Manifold hypothesis

The manifold hypothesis posits that many <a href="/facts/High-dimensional/0ktmUyac">high-dimensional</a> data sets that occur in the real world actually lie along low-dimensional <a href="/facts/Latent_manifold/IDYfRCU7">latent manifolds</a> inside that high-dimensional space. As a consequence of the manifold hypothesis, many data sets that appear to initially require many variables to describe, can actually be described by a comparatively small number of variables, likened to the local <a href="/facts/Coordinate_system/Mtd0q4SP">coordinate system</a> of the underlying manifold. It is suggested that this principle underpins the effectiveness of machine learning algorithms in describing high-dimensional data sets by considering a few common features.
The manifold hypothesis is related to the effectiveness of <a href="/facts/Nonlinear_dimensionality_reduction/1lL1Y0vN">nonlinear dimensionality reduction</a> techniques in machine learning. Many techniques of dimensional reduction make the assumption that data lies along a low-dimensional submanifold, such as <a href="/facts/Nonlinear_dimensionality_reduction/1lL1Y0vN">manifold sculpting</a>, <a href="/facts/Manifold_alignment/GnmLIeWr">manifold alignment</a>, and <a href="/facts/Manifold_regularization/foBI9c7S">manifold regularization</a>.
The major implications of this hypothesis is that

<ul><li><a href="/facts/Machine_learning/e0w0XJTu">Machine learning</a> models only have to fit relatively simple, low-dimensional, highly structured subspaces within their potential input space (latent manifolds).</li>
<li>Within one of these manifolds, it’s always possible to <a href="/facts/Interpolation/8PyetX4f">interpolate</a> between two inputs, that is to say, morph one into another via a continuous path along which all points fall on the manifold.</li></ul>
The ability to interpolate between samples is the key to generalization in <a href="/facts/Deep_learning/JLuwD3ea">deep learning</a>.

Manifold hypothesis open-in-new

Manifold hypothesis