Such learning problems are formulated as latent or generative model learning assuming that observations were emerged from the low-dimensional latent states, which includes an intractable posterior inference of latent states for given input data.

An instance of natural occurrences of such constraints is when mixed strategies are used, which correspond to a probability simplex constraint.

FromTheorem 2,thevariational gap is strictly positive as long as the rotation part of the score network remainstobenonzero.

So we can check that ddtE(qt,λt) (qt,λt) in both cases. Combing the two cases yield the result. Pm i=1N(θ;µi,σ2i) where m is fixed to5 in all the experiments. Monotonic Bayesian Neural Networks In this experiment, we use the COMPAS dataset (J. The task istopredict whether the individual will commit acrime againin2years.

Inthis section we provide the convergence analysis for our algorithms.

SB-FBSDE isanewclass ofgenerativemodels that, inspiring bytherecent advance of understanding deep learning through the optimal control perspective [61-63], adopts Lemma 5 to generalize the score-based diffusion models.

However, it remains a challenge to extract reliable inference from complexdatasets with uncertainty quantification.

However, it remains a challenge to extract reliable inference from complexdatasets with uncertainty quantification.

In the following, we provide a brief overview of Riemannian geometry and constant curvature manifolds, specifically the Poincaré ball and the hypersphere models. Sphere In the two-dimensional settingd = 2, we rely on polar coordinates to parametrize the sphere S2. In the following subsection we remind that this regularization term can also be motivated from an estimator'svarianceperspective. 5 D.2 Frobeniusnorm Hutchinson'sestimator Hutchinson'sestimator(Hutchinson,1990)isasimple waytoobtain a stochastic estimate ofthetrace ofamatrix. The variance of this estimator thus depends on the Frobenius norm of the vector's field Jacobian Thenγ(tn) is also a Cauchy sequence by Equation 16. So for every sequence (tn) in (a,b) that converges tob, we have that(γ(tn)) converges top.