An SVD-like Decomposition of Bounded-Input Bounded-Output Functions

TL;DR

The paper addresses the problem of extending the singular value decomposition (SVD) framework to nonlinear bounded-input bounded-output (BIBO) functions $f:\mathbb{R}^n\to\mathbb{R}^p$ by introducing a finite-dimensional, norm-preserving lifting $v:\mathbb{R}^n\to\mathbb{R}^m$ so that $f(x)=U\Sigma v(x)$. This SVD-like representation leverages a norm-preserving, injective mapping to maintain meaningful linearization of the nonlinear function and yields an upper bound on the 2-induced norm, $\\|f\\|_{2-2} \le \sigma_1$, with a strict bound under the construction. The main contributions include (i) a constructive proof of the decomposition with $m\ge p+n$, (ii) a bound on the induced norm and conditions ensuring a real, feasible lifting, (iii) demonstrations on numerical examples showing the liftings recover $f$ through a linear operator $K$, and (iv) a discussion extending Reisz representation to $p=1$ and potential implications for nonlinear optimization; the Appendix shows a fundamental limitation: a universal injective norm-preserving lifting $g$ to represent all BIBO functions is impossible. The work provides a pathway to apply linear tools to nonlinear maps and to reason about generalized row/null spaces in a bounded-input setting.

Abstract

The Singular Value Decomposition (SVD) of linear functions facilitates the calculation of their 2-induced norm and row and null spaces, hallmarks of linear control theory. In this work, we present a function representation that, similar to SVD, provides an upper bound on the 2-induced norm of bounded-input bounded-output functions, as well as facilitates the computation of generalizations of the notions of row and null spaces. Borrowing from the notion of "lifting" in Koopman operator theory, we construct a finite-dimensional lifting of inputs that relaxes the unitary property of the right-most matrix in traditional SVD, $V^*$, to be an injective, norm-preserving mapping to a slightly higher-dimensional space.

Figures (4)

• Figure 1: An example of a norm-preserving mapping: The unit disc ${\@fontswitch\mathcal{H}}$ in $\mathbb{R}^2$ is depicted in blue. Two norm-preserving mappings $g_1:\mathcal{H} \to {\@fontswitch\mathcal{G}}_1$ (black and dashed) and $g_2: \mathcal{H} \to {\@fontswitch\mathcal{G}}_2$ (orange and solid) are shown. Both are liftings for distinct hypothetical functions, $f_1(x) = K \circ g_1(x)$ and $f_2(x) = K \circ g_2(x)$ (per the notation in Remark \ref{['remark:V']}). Since $f_1$ and $f_2$ are functionals, $K$ can only have one non-zero singular value. The right singular vector corresponding to this non-zero singular value is shown in red and is denoted as $v^*_1$. Note then that $g_2$ would correspond to a function $f$ that stretches all elements of $\mathcal{H}$ uniformly.
• Figure 2: Numerical example of the lifting proposed in this paper: The top panel represents the function (the blue curve) while the orange curve represents its numerical reconstruction using $f(x) = K \circ g(x)$, with $g(x)$ norm-preserving and injective. The black dotted diagonal lines represent the upper bound on the induced norm of the function, given by $\pm \sigma_1 \|x\|_2$ (see Corollary \ref{['corollary:upperBound']}). The bottom panel shows the norm-preserving, injective lifting that is given in the construction of the Theorem (\ref{['theorem:sufficient']}).
• Figure 3: Numerical example of the lifting proposed in this paper: The top panel represents a multi-input single-output bounded-input bounded-output function. The bottom panel shows the norm-preserving lifting computed using the construction given in this paper. Let $K = U \Sigma V^*$ be the SVD of $K$. Then the functional $g_3(x)$ is the projection of the lifting $g$ onto $v_1^*$, which corresponds to $\sigma_1 = 1$ in this instance, and thus lifting further in $g_3(x)$ corresponds to increased stretching of $x$. In contrast, $g_2(x)$ and $g_3(x)$ correspond with the projections of the lifting onto $v_2^*$ and $v_3^*$, which correspond to $\sigma_2 = \sigma_3 = 0$, i.e. $g_1$ and $g_2$ store information that is lost from $x \rightarrow f(x)$.
• Figure 4: 2-dimensional demonstration of the geometric reasoning in Lemma \ref{['lemma:nec']}: In this example, $g:\mathbb{R}^2 \to \mathbb{R}^2$, with $f:\mathbb{R}^2 \to \mathbb{R}^2$. The blue, yellow, and red curves correspond to the sets $\mathcal{H}_r$, $\mathcal{Y}_1$, and $\mathcal{Y}_2$ mentioned in the proof. The key points of the proof are 1) to be norm-preserving, $g$ must map $\mathcal{H}_r \to \mathcal{H}_r$, 2) to satisfy representation of $f(x) = a \|x\|_2$ (an arbitrary BIBO function) $g$ must map $\mathcal{H}_r \to \mathcal{Y}_1$ and simultaneously $\mathcal{H}_r \to \mathcal{Y}_2$, and 3) the cardinality of the intersection of $\mathcal{H}_r$, $\mathcal{Y}_1$, and $\mathcal{Y}_2$ is at most 1. This contradicts the injectivity of $g$, since, in order for $g$ to be injective, the set $\{g(x)|x \in \mathcal{H}_r\}$ must have cardinality of the continuum, $2^{\aleph_0}$. Therefore a norm-preserving, injective mapping from $\mathbb{R}^n \to \mathbb{R}^n$ composed with $K \in \mathbb{R}^{n \times n}$ cannot represent all bounded-input bounded-output functions.

Theorems & Definitions (9)

• Theorem 1
• proof
• Remark 1
• Remark 2
• Remark 3
• Corollary 1
• proof
• Lemma 1
• proof