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In this paper, we investigate the impact of numerical instability on thereliability of sampling, density evaluation, and evidence lower bound (ELBO)estimation in variational flows. We first empirically demonstrate that commonflows can exhibit a catastrophic accumulation of error: the numerical flow mapdeviates significantly from the exact map -- which affects sampling -- and thenumerical inverse flow map does not accurately recover the initial input --which affects density and ELBO computations. Surprisingly though, we find thatresults produced by flows are often accurate enough for applications despitethe presence of serious numerical instability. In this work, we treatvariational flows as dynamical systems, and leverage shadowing theory toelucidate this behavior via theoretical guarantees on the error of sampling,density evaluation, and ELBO estimation. Finally, we develop and empiricallytest a diagnostic procedure that can be used to validate results produced bynumerically unstable flows in practice.

Embracing the chaos: analysis and diagnosis of numerical instability in variational flows