1) The document discusses using knowledge of signal structure to improve compressive sensing measurements and reconstruction. 2) It presents a method called LASeR (Learning Adaptive Sensing Representations) that learns a tree-structured dictionary to sparsely represent signals and enable more efficient compressive sensing. 3) Results show that LASeR provides gains over other compressive sensing methods, achieving higher reconstruction quality with fewer measurements. This demonstrates that exploiting prior knowledge of signal structure can enhance compressive sensing performance.

1. Knowledge Enhanced Compressive Measurements
Training'
Data'
Structured'
Sparsity'
Adap3ve'
Sensing'
LASeR'
LASeR:'Learning'Adap3ve'Sensing'Representa3ons'
a`(1)
a`(2)
a`(5)
a`(3)
a`(4) a`(6) a`(7)
Akshay
Soni
University
of
Minnesota
www.tc.umn.edu/~sonix022
KECoM
Student
Workshop
2012
ExploiEng
Tree
Priors

2. A Sparse Signal Model

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25
30
#projections
ReconstructionSNR(dB)
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0
2
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6
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10
12
14
16
#projections
ReconstructionSNR(dB)
Can knowledge buy something?
12
dB
gain
CS
DCT
Lasso
CS
Random
Lasso

4. 0 20 40 60 80 100 120
0
2
4
6
8
10
12
14
16
#projections
ReconstructionSNR(dB)
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0
5
10
15
20
25
30
#projections
ReconstructionSNR(dB)
Can knowledge buy something?
15
dB
gain
CS
DCT
Lasso
CS
Random
Lasso

5. A Sparse Signal Model
|xi|
⇢
> µ > 0 i 2 S,
0 i /2 S.

6. Exact Support Recovery (ESR)
CS: Non adaptive & Non structured
|xi|
⇢
> µ > 0 i 2 S,
0 i /2 S.

7. The Big Picture: Minimum Signal Amplitudes for ESR
Can we exploit structure or adaptivity or both?
[*]
D.
Donoho
and
J.
Jin,
“Higher
criEcism
for
detecEng
sparse
heterogeneous
mixtures,”
Ann.
StaEst.,
vol.
32,
no.
3,
pp.
962–994,
2004.
[*]
[*]
S.
Aeron,
V.
Saligrama,
and
M.
Zhao,
"InformaEon
TheoreEc
Bounds
for
Compressed
Sensing,"
InformaEon
Theory,
IEEE
TransacEons
on
,
vol.56,
no.10,
pp.5111-­‐5130,
Oct.
2010
Uncompressed
/
compressed
µ
q
n
R log n

8. M.
Malloy
and
R.
Nowak,
“On
the
limits
of
sequenEal
tesEng
in
high
dimensions,”
preprint,
2011.
[*]
[*]
SequenEal
but
non
structured
/
uncompressed
The Big Picture: Minimum Signal Amplitudes for ESR
J.
Haupt,
R.
Baraniuk,
R.
Castro
and
R.
Nowak,
“SequenEally
Designed
Compressed
Sensing,”
SSP,
2012.
[*]
µ
q
n
R log n
µ
q
n
R log k

9. Tree Sparse Signal Model
Can
we
exploit
this
tree
structure
for
ESR
problem?

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5
10
15
20
25
30
#projections
ReconstructionSNR(dB)
Can structure buy something?
Tree
Structured
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2
4
6
8
10
12
14
16
#projections
ReconstructionSNR(dB)
Random
CS
DCT
CS

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0
2
4
6
8
10
12
14
16
#projections
ReconstructionSNR(dB)
0 20 40 60 80 100 120
0
5
10
15
20
25
30
#projections
ReconstructionSNR(dB)
Can structure buy something?
Random
CS
DCT
CS

12. [*]
[*]
The Big Picture: Minimum Signal Amplitudes for ESR
Arias-­‐Castro,
E.,
Candès,
E.
J.,
Helgason,
H.
and
Zeitouni,
O.
(2008).
Searching
for
a
trail
of
evidence
in
a
maze.
Ann.
StaEst.
36
1726–1757.
Uncompressed
search
for
simple
trail
µ
q
n
R log k
µ
q
n
R log n µ
q
n
R

13. The Big Picture: Minimum Signal Amplitudes for ESR
[*]
A.
Soni
and
J.
Haupt,
“Efficient
adapEve
compressive
sensing
using
sparse
hierarchical
learned
dicEonaries,”
in
Proc.
Asilomar
Conf.
on
Signals,
Systems,
and
Computers,
2011,
pp.
1250–1254.
µ
q
n
R log k
µ
q
n
R log n µ
q
n
R
µ
q
k
R log k

14. Structure Dependent Adaptive Support Recovery – An Example
1
2 5
3 4 6 7
Stack&/&Queue&(both&ini1alized&to&index&of&root)&
!
Repeat&&&&&&&&&&for&next&queue/
stack&element.&
&
Pop if Queue/Stack not empty
Queue: Insert indices of
children of node
Unknown signal
1&
No
1&
|y(i, k)| ?y(j) = ( dj)T
x + N(0, 1)

15. Theorem
(2011):
A.
Soni
&
J.
Haupt
Tree Structured Adaptive Support Recovery

16. 0 5 10 15 20 25
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
αm in
Pr(S=ˆS)
The Big Picture: Minimum Signal Amplitudes for ESR
µ
q
n
R log k
µ
q
n
R log n µ
q
n
R
µ
q
k
R log k

17. 0 5 10 15 20 25
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
αm in
Pr(S=ˆS)
The Big Picture: Minimum Signal Amplitudes for ESR
Sufficient
condiEon
May
we
improve?
necessary
condiEons

18. Tree Structured Signal Reconstruction
Two-­‐step
ReconstrucEon
AdapEve
Support
Recovery
Measure
Support
LocaEons
Corollary
(2011):
A.
Soni
&
J.
Haupt

19. Learning Adaptive Sensing Representations (LASeR)
Learning
Tree
Sparsifying
DicEonary
[hop://spams-­‐devel.gforge.inria.fr/]

20. R
=
(128
x
128)
Qualitative Results - I
Direct
Wavelet
Sensing
PCA
CS
LASSO
CS
Tree
LASSO
LASeR
m
=
20
m
=
50
m
=
80
Image
from
PICS
database

21. R
=
(128
x
128)/32
Qualitative Results - II
Direct
Wavelet
Sensing
PCA
CS
LASSO
CS
Tree
LASSO
LASeR
m
=
50
m
=
80
m
=
20
Image
from
PICS
database

22. Quantitative Results
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35
#projections
ReconstructionSNR(dB)
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16
18
#projections
ReconstructionSNR(dB)
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0
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25
#projections
ReconstructionSNR(dB)

23. Future Directions for Tree Sensing
Thank You.
Contact:
Akshay
Soni
sonix022@umn.edu
1. LASeR with clutter signal model:
y = (x + c) + w
(clever regularization for di↵erent signal classes – eg., di↵usion of clutter
over whole signal space using `2 rather that `1 penalty)
2. LASeR with non-orthonormal learned dictionaries.
3. Exploiting signal amplitude correlation in LASeR.