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