In this section we discuss recent work on self-calibration to provide context for our numerical experiments.
7.2.1
Linear least squares
Bolzano and Nowak [BN07;BN08] consider the problem of self-calibration in sensor networks. They assume an array ofnsensors that take measurements
{xj}nj=1. Each sensor, however, reports its measurements subject to an unknown gainαjand offset βj resulting in the observations
yj =
xj − βj αj
.
In vector notation this becomes
x =Yα+ β,
whereY is ann×ndiagonal matrix with entriesyj. They refer to this as a single snapshot—a report from all the sensors at a particular time.
To recoverx from the observationsY, they assume that the true signalx lies in a low-dimensional subspace. For instance, the signal may be bandlimited
and therefore lie in the span of a smaller number of sinusoids. LettingP be the orthogonal projector onto the complement of this subspace, we see that
P x =Yα+ β = 0.
By consideringk snapshots they obtain a linear system
P(Yiα+ β)= 0, fori =1, . . . ,k.
If the signal subspace has dimensionr, then this system representsk(n−r)equa- tions in 2nunknowns. They show that it is possible to perform self-calibration fork large enough. The subsequent work of Lipor and Balzano [LB14] allows for noisy measurements.
The very recent work of Ling and Strohmer [LS16] extends this approach of linearizing the bilinear measurements in various self-calibration models. They provide rigorous guarantees for recovery and address the question of stability with noisy measurements.
7.2.2
Calibrating compressed sensing
Consider the compressed sensing problem, where we observe
b =T y, (7.2)
withT ∈RL×N andy ∈Rna sparse vector. A typical compressed sensing setting would assume thatT is a known measurement matrix (withL <N) and attempt to recover the sparse vectory. If, however, the sensing matrixT is unknown (or partially known) we have a calibration problem.
Gribonval et al. [GCD12] consider the case whereT is known but subject to unknown gains. That is, the observations take the form
b =DT y, (7.3)
whereD is a diagonal matrix. In other words, each observationbi in (7.3) corre- sponds to that of (7.2) multiplied by an unknown gaindi.
To blindly calibrate this compressed sensing problem, they assume that they see the observations from multiple signals in the form
where the matrixY ∈MN×Q hasQ unknown sparse signals inRN as its columns. Provided that none of the gainsdi is zero, the diagonal matrixD is invertible with∆ =D−1. Then the self-calibration problem becomes
minimize (Y,∆)
kYk`
1 subject to ∆B =TY and tr(∆)= δ,
where the trace constraint serves to exclude the trivial solution(0,0). This prob- lem is now convex, and they perform numerical experiments to demonstrate that this approach to self-calibration works for compressed sensing problems provided that the numberQ of training signals is large enough.
Bilen et al. [BPGD14] extend this work to the space of complex signals. That is, they consider both the gain/amplitude calibration here as well as phase cali- bration. Recent work of Wang and Chi [Wan16] provide theoretical guarantees on the calibration procedure when the signals are sparse in the Fourier basis (i.e.,T is a Fourier matrix).
7.2.3
A lifting approach
Our motivation is similar to the preceding work in that we wish to convexify the self-calibration problem. We, however, frame the problem by lifting to the space of operators. In particular, we follow the work of Ling and Strohmer [LS15b], and assume that the measurements (7.1) take the special form
b =DT y +z, and D =diag(Sx), (7.4) where we now assume thatT ∈ RL×N andS ∈ RL×M are known. That is, the gainsD belong to a low-dimensional subspace. The goal now is to recover the vectorx ∈RM that determines the gains along with the signaly.
We see that this model is a generalization of the blind deconvolution model discussed in Section2.1.2. Indeed, Ling and Strohmer took inspiration from the lifting procedure of Ahmed et al. [ARR14].
Assume that the measurements have no noise. We can write the entries ofb as
bl = tr(Mlt(x yt))= hMl, x yti, where Ml =sltlt, whereslis thelth row ofS, andTl is thelth row ofT.
If we setA =x yt, then we can write the measurement vectorb as
whereµis a linear measurement map such that thelth entry ofµ(A)is tr(MltA). That is, the bilinear measurement model becomes a linear measurement model on matrices.
They assume that the true signaly is sparse and solve the self-calibration prob- lem
minimize
A kAk`1 subject to µ(A)=b. (7.5) They call this procedure SparseLift.
They also provide a result on the probability of recovery.
Theorem 7.2.1(`1-minimization for self-calibration [LS15b, Thm. 3.1]). In the
model(7.4), assume thatS ∈RL×M satisfiesS∗S = IM (withM ≤ L),x ∈ RM is sparse, andy ∈Rn iss-sparse. Further assume thatT ∈ML×N withL < N has independent standard Gaussian entries. Then the solutionbA to(7.5)equalsx yt
with probability at least1−O(L−α+1)provided that
L
log2L
≥ Cαµ2
maxM slog(M s),
where the constantCα grows linearly withα, and µmax is the largest absolute entry in√LS.
Therefore, the numberLof measurements required scales roughly withM s, the size of the subspace containing the unknown gains multiplied by the sparsity of the signal. A similar result holds whenT is a partial Fourier matrix.
In their numerical experiments, they also consider using thek·k1,2norm—the
sum of the Euclidean norms of the rows—to enforce column-sparsity ofA. In the nuclear norm framework this is the`2⊗`1nuclear norm, and its use makes
sense asA = x yt =x ⊗y, wherey is sparse butx is not.
Flinth [Fli16] extends the theoretical results of Ling and Strohmer [LS15b;LS15a] to handle self-calibration and demixing problems with the k·k1,2norm. He
obtains qualitatively similar guarantees, but his results on stability suggest that the mixed norm will perform better than the`1norm.
7.2.4
Our work
We also follow the lifting approach of Ling and Strohmer [LS15b], but we con- sider a lifting to the operator space to allow for two different extensions: multiple snapshots and two-dimensional signals. Under the lifted operator model, we
can replace the`1regularizer in (7.5) with a nuclear norm that can account for
the shared structure between snapshots, or the two-dimensional structure of a signal. While the above-mentioned approach in [GCD12;BPGD14] may be able to accommodate such structured signals, the authors do not discuss this possibility.
The remainder of this chapter discusses the operator lifting model and its im- plementation inoperfactalong with numerical experiments to demonstrate its use.