  In this paper we revisit an idea originally proposed by Mandelbrot about the
possibility to observe ``negative dimensions'' in random multifractals. For
that purpose, we define a new way to study scaling where the observation scale
$\tau$ and the total sample length $L$ are respectively going to zero and to
infinity. This ``mixed'' asymptotic regime is parametrized by an exponent
$\chi$ that corresponds to Mandelbrot ``supersampling exponent''. In order to
study the scaling exponents in the mixed regime, we use a formalism introduced
in the context of the physics of disordered systems relying upon traveling wave
solutions of some non-linear iteration equation. Within our approach, we show
that for random multiplicative cascade models, the parameter $\chi$ can be
interpreted as a negative dimension and, as anticipated by Mandelbrot, allows
one to uncover the ``hidden'' negative part of the singularity spectrum,
corresponding to ``latent'' singularities. We illustrate our purpose on
synthetic cascade models. When applied to turbulence data, this formalism
allows us to distinguish two popular phenomenological models of dissipation
intermittency: We show that the mixed scaling exponents agree with a log-normal
model and not with log-Poisson statistics.
