  Real-world data usually have high dimensionality and it is important to
mitigate the curse of dimensionality. High-dimensional data are usually in a
coherent structure and make the data in relatively small true degrees of
freedom. There are global and local dimensionality reduction methods to
alleviate the problem. Most of existing methods for local dimensionality
reduction obtain an embedding with the eigenvalue or singular value
decomposition, where the computational complexities are very high for a large
amount of data. Here we propose a novel local nonlinear approach named Vec2vec
for general purpose dimensionality reduction, which generalizes recent
advancements in embedding representation learning of words to dimensionality
reduction of matrices. It obtains the nonlinear embedding using a neural
network with only one hidden layer to reduce the computational complexity. To
train the neural network, we build the neighborhood similarity graph of a
matrix and define the context of data points by exploiting the random walk
properties. Experiments demenstrate that Vec2vec is more efficient than several
state-of-the-art local dimensionality reduction methods in a large number of
high-dimensional data. Extensive experiments of data classification and
clustering on eight real datasets show that Vec2vec is better than several
classical dimensionality reduction methods in the statistical hypothesis test,
and it is competitive with recently developed state-of-the-art UMAP.
