  We present measurements of partial branching fractions of inclusive
semileptonic $B \to X_u \, \ell^+\, \nu_{\ell}$ decays using the full Belle
data set of 711 fb$^{-1}$ of integrated luminosity at the $\Upsilon(4S)$
resonance and for $\ell = e, \mu$. Inclusive semileptonic $B \to X_u \,
\ell^+\, \nu_{\ell}$ decays are CKM suppressed and measurements are complicated
by the large background from CKM-favored $B \to X_c \, \ell^+\, \nu_{\ell}$
transitions, which have a similar signature. Using machine learning techniques,
we reduce this and other backgrounds effectively, whilst retaining access to a
large fraction of the $B \to X_u \, \ell^+\, \nu_{\ell}$ phase space and high
signal efficiency. We measure partial branching fractions in three phase-space
regions covering about $31\%$ to $86\%$ of the accessible $B \to X_u \,
\ell^+\, \nu_{\ell}$ phase space. The most inclusive measurement corresponds to
the phase space with lepton energies of $E_\ell^B > 1 $ GeV, and we obtain
$\Delta \mathcal{B}(B \to X_u \ell^+ \, \nu_\ell) = \left( 1.59 \pm 0.07 \pm
0.16 \right) \times 10^{-3}$ from a two-dimensional fit of the hadronic mass
spectrum and the four-momentum-transfer squared distribution, with the
uncertainties denoting the statistical and systematic error. We find $\left|
V_{ub} \right| = \left( 4.10 \pm 0.09 \pm 0.22 \pm 0.15 \right) \times 10^{-3}$
from an average of four calculations for the partial decay rate with the third
uncertainty denoting the average theory error. This value is higher but
compatible with the determination from exclusive semileptonic decays within 1.3
standard deviations. In addition, we report charmless inclusive partial
branching fractions separately for $B^+$ and $B^0$ mesons as well as for
electron and muon final states. No isospin breaking or lepton flavor
universality violating effects are observed.
