We describe a simple and effective method (Spectral Attribute removaL; SAL) to remove private or guarded information from neural representations. Our method uses matrix decomposition to project the input representations into directions with reduced covariance with the guarded information rather than maximal covariance as factorization methods normally use. We begin with linear information removal and proceed to generalize our algorithm to the case of nonlinear information removal using kernels. Our experiments demonstrate that our algorithm retains better main task performance after removing the guarded information compared to previous work. In addition, our experiments demonstrate that we need a relatively small amount of guarded attribute data to remove information about these attributes, which lowers the exposure to sensitive data and is more suitable for low-resource scenarios.