What are high dimensional problems?

What are high dimensional problems?

Dimensionally cursed phenomena occur in domains such as numerical analysis, sampling, combinatorics, machine learning, data mining and databases. The common theme of these problems is that when the dimensionality increases, the volume of the space increases so fast that the available data become sparse.

What is a high dimensional system?

High-dimensional chaos is the term used to describe systems which possess chaos with more than one positive Lyapunov exponent, while on the contrary Low-dimensional chaos correspond to only one positive Lyapunov exponent.

Which is the best description of high dimensional statistics?

High-dimensional statistics. In statistical theory, the field of high-dimensional statistics studies data whose dimension is larger than dimensions considered in classical multivariate analysis. High-dimensional statistics relies on the theory of random vectors.

Is it hard to do Statistics in high dimensions?

In fact, statistical inference in high dimensions is intrinsically hard, a phenomenon known as the curse of dimensionality, and it can be shown that no estimator can do better in a worst-case sense without additional information (see Example 15.10 ).

What are the assumptions for high dimensional regression?

One common assumption for high-dimensional linear regression is that the vector of regression coefficients is sparse, in the sense that most coordinates of are zero. Many statistical procedures, including the Lasso, have been proposed to fit high-dimensional linear models under such sparsity assumptions.

How are estimators used in high dimensional settings?

Indeed, the James-Stein estimator provided the insight that in high-dimensional settings, one may obtain improved estimation performance through shrinkage, which reduces variance at the expense of introducing a small amount of bias.