Sparse power factorization: balancing peakiness and sample complexity

Abstract

In many applications, one is faced with an inverse problem, where the known signal depends in a bilinear way on two unknown input vectors. Often at least one of the input vectors is assumed to be sparse, i.e., to have only few non-zero entries. Sparse power factorization (SPF), proposed by Lee, Wu, and Bresler, aims to tackle this problem. They have established recovery guarantees for a somewhat restrictive class of signals under the assumption that the measurements are random. We generalize these recovery guarantees to a significantly enlarged and more realistic signal class at the expense of a moderately increased number of measurements.

Appendix: Proof of Lemma 5

For the proof of Lemma 5, we will use the following result.

Lemma 1 (Lemma 17 and Lemma 18 in 23)

Assume that the (3s₁, 3s₂, 2)-restrictedisometry property is fulfilled for some restricted isometry constantδ > 0.Assume that the cardinality of$\widetilde J_{1} \subseteq \left [ n_{1} \right ]$,respectively$\widetilde J_{2} \subseteq \left [ n_{2}\right ]$isat most 2s₁,respectively 2s₂.Then, whenever$u\in \mathbb {C}^{n_{1}}$isat most 2s₁-sparseand$v \in \mathbb {C}^{n_{2}}$isat most 2s₂-sparse,we have that

$$ \|{\Pi}_{\widetilde J_{1}}[(\mathcal{A}^{\ast}\mathcal{A}- I)(uv^{*})]{\Pi}_{\widetilde J_{2}}\| \leq \delta \|uv^{*}\|_{\mathrm{F}}. $$

Furthermore for all$z \in \mathbb {C}^{n}$and for all$\widetilde J_{1} \subseteq \left [ n_{1} \right ]$, respectively$\widetilde J_{2} \subseteq \left [ n_{2}\right ]$, with cardinality at mosts₁, respectivelys₂, we have that

$$ \|{\Pi}_{\widetilde J_{1}}[\mathcal{A}^{\ast}(z)]{\Pi}_{\widetilde J_{2}}\| \leq \sqrt{1+\delta}\|z\|_{\ell_{2}}. $$

Proof of Lemma 5

Recall that $b = \mathcal {A}\left (X\right ) + z$ and define k₁ and k₂ by

$$ \begin{array}{llll} k_{1} &:= \underset{ k \in \left[ n_{2} \right] }{\text{arg max}} \ \vert v_{k} \vert\\ k_{2} & := \underset{ k \in \left[ n_{2} \right] }{\text{arg max}} \big\|{\Pi}_{\widehat J_{1}}[\mathcal{A}^{\ast}(b)]{\Pi}_{\left\{ k \right\} }\big\|_{\mathrm{F}}. \end{array} $$

(23)

The starting point of our proof is the observation that

$$ \big\|{\Pi}_{\widehat J_{1}}[\mathcal{A}^{\ast}(b)]{\Pi}_{\left\{ k_{2} \right\} }\big\|_{\mathrm{F}} \ge \big\|{\Pi}_{\widehat J_{1}}[\mathcal{A}^{\ast}(b)]{\Pi}_{\left\{ k_{1} \right\} }\big\|_{\mathrm{F}} \ge \big\|{\Pi}_{\widetilde{J_{1}}}[\mathcal{A}^{\ast}(b)]{\Pi}_{\left\{ k_{1} \right\} }\big\|_{\mathrm{F}}, $$

(24)

where the first inequality is due to the definition of k₂ and the second one follows from $\widetilde {J_{1}} \subset \widehat J_{1}$, which is due to Lemma 4. The right-hand side of the inequality chain can be estimated from below by

$$ \begin{array}{llll} &\big\|{\Pi}_{\widetilde{J_{1}}}[\mathcal{A}^{\ast}(b)]{\Pi}_{\left\{ k_{1} \right\} }\big\|_{\mathrm{F}}\\ \geq & \big\|{\Pi}_{\widetilde{J_{1}} } uv^{*} {\Pi}_{\left\{ k_{1} \right\} }\big\|_{\mathrm{F}} - \big\|{\Pi}_{\widetilde{J_{1}}} \left[ \left( \mathcal{A}^{*} \mathcal{A} - I \right) \left( uv^{*}\right) \right] {\Pi}_{\left\{ k_{1} \right\} }\big\|_{\mathrm{F}} - \big\|{\Pi}_{\widetilde{J_{1}} } \mathcal{A}^{*} \left( z\right) {\Pi}_{\left\{ k_{1} \right\} }\big\|_{\mathrm{F}} \\ \geq & \big\|{\Pi}_{\widetilde{J_{1}} } uv^{*} {\Pi}_{\left\{ k_{1} \right\} }\big\|_{\mathrm{F}} - \left( \delta \Vert uv^{*} \Vert_{F} + \sqrt{1 + \delta} \Vert z \Vert \right) \\ \ge & \big\|{\Pi}_{\widetilde{J_{1}} } u \big\| \Vert v \Vert_{\infty} - \left( \delta + \nu +\delta \nu \right). \end{array} $$

(25)

In the first inequality, we used $b= \mathcal {A} \left (uv^{*}\right ) + z$ and the triangle inequality. The second inequality follows from Lemma 7. The last line follows from ∥uv^∗∥_F = 1 and ∥z∥ = ν. Next, we will estimate the left-hand side of (24) by

$$ \begin{array}{llll} &\big\|{\Pi}_{\widehat J_{1}}[\mathcal{A}^{\ast}(b)]{\Pi}_{\left\{ k_{2} \right\} }\big\|_{\mathrm{F}}\\ \leq & \big\|{\Pi}_{\widehat J_{1} } uv^{*} {\Pi}_{\left\{ k_{2} \right\} }\big\|_{\mathrm{F}} + \left( \delta \Vert uv^{*} \Vert_{F} + \sqrt{1 + \delta} \Vert z \Vert \right) \\ \le & \big\|{\Pi}_{\widehat J_{1} } u \big\| \big\| {\Pi}_{\left\{ k_{2} \right\} } v \big\| + \left( \delta + \nu +\delta \nu \right) \\ \le & \big\|{\Pi}_{\widehat J_{1} } u \big\| \big\| {\Pi}_{\widehat J_{2} } v \big\| + \left( \delta + \nu +\delta \nu \right). \end{array} $$

(26)

The first two lines are obtained by an analogous reasoning as for (25). The last line is due to $\left \{ k_{2} \right \} \subset \widehat {J_{2}}$, which is a consequence of the definition of $\widehat J_{2}$ (3) and the definition of {k₂} (23). We finish the proof by combining the inequality chains (24), (25), and (26). □