  The clustering of galaxies and their connections to their initial conditions
is a major means by which we learn about cosmology. However, the stochasticity
between galaxies and their underlying matter field is a major limitation for
precise measurements of galaxy clustering. Efforts have been made with an
optimal weighting scheme to reduce this stochasticity using the mass-dependent
clustering of dark matter haloes. Here, we show that this is not optimal. We
demonstrate that the cosmic-web environments (voids, sheets, filaments \&
knots) of haloes, when combined linearly with the linear bias, provide extra
information for reducing stochasticity in terms of two-point statistics. Using
the environmental information alone can increase the signal-to-noise of
clustering by a factor of 3 better than the white-noise level at the scales of
the baryon acoustic oscillations. The information about the environment and
halo mass are complementary. Their combination increases the signal-to-noise by
another factor of 2-3. The information about the cosmic web correlates with
other properties of haloes, including halo concentrations and tidal forces --
all are related to the assembly bias of haloes.
