Mean-field variational inference (MFVI) is a widely used method for approximating high-dimensional probability distributions by product measures. It has been empirically observed that MFVI optimizers often suffer from mode collapse. For example, when the target measure π is a mixture π=wP0+(1−w)P1, the MFVI optimizer tends to place most of its mass near a single component of the mixture. This work provides the first theoretical explanation of mode collapse in MFVI. Starting from two-component mixtures, we introduce the notion of \emph{ε-separateness} to quantify the separation of the mixture components. We derive explicit bounds on the mass that any MFVI optimizer assigns to each component, when P0 and P1 are ε-separated for sufficiently small ε. We further extend this analysis to mixture target distributions with multiple components. Our results reveal that the occurrence of mode collapse crucially depends on the relative position of the components. To address this issue, we propose rotational variational inference (RoVI), which augments MFVI with a rotation matrix. The numerical studies support our theoretical findings and illustrate improved mode coverage with RoVI in Bayesian mixture posteriors, including a real-data example.
Figures & tables
Figure 1: π=21N([−m,−m]⊤,I2)+21N([m,m]⊤,I2) . The above figure shows the contour plots of π and a fitted MFVI approximation when m∈{1,2,3} . We observe the mode-collapse is prominent when m=3 . Moreover, when m=3 , the fitted approximation can switch from one mode to the other depending on the initialization.
Figure 2: Example of well-separated P0 and P1 . The blue and red regions represent the supports of P0 and P1 , respectively.
Figure 3: Comparison of LMC, MFVI, and RoVI across six Gaussian mixture settings with varying means and covariances. RoVI improves approximation in several settings where MFVI is underdispersed or collapses.
Figure 4: The target is a symmetric Gaussian mixture: 21N([−m,−m]⊤,I2)+21N([m,m]⊤,I2) . RoVI captures both components and the marginal distributions, while MFVI collapses to a single mode in the sufficiently separated cases.
Target
Method
m(q)
ℓ(q)
3D
MFVI
0.000±0.000
0.000±0.000
3D
RoVI
0.267±0.098
0.252±0.091
Old Faithful
MFVI
0.000±0.000
0.000±0.000
Old Faithful
RoVI
0.302±0.137
0.285±0.131
Table 1: Comparison of mode coverage for the simulated mixture (3D) and Old Faithful posterior. Reported values are means ± standard deviations across five paired runs following validation-based restart selection.
Figure 5: Joint densities for the Old Faithful posterior. We also report the probability P(μ1<μ2) . The columns show the reference and the fitted MFVI and RoVI approximations. Values of P(μ1<μ2) closer to 0.5 indicate more balanced coverage of the two symmetric modes.
Appendix figures & tables1 asset
Supplementary material from the paper’s appendix.
Appendix
Figure 6: Joint densities for the separated-mixture posterior (3D). The columns show the reference and the fitted MFVI and RoVI approximations, with the probability P(μ1<μ2) above each panel. RoVI retains both modes, while MFVI concentrates on one.