  Recently, the computer vision and machine learning community has been in
favor of feature extraction pipelines that rely on a coding step followed by a
linear classifier, due to their overall simplicity, well understood properties
of linear classifiers, and their computational efficiency. In this paper we
propose a novel view of this pipeline based on kernel methods and Nystrom
sampling. In particular, we focus on the coding of a data point with a local
representation based on a dictionary with fewer elements than the number of
data points, and view it as an approximation to the actual function that would
compute pair-wise similarity to all data points (often too many to compute in
practice), followed by a Nystrom sampling step to select a subset of all data
points.
  Furthermore, since bounds are known on the approximation power of Nystrom
sampling as a function of how many samples (i.e. dictionary size) we consider,
we can derive bounds on the approximation of the exact (but expensive to
compute) kernel matrix, and use it as a proxy to predict accuracy as a function
of the dictionary size, which has been observed to increase but also to
saturate as we increase its size. This model may help explaining the positive
effect of the codebook size and justifying the need to stack more layers (often
referred to as deep learning), as flat models empirically saturate as we add
more complexity.
