Exact Support Recovery for Sparse Spikes Deconvolution

Abstract

This paper studies sparse spikes deconvolution over the space of measures. We focus on the recovery properties of the support of the measure (i.e., the location of the Dirac masses) using total variation of measures (TV) regularization. This regularization is the natural extension of the $\ell ^1$ norm of vectors to the setting of measures. We show that support identification is governed by a specific solution of the dual problem (a so-called dual certificate) having minimum $L^2$ norm. Our main result shows that if this certificate is non-degenerate (see the definition below), when the signal-to-noise ratio is large enough TV regularization recovers the exact same number of Diracs. We show that both the locations and the amplitudes of these Diracs converge toward those of the input measure when the noise drops to zero. Moreover, the non-degeneracy of this certificate can be checked by computing a so-called vanishing derivative pre-certificate. This proxy can be computed in closed form by solving a linear system. Lastly, we draw connections between the support of the recovered measure on a continuous domain and on a discretized grid. We show that when the signal-to-noise level is large enough, and provided the aforementioned dual certificate is non-degenerate, the solution of the discretized problem is supported on pairs of Diracs which are neighbors of the Diracs of the input measure. This gives a precise description of the convergence of the solution of the discretized problem toward the solution of the continuous grid-free problem, as the grid size tends to zero.

Appendices

Appendix 1: Auxiliary results

For the convenience of the reader, we give here the proofs of several auxiliary results which are needed in the discussion.

Proposition 12

(Subdifferential of the total variation) Let us endow $\mathcal {M}(\mathbb {T})$ with the weak-* topology and $C(\mathbb {T})$ with the weak topology. Then, for any $m\in \mathcal {M}(\mathbb {T})$, we have:

$$\begin{aligned} \partial || m ||_{\text {TV}} = \left\{ \eta \in C(\mathbb {T}) \;;\; || \eta ||_{\infty } \leqslant 1 \quad \text {and} \quad \int \eta \, \mathrm {d}m =|| m ||_{\text {TV}} \right\} . \end{aligned}$$

Proof

Let $A= \left\{ \eta \in C(\mathbb {T}) \;;\; \forall m\in \mathcal {M}(\mathbb {T}), \ \langle \eta ,\,m\rangle \leqslant || m ||_{\text {TV}} \right\}$. It is clear that $B_\infty (0,1)\subset A$, where $B_\infty (0,1)$ is the $L^\infty (\mathbb {T})$ closed unit ball. Conversely, we observe that $A\subset B_\infty (0,1)$ by considering the Dirac masses $(\pm \delta _t)_{t\in \mathbb {T}}$.

Let us write $J(m):= || m ||_{\text {TV}}$. The function $J:\mathcal {M}(\mathbb {T})\rightarrow \mathbb {R}\cup \{+\infty \}$ is convex, proper, lower semi-continuous (for the weak-* topology), positively homogeneous and:

$$\begin{aligned} J^*(\eta )&=\sup _{m\in \mathcal {M}(\mathbb {T})} \sup _{t>0} \left( \langle \eta ,\,t m\rangle -J(t m) \right) \\&=\sup _{t>0} t\left( \sup _{m\in \mathcal {M}(\mathbb {T})} \langle \eta ,\,m\rangle -J(m) \right) \\&=\left\{ \begin{array}{ll}0 &{}\quad \text{ if } \eta \in A, \\ +\infty &{}\quad \text{ otherwise. } \end{array}\right. \end{aligned}$$

By Proposition I.5.1 in [17], for any $\eta \in C(\mathbb {T})$:

$$\begin{aligned} \eta \in \partial J(m) \Longleftrightarrow \langle \eta ,\,m\rangle = J(m) +J^*(\eta ), \end{aligned}$$

which is equivalent to $|| \eta ||_{\infty }\leqslant 1$ and $\int \eta \mathrm {d}m = || m ||_{\text {TV}}$. $\square$

Proposition 13

There exists a solution to $(\mathcal {P}_0(y_0))$ and the strong duality holds between $(\mathcal {P}_0(y_0))$ and $(\mathcal {D}_0(y_0))$, i.e.,

$$\begin{aligned} \underset{\varPhi (m)=y_0}{\min }\; || m ||_{\text {TV}} = \underset{|| \varPhi ^* p ||_{\infty }\leqslant 1}{\sup }\; \langle y_0,\,p\rangle . \end{aligned}$$

(48)

Moreover, if a solution $p^\star$ to $(\mathcal {D}_0(y_0))$ exists,

$$\begin{aligned} \varPhi ^* p^\star \in \partial {|| m^\star ||_{\text {TV}}} \end{aligned}$$

(49)

where $m^\star$ is any solution to $(\mathcal {P}_0(y_0))$. Conversely, if (49) holds, then $m^\star$ and $p^\star$ are solutions of, respectively, $(\mathcal {P}_0(y_0))$ and $(\mathcal {D}_0(y_0))$.

Proof

We apply [17, Theorem II.4.1] to $(\mathcal {D}_0(y_0))$ (and not to $(\mathcal {P}_0(y_0))$ as would be natural) rewritten as

$$\begin{aligned} \inf _{|| \varPhi ^* p ||_{\infty } \leqslant 1} \langle -y_0,\,p\rangle , \end{aligned}$$

The infimum is finite since for any admissible $p$, $\langle -y_0,\,p\rangle =\langle m_0,\,\varPhi p\rangle \geqslant - || m_0 ||_{\text {TV}}$. Let $V=L^2(\mathbb {T})$, $Y=C(\mathbb {T})$ (endowed with the strong topology), $Y^*=\mathcal {M}(\mathbb {T})$, $F(u)=\langle -y_0,\,u\rangle$ for $u\in V$, $G(\psi )=\iota _{|| \cdot ||_{\infty }\leqslant 1}(\psi )$ for $\psi \in Y$ and $\Lambda =\varPhi ^*$. It is clear that $F$ and $G$ are proper convex lower semi-continuous functions. Eventually, $F$ is finite at 0, G is finite and continuous at $0=\Lambda 0$. Hence the result. $\square$

Appendix 2: Proof of Proposition 6

Assume that for some $(u,v)\in \mathbb {R}^N\times \mathbb {R}^N$, $\varGamma _x (u,v)=0$. Then,

$$\begin{aligned} \forall \,t\in \mathbb {T}, \quad 0&=\sum _{j=1}^N \left( u_j\varphi (t-x_j)+v_j \varphi '(t-x_j)\right) \\&=\sum _{k=-f_c}^{f_c} \left( \sum _{j=1}^N (u_j + 2ik\pi v_j)e^{-2ik\pi x_j} \right) e^{2ik\pi t} \end{aligned}$$

We deduce that

$$\begin{aligned} \forall \,k\in \{-f_c,\ldots f_c \}, \quad \sum _{j=1}^N (u_j + k \tilde{v}_j)r_j^k =0 \quad \text {where} \quad \left\{ \begin{array}{l} r_j=e^{-2i\pi x_j},\\ \tilde{v}_j=2i\pi v_j. \end{array} \right. \end{aligned}$$

It is therefore sufficient to prove that the columns of the following matrix are linearly independent

$$\begin{aligned} \begin{pmatrix} r_1^{-f_c} &{} \ldots &{}r_N^{-f_c} &{} (-f_c)r_1^{-f_c}&{} \ldots &{}(-f_c)r_N^{-f_c} \\ \vdots &{} &{}\vdots &{} \vdots &{} &{}\vdots \\ r_1^{k} &{} \ldots &{}r_N^{k} &{} kr_1^{k}&{} \ldots &{}k r_N^{k} \\ \vdots &{} &{}\vdots &{} \vdots &{} &{}\vdots \\ r_1^{f_c} &{} \ldots &{}r_N^{f_c} &{} (f_c)r_1^{f_c}&{} \ldots &{}(f_c)r_N^{f_c} \\ \end{pmatrix}. \end{aligned}$$

If $N<f_c$, we complete the family $\{r_1, \ldots r_N\}$ in a family $\{r_0,r_1,\ldots r_{f_c}\}\subset \mathbb {S}^1$ such that the $r_i$’s are pairwise distinct. We obtain a square matrix $M$ by inserting the corresponding columns

$$\begin{aligned} M = \begin{pmatrix} r_1^{-f_c} &{} \ldots &{}r_{f_c}^{-f_c} &{} r_0^{-f_c} &{}(-f_c)r_1^{-f_c}&{} \ldots &{}(-f_c)r_{f_c}^{-f_c} \\ \vdots &{} &{}\vdots &{}\vdots &{} \vdots &{} &{}\vdots \\ r_1^{k} &{} \ldots &{}r_{f_c}^{k} &{} r_0^k &{}kr_1^{k}&{} \ldots &{}k r_{f_c}^{k} \\ \vdots &{} &{}\vdots &{}\vdots &{} \vdots &{} &{}\vdots \\ r_1^{f_c} &{} \ldots &{}r_{f_c}^{f_c} &{} r_0^{f_c}&{}(f_c)r_1^{f_c}&{} \ldots &{}(f_c)r_{f_c}^{f_c} \\ \end{pmatrix}. \end{aligned}$$

We claim that $M$ is invertible. Indeed, if there exists $\alpha \in \mathbb {C}^{(2f_c+1)}$ such that $M^T \alpha =0$, then the rational function $F(z) = \sum _{k=-f_c}^{f_c} \alpha _k z^{k}$ satisfies:

$$\begin{aligned} F(r_j)&=0 \quad \text {and} \quad F'(r_j)=0 \ \text{ for } 1\leqslant j \leqslant f_c,\\ F(r_0)&=0. \end{aligned}$$

Hence, $F$ has at least $2f_c+1$ roots in $\mathbb {S}^1$, counting the multiplicities. This imposes that $F=0$, thus $\alpha =0$, and $M$ is invertible. The result is proved.

Appendix 3: Proof of Proposition 9

Let us denote by $P_{C_n}(x)$ the projection of $x\in L^2(\mathbb {T})$ onto $C_n$. We have:

$$\begin{aligned} \left\| P_{C_n}\left( \frac{y_0}{\lambda }\right) -P_{C_n}(0)\right\| _2 \leqslant \left\| \frac{y_0}{\lambda } - 0\right\| _2, \end{aligned}$$

so that the sequence $p_{\lambda }^{\mathcal {G}_n}=P_{C_n}(\frac{y_0}{\lambda })$ is bounded in $L^2(\mathbb {T})$, and we may extract a subsequence $p_{\lambda }^{\mathcal {G}_n'}$ which weakly converges to some $p_\lambda ^\star \in L^2(\mathbb {T})$. Since $C_{n'}$ is (weakly) closed for all $n'$, $p_\lambda ^\star \in \bigcap _{n'} C_{n'}=C$.

Moreover, by the characterization of the projection onto convex sets:

$$\begin{aligned} \forall z\in C\subset C_n', \ \left\langle \frac{y_0}{\lambda } - p_{\lambda }^{\mathcal {G}_n'},z\right\rangle - \left\langle \frac{y_0}{\lambda },p_{\lambda }^{\mathcal {G}_n'}\right\rangle +|| p_{\lambda }^{\mathcal {G}_n'} ||_2^2&\leqslant 0.\\ \text{ Passing } \text{ to } \text{ the } \text{ limit } n'\rightarrow +\infty , \ \left\langle \frac{y_0}{\lambda } - p_{\lambda }^\star ,z\right\rangle - \left\langle \frac{y_0}{\lambda },p_{\lambda }^\star \right\rangle + \liminf _{n'} || p_{\lambda }^{\mathcal {G}_n'} ||_2^2&\leqslant 0,\\ \left\langle \frac{y_0}{\lambda } - p_{\lambda }^\star ,z\right\rangle - \left\langle \frac{y_0}{\lambda },p_{\lambda }^\star \right\rangle + || p_{\lambda }^\star ||_2^2&\leqslant 0,\\ \left\langle \frac{y_0}{\lambda } - p_\lambda ^\star ,z - p_\lambda ^\star \right\rangle&\leqslant 0.\\ \end{aligned}$$

Thus, $p_\lambda ^\star$ is the orthogonal projection of $\frac{y_0}{\lambda }$ on $C$: $p_\lambda ^\star =P_{C}\left( \frac{y_0}{\lambda }\right) =p_\lambda$. Since this is true for any subsequence, the whole sequence $p_{\lambda }^{\mathcal {G}_n}$ weakly converges to $p_\lambda$.

Moreover, by lower semi-continuity and the inclusion $C\subset C_n$ we have:

$$\begin{aligned} \left\| \frac{y_0}{\lambda }-p_\lambda \right\| _2&\leqslant \liminf _{n\rightarrow +\infty } \left\| \frac{y_0}{\lambda }-p_{\lambda }^{\mathcal {G}_n}\right\| _2\leqslant \limsup _{n\rightarrow +\infty }\left\| \frac{y_0}{\lambda }-p_{\lambda }^{\mathcal {G}_n}\right\| _2 \leqslant \left\| \frac{y_0}{\lambda }-p_{\lambda }\right\| _2, \end{aligned}$$

so that $\frac{y_0}{\lambda }-p_{\lambda }^{\mathcal {G}_n}$ converges strongly to $\frac{y_0}{\lambda }-p_{\lambda }$, hence the strong convergence of $p_{\lambda }^{\mathcal {G}_n}$ to $p_\lambda$.

The rest of the statement follows from Proposition 1.