Data-independent methods for dimensionality reduction such as random projections, sketches, and feature hashing have become increasingly popular in recent years. These methods often seek to reduce dimensionality while preserving the hypothesis class, resulting in inherent lower bounds on the size of projected data. For example, preserving linear separability requires Ω(1/γ2 ) dimensions, where γ is the margin, and in the case of polynomial functions, the number of required dimensions has an exponential dependence on the polynomial degree. Despite these limitations, we show that the dimensionality can be reduced further while maintaining performance guarantees, using improper learning with a slightly larger hypothesis class. In particular, we show that any sparse polynomial function of a sparse binary vector can be computed from a compact sketch by a single-layer neural network, where the sketch size has a logarithmic dependence on the polynomial degree. A practical consequence is that networks trained on sketched data are compact, and therefore suitable for settings with memory and power constraints. We empirically show that our approach leads to networks with fewer parameters than related methods such as feature hashing, at equal or better performance.