Complete set of translation invariant measurements with Lipschitz bounds

Abstract: In image and audio signal classification, a major problem is to build stable representations that are invariant under rigid motions and, more generally, to small diffeomorphisms. Translation invariant representations of signals in $\mathbb{C}^n$ are of particular importance. The existence of such representations is intimately related to classical invariant theory, inverse problems in compressed sensing and deep learning. Despite an impressive body of litereature on the subject, most representations available are either: i) not stable due to the presence of high frequencies; ii) non discriminative; iii) non invariant when projected to finite dimensional subspaces. In the present paper, we construct low dimensional representations of signals in $\mathbb{C}^n$ that are invariant under finite unitary group actions, as a special case we establish the existence of low-dimensional and complete $\mathbb{Z}_m$-invariant representations for any $m\in\mathbb{N}$. Our construction yields a stable, discriminative transform with semi-explicit Lipschitz bounds on the dimension; this is particularly relevant for applications. Using some tools from Algebraic Geometry, we define a high dimensional homogeneous function that is injective. We then exploit the projective character of this embedding and see that the target space can be reduced significantly by using a generic linear transformation. Finally, we introduce the notion of {\it non-parallel} map, which is enjoyed by our function and employ this to construct a Lipschitz modification of it.