Identifiability of an X-Rank Decomposition of Polynomial Maps

Abstract

Abstract. In this paper, we study a polynomial decomposition model that arises in problems of system identification, signal processing, and machine learning. We show that this decomposition is a special case of the X-rank decomposition-a powerful novel concept in algebraic geometry that generalizes the tensor CP decomposition. We prove new results on generic/maximal rank and on identifiability of a particular polynomial decomposition model. We try to make the results and basic tools accessible to a general audience (i.e., assuming no knowledge of algebraic geometry or its prerequisites).

Decoupling multivariate polynomials: Interconnections between tensorizations

“…where v k and w k are the columns of V and W, respectively. As shown in [32,33], the decomposition (3) is a special case of the X-rank decomposition [34, §5.2.1], where the set of "rank-one" terms is the set of polynomial maps of the form wg(v ⊤ u). The X-rank framework is useful [33] for studying the identifiability of the model (3).…”

“…The fact that rank Ψ( f ) ≤ 1 implies f (u) = g(v ⊤ u) also can be proved alternatively, by noting that the matrix Ψ f , after removing duplicate columns, can be reduced to the form S(f ) in [33,Proposition 22]. Hence, by [33,Proposition 4.1], the polynomial f has necessarily the form f (u) = g(v ⊤ u). However, this alternative proof requires introducing extra notation, which would be much longer that the proof presented in this paper.…”

Decoupling Multivariate Functions Using Second-Order Information and Tensors

“…Nonlinear models are used in a wide variety of science and engineering fields, such as data analytics, signal processing, system identification, and control engineering. While nonlinear models are able to capture wild nonlinear effects, this often comes at the cost of high parametric complexity, and a lack of 'model interpretability '. This paper studies the question how a given nonlinear multivariate vector function f : R m → R n can be decomposed into a simpler structure, as in [5,9,18,19,21]. In particular, we investigate a structure of the form…”

“…By involving the second-order derivatives, we impose additional constraints on the (joint) tensor decompositions, hence it is expected to enjoy more relaxed uniqueness conditions. In the article, we assume that an exact and uniquely identifiable [5] representation of f (x) exists. Nevertheless, the resulting joint tensor decomposition will ultimately be phrased as an optimization problem, and provides a natural starting point for studying both the noisy decoupling problem, as well as a model reduction interpretation, but this is beyond the scope of the current paper.…”

“…It can be verified that the Jacobian-based CPD method is not able to retrieve the underlying model. However, [5] asserts that r = 3 is still identifiable for polynomial models (with degree d ≥ 3). Considering the second-order information then leads to a CPD that is generically unique.…”