  Autoencoders (AE) provide a useful method for nonlinear dimensionality
reduction but are ill-suited for low data regimes. Conversely, Principal
Component Analysis (PCA) is data-efficient but is limited to linear
dimensionality reduction, posing a problem when data exhibits inherent
nonlinearity. This presents a challenge in various scientific and engineering
domains such as the nanophotonic component design, where data exhibits
nonlinear features while being expensive to obtain due to costly real
measurements or resource-consuming solutions of partial differential equations.
  To address this difficulty, we propose a technique that harnesses the best of
both worlds: an autoencoder that leverages PCA to perform well on scarce
nonlinear data. Specifically, we outline a numerically robust PCA-based
initialization of AE, which, together with the parameterized ReLU activation
function, allows the training process to start from an exact PCA solution and
improve upon it. A synthetic example is presented first to study the effects of
data nonlinearity and size on the performance of the proposed method. We then
evaluate our method on several nanophotonic component design problems where
obtaining useful data is expensive. To demonstrate universality, we also apply
it to tasks in other scientific domains: a benchmark breast cancer dataset and
a gene expression dataset.
  We show that our proposed approach is substantially better than both PCA and
randomly initialized AE in the majority of low-data regime cases we consider,
or at least is comparable to the best of either of the other two methods.
