Themanifold hypothesis posits that manyhigh-dimensional data sets that occur in the real world actually lie along low-dimensionallatent manifolds inside that high-dimensional space.^[1][2][3][4] 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, linked to the localcoordinate system 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 ofnonlinear dimensionality reduction techniques in machine learning. Many techniques of dimensional reduction make the assumption that data lies along a low-dimensional submanifold, such asmanifold sculpting,manifold alignment, andmanifold regularization.

The major implications of this hypothesis is that

• Machine learning models only have to fit relatively simple, low-dimensional, highly structured subspaces within their potential input space (latent manifolds).
• Within one of these manifolds, it's always possible tointerpolate between two inputs, that is to say, morph one into another via a continuous path along which all points fall on the manifold.

The ability to interpolate between samples is the key to generalization indeep learning.^[5]

An empirically-motivated approach to the manifold hypothesis focuses on its correspondence with an effective theory for manifold learning under the assumption that robust machine learning requires encoding the dataset of interest using methods for data compression. This perspective gradually emerged using the tools of information geometry thanks to the coordinated effort of scientists working on theefficient coding hypothesis,predictive coding andvariational Bayesian methods.

The argument for reasoning about the information geometry on the latent space of distributions rests upon the existence and uniqueness of theFisher information metric.^[6] In this general setting, we are trying to find a stochastic embedding of a statistical manifold. From the perspective of dynamical systems, in thebig data regime this manifold generally exhibits certain properties such as homeostasis:

1. We can sample large amounts of data from the underlying generative process.
2. Machine Learning experiments are reproducible, so the statistics of the generating process exhibit stationarity.

In a sense made precise by theoretical neuroscientists working on thefree energy principle, the statistical manifold in question possesses aMarkov blanket.^[7]

• Kolmogorov complexity
• Minimum description length
• Solomonoff's theory of inductive inference

• Machine learning
• Theoretical computer science

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