IEEE Circuits and Systems Magazine - Q1 2020 - 44

should choose the matrix A so that the mutual coherence
of the columns of B = AD is as low as possible to keep the
projections of the axes as far as possible from each other.
C. Explaining CS With Polytopes
An interesting alternative view on noiseless recovery
comes from a polytope interpretation of BP whose working
principle can be exemplified using the same toy case used
above. The 1-norm sphere of radius r, i.e., " p ! R n p 1 # r ,,
is the so-called n-dimensional cross-polytope. For n = 3 is
its the blue diamond-like shape in Figure 3.
In that figure, the radius of the 1-norm sphere is the
minimum allowing a non-empty intersection between
the sphere itself and the yellow line, an intersection that
is the solution of BP. Note that this intersection contains
the true pr but only a properly designed B can guarantee that other solutions do not exist.
In particular, since l = 1, the solution is on a vertex
of the cross-polytope that must be identified starting
from the projection B + y on the green plane.

⊥

imB

ξ∗

y = Bξ

B +y

(a)

"

ξ ≠ ξ∗
ξ∗

⊥

imB

B +y

(b)
Figure 3. The polytope point of view of CS.
44

IEEE CIRCUITS AND SYSTEMS MAGAZINE

In the case of Figure 3(a) the projection of the crosspolytope on the green plane yields the red hexagon
in which the 6 vertices of the cross-polytope are still
distinguishable.
On the contrary, in Figure 3(b) the projection of crosspolytope on the green plane is the red rectangle in which
2 of the 6 vertices of the cross-polytope disappear.
When this happens, more than one point of the minimum-radius cross-polytope projects on the same B + y
and BP is unable to ensure that its solution is sparse and
thus coincides with the true signal.
This can be generalized to generic l-sparse signals sitting on l-dimensional facets of the cross-polytope, which
in this case should be projected on the subspace imB <
and remain distinguishable. This leads to the estimation
of the probability that reconstruction is possible as the
ratio between the number of facets that are still recognizable after projection over the total number of facets.
Overall, if one adopts this point of view, good matrices A are those for which B = AD preserves distinguishability of l-dimensional face t s of t he n - dimensional
cross-polytope.
Adaptation methods start from the theoretical developments that we have just sketched and identify promising merit figures that are related with the capability
of effectively reconstructing pr from y. Then, propose
heuristics to improve such merit figures.
III. Adaptation at the Encoder Side
In describing all the methods we adopt some common
notation. Given a possible sensing matrix A, the matrix
B = AD and the Gram matrix G = B < B remain implicitly
defined. Given any matrix M, we indicate its Singular Value
Decomposition (SVD) as M = U M K M V 


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