Are there special techniques for conducting dimension reduction on nested data? I have 20 items rated on 25 dimensions. Each item was rated on each dimension ~30 times. Each of ~40 participants rated 15 of the 20 items on all of the dimensions.
I had originally thought to do PCA on the item means (i.e. averaging across all ratings for that item), but realized that I will have fewer observations than dimensions, which I understand is not idea for PCA.
My question is, can I do PCA at the trial level (i.e. using each individual rating [e.g. participant #1 rating item #1 on dimension #1])? It seems that I should somehow account for the fact that the data are non-independent (i.e. nested within subjects and items). I've seen PCA with random effects mentioned on here, but haven't come across a definitive way to do this. 
Is it okay to do PCA at the trial level? Are there additional steps I need to take?
 A: 
I had originally thought to do PCA on the item means (i.e., averaging across all ratings for that item), but realized that I will have fewer observations than dimensions, which I understand is not idea for PCA.

That's not necessarily the case, for example see this question.
Also you could try models that explicitly handle missing data, like probabilistic matrix factorization. The general term you could google is low-rank matrix factorization in the context of recommender systems. For an implementation you can see H2O - see this example for slides on application for missing values.
If you're into Python you could also check this example (it uses H2O on Movielens dataset).
A: You could perform a multilevel factor analysis (Muthén, 1991, 1994; see Reise et al., 2005 for an accessible tutorial, and see Kim et al., 2016, for an updated overview). From  Reise et al., the process sounds pretty straightforward: 


*

*Conduct a generic EFA, ignoring dependency (gives you a starting, albeit misinformed factor solution).

*Estimate ICCs for measurements within each item (helps you appraise to what extent the multilevel approach is needed)

*Conduct factor analysis on within-subject correlation matrix

*Conduct factor analysis on between-subject correlation matrix


Given Muthén's publishing on this approach, it's a virtual certainty that it can be implemented in Mplus. I don't recall this kind of multilevel functionality for factor analyses in SPSS, and am unsure about what options you might have for R (I know there are packages for specifying multilevel confirmatory models, but unsure about exploratory models). 
References
Kim, E. S., Dedrick, R. F., Cao, C., & Ferron, J. M. (2016). Multilevel factor analysis: Reporting guidelines and a review of reporting practices. Multivariate Behavioral Research, 51(6), 881-898.
Muthén, B. O. (1991). Multilevel factor analysis of class and student achievement components. Journal of Educational Measurement, 28(4), 338-354.
Muthén, B. O. (1994). Multilevel covariance structure analysis. Sociological Methods & Research, 22(3), 376-398.
Reise, S. P., Ventura, J., Nuechterlein, K. H., & Kim, K. H. (2005). An illustration of multilevel factor analysis. Journal of Personality Assessment, 84(2), 126-136.
