Problems tagged with "intrinsic dimension"

3.

The ambient dimension is the dimension of the space in which the data lives. Here, the helix is embedded in 3D space (it has \(x\), \(y\), and \(z\) coordinates), so the ambient dimension is 3.

1.

The intrinsic dimension is the number of parameters needed to describe a point's location on the manifold. For this helix, even though it lives in 3D space, you only need one parameter (e.g., the arc length along the curve, or equivalently, the angle of rotation) to specify any point on it. If you ``unroll'' the helix, it becomes a straight line, which is 1-dimensional. Therefore, the intrinsic dimension is 1.

3.

The ambient dimension is the dimension of the space in which the data lives. The curve shown is embedded in 3D space (with \(x\), \(y\), and \(z\) axes visible), so the ambient dimension is 3.

1.

The intrinsic dimension is the minimum number of coordinates needed to describe a point's position on the manifold itself. This zigzag path, despite twisting through 3D space, is fundamentally a 1-dimensional curve. You only need a single parameter (such as the distance traveled along the path from a starting point) to uniquely identify any location on it. If you were to ``straighten out'' the path, it would become a line segment, confirming its intrinsic dimension is 1.