Can anyone provide a definition of "factor tilt" that uses mathematical notation?
Perhaps factor tilts are based on factor regression models. Let's say that our returns vector $\mathbf{y}_t$ can be expressed in terms of a market return $x_t$: $$ \mathbf{y}_t = \mathbf{B}x_t + \mathbf{v}_t $$ where $\mathbf{B}$ is a loadings vector containing "betas" and $\mathbf{v}_t$ is iid noise.
A model like this is often advertised on its ability to reduce the number of parameters that need to be estimated in order to specify a forecast's covariance matrix. It also suggests, depending on one's strategy, a vector of portfolio weights $\mathbf{w}_t$.
Does "factor tilting" somehow manually override the weights suggested by an optimization routine? Is this idea even necessarily tied to factor-based modeling?