  Periodically-driven or Floquet systems can realize anomalous topological
phenomena that do not exist in any equilibrium states of matter, whose
classification and characterization require new theoretical ideas that are
beyond the well-established paradigm of static topological phases. In this
work, we provide a general framework to understand anomalous Floquet
higher-order topological insulators (AFHOTIs), the classification of which has
remained a challenging open question. In two dimensions (2D), such AFHOTIs are
defined by their robust, symmetry-protected corner modes pinned at special
quasienergies, even though all their Floquet bands feature trivial band
topology. The corner-mode physics of an AFHOTI is found to be generically
indicated by 3D Dirac/Weyl-like topological singularities living in the phase
spectrum of the bulk time-evolution operator. Physically, such a phase-band
singularity is essentially a "footprint" of the topological quantum
criticality, which separates an AFHOTI from a trivial phase adiabatically
connected to a static limit. Strikingly, these singularities feature
unconventional dispersion relations that cannot be achieved on any static
lattice in 3D, which, nevertheless, resemble the surface physics of 4D
topological crystalline insulators. We establish the above higher-order
bulk-boundary correspondence through a dimensional reduction technique, which
also allows for a systematic classification of 2D AFHOTIs protected by point
group symmetries. We demonstrate applications of our theory to two concrete,
experimentally feasible models of AFHOTIs protected by $C_2$ and $D_4$
symmetries, respectively. Our work paves the way for a unified theory for
classifying and characterizing Floquet topological matters.
