables are not point mass distributions. Is there a way to do the general case using variational message passing in RxInfer?

Cheers,
Alvaro

[bartvanerp]

Hi Alvaro!

Interesting question! I must honestly say that I am a bit unfamiliar with the topic you are describing, but will try to give it a shot anyway. Before diving deep into the topic, you are describing three types of nodes. As you mention the plus node can be created by chaining addition nodes as z2 := x1 + x2 + x3 + .... Although the min and max node are not implemented explicitly, you can simply call the non-linearity z1 = max(x1, x2, x3,. ...) with some approximation method as described in this example: https://reactivebayes.github.io/RxInferExamples.jl/dev/examples/Basic%20Examples/Kalman%20filtering%20and%20smoothing/Kalman%20filtering%20and%20smoothing/#Combination-2:-y-min(x,-w).
I hope this helps with your problem. If there is anything I left unaddressed or if there are further questions, please let me know:)

[apiedrafTNO]

Hi Bart,

I think that's exactly what I want, thanks a lot.
In the example you call the nonlinear min(x,y) node for two variables. Do you think it's possible to define a node that does the same trick for any number of variables or is it better to chain the 2-variable max (and min) nodes like it's done for the addition nodes?

Cheers,
Alvaro

[bartvanerp]

Good to hear! I would say that using a single min or max function would be the best. Luckily Julia supports min and max with an arbitrary amount of inputs, so you should be good to go already.

[bartvanerp]

Something just came to mind. In the past we have seen that using a so-called smooth-max or smooth-min function (https://en.wikipedia.org/wiki/Smooth_maximum) seemed to give more robust performance in some applications. One could perform annealing to make these smooth functions less smooth as time progresses

[bvdmitri]

It makes sense since Linearization depends on autodiff and assumes functions to be differentiable afair, just using max/min can seemingly work but probably will give wrong results. We should add this note to the documentation. At least in this example we use smoothed version to avoid problems with derivatives.