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 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 C n that are invariant under finite unitary group actions, as a special case we establish the existence of low-dimensional and complete Zm-invariant representations for any m ∈ 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 non-parallel map, which is enjoyed by our function and employ this to construct a Lipschitz modification of it.

Injection dimensions of projective varieties

“…An intermediate approach between these two extremes is the more recent field of study of separating invariants [DK15, §2.4], [Kem09], which maintain the full geometric information about orbit separation while remedying many of the complications of complete generating sets of invariants [Dom07; DKW08; NS09]. Separating invariants are of major importance for applications, as for example in the recent work [CCH19], see [DK15,§5] for an overview of possible application.…”

Classifying Signals Under a Finite Abelian Group Action: The Finite Dimensional Setting

“…On the other hand, for real life applications it is crucial to understand the finite dimensional setting and to have a transform with complete discriminative power. In [3] we studied this problem and drawing from algebraic tools, were able to develop a framework that allows one to obtain discriminative invariants with respect to finite group actions under certain hypotheses. Furthermore, our transforms in [3] come with explicit Lipschitz bounds and Date: November 15, 2019.…”

“…In [3] we studied this problem and drawing from algebraic tools, were able to develop a framework that allows one to obtain discriminative invariants with respect to finite group actions under certain hypotheses. Furthermore, our transforms in [3] come with explicit Lipschitz bounds and Date: November 15, 2019.…”

“…In this work we extend our previous results to include all finite Abelian group actions. The main motivation for our present work however, comes from the need to address a much more important problem: the final map in [3] has the form x F x x .…”

“…Some instances include certain generalizations of phase retrieval, image recognition, the analysis of textures, etc. In our previous work [3], based on an algebraic approach, we constructed a Lipschitz translation invariant transform-the case when G is cyclic. Here, we extend our results to include all finite Abelian groups.…”