Post on 28-Dec-2015
Richard Baraniuk Chinmay HegdeMarco DuarteMark DavenportRice University
Michael Wakin University of Michigan
Compressive Learning and Inference
Pressure is on Digital Sensors
• Success of digital data acquisition is placing increasing pressure on signal/image processing hardware and software to support
higher resolution / denser sampling» ADCs, cameras, imaging systems, microarrays, …
xlarge numbers of sensors
» image data bases, camera arrays, distributed wireless sensor networks, …
x increasing numbers of modalities
» acoustic, RF, visual, IR, UV=
deluge of datadeluge of data» how to acquire, store, fuse,
process efficiently?
Sensing by Sampling• Long-established paradigm for digital data acquisition
– sample data at Nyquist rate (2x bandwidth) – compress data (signal-dependent, nonlinear)– brick wall to resolution/performance
compress transmit/store
receive decompress
sample
sparse /compressiblewavelettransform
Compressive Sensing (CS)
• Directly acquire “compressed” data
• Replace samples by more general “measurements”
compressive sensing transmit/store
receive reconstruct
Compressive Sensing
• When data is sparse/compressible, can directly acquire a condensed representation with no/little information loss
• Random projection will work
measurements
[Candes-Romberg-Tao, Donoho, 2004]
sparsesignal
nonzeroentries
Why CS Works• Random projection not full rank, but stably embeds
signals with concise geometrical structure– sparse signal models is K-sparse– compressible signal models
with high probability provided M large enough
Why CS Works• Random projection not full rank, but stably embeds
signals with concise geometrical structure– sparse signal models is K-sparse– compressible signal models
with high probability provided M large enough
• Stable embedding: preserves structure– distances between points, angles between vectors, …
K-dim planes
K-sparsemodel
CS Signal Recovery
• Recover sparse/compressible signal x from CS measurements y via optimization
K-dim planes
K-sparsemodel
recovery
linear program
Information Scalability
• Many applications involve signal inference and not reconstruction
detection < classification < estimation < reconstruction
computationalcomplexityfor linearprogramming
Information Scalability
• Many applications involve signal inference and not reconstruction
detection < classification < estimation < reconstruction
• Good news: CS supports efficient learning, inference, processing directly on compressive measurements
• Random projections ~ sufficient statisticsfor signals with concise geometrical structure
• Extend CS theory to signal models beyond sparse/compressible
Application:
CompressiveDetection/Classification
viaMatched Filtering
Matched Filter• Detection/classification with K unknown
articulation parameters– Ex: position and pose of a vehicle in an image– Ex: time delay of a radar signal return
• Matched filter: joint parameter estimation and detection/classification– compute sufficient statistic for each potential target and
articulation– compare “best” statistics to detect/classify
Matched Filter Geometry
• Detection/classification with K unknown articulation parameters
• Images are points in
• Classify by finding closesttarget template to datafor each class (AWG noise)
– distance or inner product
data
target templatesfrom
generative modelor
training data (points)
Matched Filter Geometry
• Detection/classification with K unknown articulation parameters
• Images are points in
• Classify by finding closesttarget template to data
• As template articulationparameter changes, points map out a K-dimnonlinear manifold
• Matched filter classification = closest manifold search articulation parameter space
data
CS for Manifolds
• Theorem: random measurements preserve manifold structure[Wakin et al, FOCM ’08]
• Enables parameter estimation and MFdetection/classificationdirectly on compressivemeasurements– K very small in many
applications
Example: Matched Filter
• Detection/classification with K=3 unknown articulation parameters1. horizontal translation2. vertical translation3. rotation
Smashed Filter
• Detection/classification with K=3 unknown articulation parameters (manifold structure)
• Dimensionally reduced matched filter directly on compressive measurements
Smashed Filter
• Random shift and rotation (K=3 dim. manifold)• Noise added to measurements• Goal: identify most likely position for each image class
identify most likely class using nearest-neighbor test
number of measurements Mnumber of measurements M
avg
. sh
ift
est
imate
err
or
class
ifica
tion
rate
(%
)more noise
more noise
Application:
CompressiveData Fusion
Multisensor Inference• Example: Network of J cameras observing
an articulating object
• Each camera’s images lie on K-dim manifold in• How to efficiently fuse imagery from J cameras
to maximize classification accuracy and minimize network communication?
Multisensor Fusion• Fusion: stack corresponding image vectors
taken at the same time
• Fused images still lie on K-dim manifold in
Joint Articulation Manifold (JAM)
• Can take CS measurements of stacked imagesand process or make inferences
CS + JAM
w/ unfused sensing
• Can compute CS measurements in-networkas we transmit to collection/processing point
CS + JAM
Simulation Results
• J=3 CS cameras, each N=320x240 resolution• M=200 random measurements per camera
• Two classes1. truck w/ cargo2. truck w/ no cargo
class 1 class 2
Simulation Results
• J=3 CS cameras, each N=320x240 resolution• M=200 random measurements per camera
• Two classes– truck w/ cargo– truck w/ no cargo
• Smashed filtering– independent– majority vote– JAM fused
Application:
Compressive Manifold Learning
Manifold Learning
• Given training points in , learn the mapping to the underlying K-dimensional articulation manifold
• ISOMAP, LLE, HLLE, …
• Ex: images of rotating teapot
articulation space= circle
Compressive Manifold Learning
• ISOMAP algorithm based on geodesic distances between points
• Random measurements preserve these distances
• Theorem: If , then theISOMAP residual variance in the projected domain is bounded by
the additive error factor
full data (N=4096) M = 100 M = 50 M = 25
translatingdisk manifold
(K=2)
[Hegde et al ’08]
Conclusions
• Why CS works: stable embedding for signals with concise geometric structure
– sparse signals (K-planes), compressible signals ( balls)– smooth manifolds
• Information scalability– detection< classification < estimation < reconstruction– compressive measurements ~ sufficient statistics– many fewer measurements may be required to
detect/classify/estimate than to reconstruct– leverages manifold structure and not sparsity
• Examples– smashed filter– JAM for data fusion– manifold learning
dsp.rice.edu/cs
• Partnership on open educational resources– content development– peer review
• Contribute your course notes, tutorial article, textbook draft, out-of-print textbook, …
(you must own the copyright)
• MS Word and LaTeX importers
• For more info: IEEEcnx.org
capetowndeclaration.org
Why CS Works (#3)• Random projection not full rank, but stably embeds
– sparse/compressible signal models– smooth manifolds – point clouds
into lower dimensional space with high probability• Stable embedding: preserves structure
– distances between points, angles between vectors, …
provided M is large enough: Johnson-Lindenstrauss
Q points