  A nonconventional $U(1)_{e-\mu}$ gauge model is proposed to explain the
unexpected anomalous magnetic moments of the electron and muon (lepton $g-2$),
where only the right-handed electron and muon in the standard model carry the
$U(1)_{e-\mu}$ charge. Although the light lepton masses are suppressed when the
gauge symmetry is spontaneously broken, they can be generated through the
Yukawa couplings to newly introduced particles, such as vector-like lepton
doublets and singlet, and scalar singlets. It is found that the same Yukawa
couplings combined with the new scalar couplings to the Higgs can induce the
radiative lepton-flavor violation processes $\ell' \to \ell \gamma$ and lepton
$g-2$, where the lepton $g-2$ is proportional to $m_{\ell}$. When Majorana
fermions and a scalar singlet are further added into the model, the active
neutrinos can obtain masses via the radiative seesaw mechanism. When the bounds
from the $m_e$ and $m_\mu$ and the neutrino data are satisfied, we find that
the electron $g-2$ can reach an order of $-10^{-12}$, and the muon $g-2$ can be
an order of $10^{-9}$. In addition, when the $\mu\to e \gamma$ decay is
suppressed, the resulting branching ratio for $\tau\to e \gamma$ can be of
$O(10^{-8})$, and that for $\tau\to \mu \gamma$ can be as large as the current
upper limit.
