  In this paper, we develop a new technique which we call representation theory
of the real hyperrectangle, which describes how to compute the eigenvectors and
eigenvalues of certain matrices arising from hyperrectangles. We show that
these matrices arise naturally when analyzing a number of different algorithmic
tasks such as kernel methods, neural network training, natural language
processing, and the design of algorithms using the polynomial method. We then
use our new technique along with these connections to prove several new
structural results in these areas, including:
  $\bullet$ A function is a positive definite Manhattan kernel if and only if
it is a completely monotone function. These kernels are widely used across
machine learning; one example is the Laplace kernel which is widely used in
machine learning for chemistry.
  $\bullet$ A function transforms Manhattan distances to Manhattan distances if
and only if it is a Bernstein function. This completes the theory of Manhattan
to Manhattan metric transforms initiated by Assouad in 1980.
  $\bullet$ A function applied entry-wise to any square matrix of rank $r$
always results in a matrix of rank $< 2^{r-1}$ if and only if it is a
polynomial of sufficiently low degree. This gives a converse to a key lemma
used by the polynomial method in algorithm design.
  Our work includes a sophisticated combination of techniques from different
fields, including metric embeddings, the polynomial method, and group
representation theory.
