Maximum Variance Unfolding MVU

The solution to the maximum variance unfolding problem is found by constructing a Gram matrix, F, whose top eigenvectors give rise to the low-dimensional representation of the data. MVU seeks to maximise llyi yj II with yij e Y, subject to the following constraints 

Constraint (1) ensures that the matrix is positive semideflnite and Constraint (3) ensures local isometry [10], 

MVU differs from other spectral techniques in that rather than constructing a feature matrix from measurable properties (i.e. covariance, Euclidean distance), it directly learns the feature matrix by solving a convex optimisation problem. Once the feature matrix has been learnt however, MVU fits in with other spectral techniques as the low-dimensional embedding is given as the top eigenvectors of Eq.(2.1). 

The iow-dimensional embedding of the S-Curve dataset found using MVU is shown in Fig. 2.5. 

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