Talk:Intrinsic dimension

Intrinsic dimension of a dataset

The article describes the intrinsic dimension of a function. But in data mining it is possible to talk of the intrinsic dimension of a dataset, and this should also be described, either here or perhaps in a separate article.

The idea is that data points in a high-dimensional space may all lie close to a submanifold of that space, and the intrinsic dimension is then the dimension of that submanifold. This is relevant to understanding the curse of dimensionality for datasets. There may be a probability density function for the data points, but the intrinsic dimension of that function is not the same as that of the dataset.

Sources that could be used for the article include:

• E. Chavez et al., "Searching in Metric Spaces", ACM Computing Surveys, 33, 2001, 273-321.

• V. Pestov, "An axiomatic approach to intrinsic dimension of a dataset", Neural Networks, 21, 2008, 204-213.

JonH (talk) 11:11, 10 April 2010 (UTC)

Generalizations

The "Generalizations" subsection as it stands is strange if taken at face value:

"In a general case, f has intrinsic dimension M is there exist M functions a₁, a₂, ..., a_M and an M-variable function g such that

• f(x) = g(a₁(x),a₂(x),...,a_M(x)) for all x
• M is the smallest number of functions which allows the above transformation"

This makes the intrinsic dimension of every non-constant function 1, because you can take a₁ = f and g(x) = x. Presumably the functions a_i are supposed to be restricted to some class of functions, such as functions expressible in closed form. It would be better to somehow characterize the classes of functions used in practice. -- Coffee2theorems (talk) 16:27, 5 August 2012 (UTC)

Variable transformation correct?

Can someone explain the transformation in the example? Is f(y₁,y₂) = g(y₁) really the correct answer? — Preceding unsigned comment added by 178.10.183.34 (talk) 06:57, 17 July 2015 (UTC)