When you do PCA (or any dimensionality reduction), what is "the number of dimensions"? I always thought that the thing you measure (ie, the variable) is the number of dimensions: eg, if you measure the length, width, height of a box, that's 3 dimensions (3 variables); if you measure the abundance of 10,000 genes in 200 cells, that's 10,000 dimensions (not 200 dimensions).
More specific question
In regard to image 1 (below), what is the "correct" interpretation of number of dimensions (before PCA); is it the number of cells (200), or the number of genes (10,000)?
Note: I think it's possible to use either the number of cells or the number of genes as the number of dimensions, with obviously different interpretations. Additionally, there are some other good discussions of PCA on Cross Validated; however, my question is a little bit different: I'm really hoping for a response regarding my confusion after watching this video on PCA. Here's a brief explanation of my confusion.
The narrator is trying to explain PCA in context of this experiment (Image 1, below):
This graph was drawn from single-cell RNA-seq. There were about 10,000 transcribed genes in each cell.
Each dot represents a single-cell and its transcription profile. The general idea is that cells with similar transcription should cluster.
As I thought I understood PCA, in this experiment, the genes are the "dimensions" and the cells are the observations; ie, if there are 10,000 genes, there are 10,000 dimensions. This understanding seems to coincide with a different example (see references) using the Iris data set (Image 2, below); as you can see, the number of dimensions is the number of features of the flowers that were measured.
However, in the video, the narrator goes on to describe the number of dimensions as the number of cells for which the experiment measured the gene abundance (see Images 3 and 4, below):
Q: In regard to the experiment in Image 1 where "each dot represents a single cell", was the number of dimensions (before PCA) the number of cells or the number of genes?
Link to video: https://www.youtube.com/watch?v=_UVHneBUBW0
For link to reference on the Iris data, google "Principal Component Analysis in 3 Simple Steps Sebastian Raschka" (I don't have enough reputation on this site to include more links in this question).