I read about the multivariate and high-dimensional data set. I found that the multivariate data is the data with more than 3 variables. In addition, the high-dimensional data is the data with a large number of variables. So, for me, the multivariate and high-dimensional data set are the same, is that correct? I think, do not know if it is correct or not, the multivariate data is a data with more than 3 variables but the number of observation is quite small (300 to 500 observations). However, high-dimensional data is the data with a large number of variables (10 to 100 or even larger) and a large number of observations (1000 observations). Is that correct?
What is considered 'high' in high-dimensional is a futile discussion in my opinion. However, univariate, bivariate and multivariate usually refer to the number of outcome variables.
High-dimensional data analysis is a term more often used for dealing with things specific to a large number of explanatory variables, or parameters. For example:
- Inability to perform matrix inversion and most forms of decomposition when $n < p$;
- Increasingly sparser coverage of the sampling space with growing dimensions;
- The combinatorial explosion
A multivariate approach, can but need not involve these issues. You can have multiple outcome variables with only a handful of explanatory variables. You can also have a high-dimensional multivariate problem.
Tangentially related, for a classification task consisting of two classes, the outcome can be argued to follow a binomial distribution, whereas a multi-class classification task can be argued to follow a multinomial distribution.