Dimension Reduction for UQ
High-dimensional uncertainty problems become tractable only after reducing inputs or outputs to the directions that actually matter.
The curse of dimensionality
The number of samples needed to characterize a function grows explosively with input dimension. Grid-based and many surrogate methods become infeasible beyond a handful of dimensions. Dimension reduction exploits the observation that real problems usually have far fewer effective degrees of freedom than nominal ones.
Input reduction
- Screening: identify and drop unimportant inputs (see Morris Screening)
- Active subspaces: find important linear combinations from gradients
- PCA / Karhunen-Loeve: compress correlated or field-valued inputs
- Sensitivity ranking: retain only inputs with meaningful Sobol indices
Output reduction
Field, image, or time-series outputs are compressed with POD, PCA, or autoencoders so a surrogate predicts a few coefficients. The truncation error becomes part of the emulator's error budget and must be reported alongside the surrogate's own uncertainty.
Supervised vs unsupervised
Unsupervised methods (PCA, KL) reduce based on variance in the data alone, ignoring the target quantity. Supervised methods (active subspaces, sliced inverse regression, sufficient dimension reduction) use the response, so they keep directions that matter for prediction even if they carry little input variance. Choose supervised reduction when a specific output is the goal.
Verification
Any reduction is an approximation. Validate by predicting held-out cases in the reduced space and checking error against the full-dimensional model. Report the retained dimension and the fraction of variance or sensitivity captured. When no low-dimensional structure exists, say so rather than forcing a reduction that discards real behavior.