  The performance demands of future particle-physics experiments investigating
the high-energy frontier pose a number of new challenges, forcing us to find
improved solutions for the detection, identification, and measurement of
final-state particles in subnuclear collisions. One such challenge is the
precise measurement of muon momentum at very high energy, where an estimate of
the curvature provided by conceivable magnetic fields in realistic detectors
proves insufficient for achieving good momentum resolution when detecting,
e.g., a narrow, high mass resonance decaying to a muon pair.
  In this work we study the feasibility of an entirely new avenue for the
measurement of the energy of muons based on their radiative losses in a dense,
finely segmented calorimeter. This is made possible by exploiting spatial
information of the clusters of energy from radiated photons in a regression
task. The use of a task-specific deep learning architecture based on
convolutional layers allows us to treat the problem as one akin to image
reconstruction, where images are constituted by the pattern of energy released
in successive layers of the calorimeter. A measurement of muon energy with
better than 20% relative resolution is shown to be achievable for ultra-TeV
muons.
