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Richard Baraniuk, Volkan Cevher Rice University Ron DeVore Texas A&M University Martin Wainwright

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New Theory and Algorithms for Scalable Data Fusion. Richard Baraniuk, Volkan Cevher Rice University Ron DeVore Texas A&M University Martin Wainwright University of California-Berkeley Michael Wakin Colorado School of Mines. Networked Sensing. Goals sense communicate fuse

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slide1

New Theory and Algorithms

for ScalableData Fusion

Richard Baraniuk, Volkan Cevher

Rice University

Ron DeVore

Texas A&M University

Martin Wainwright

University of California-Berkeley

Michael Wakin

Colorado School of Mines

networked sensing
Networked Sensing

Goals

  • sense
  • communicate
  • fuse
  • infer (detect, recognize, etc.)
  • predict
  • actuate/navigate

networkinfrastructure

humanintelligence

networked sensing1
Networked Sensing

Challenges

  • growing volumes of sensor data
  • increasingly diverse data
  • diverse and changing operating conditions
  • increasing mobility

networkinfrastructure

humanintelligence

research challenges
Research Challenges
  • Shear amount of data that must be acquired, communicated, processed

J sensors

N samples/pixels per sensor

  • Amount of data grows as O(JN)
    • can lead to communication and computation collapse
  • Must fuse diverse data types
research program
Research Program
  • Thrust 1: Scalable data models
  • Thrust 2: Randomized dimensionality reduction
  • Thrust 3: Scalable inference algorithms
  • Thrust 4: Scalable data fusion
  • Thrust 5: Scalable learning algorithms
thrust 1 scalable data models
Thrust 1: Scalable Data Models
  • Unifying theme: low-dimensional signal structure
        • Sparse signal models
        • Graphical models
        • Manifold models
  • Exploit geometry of these models
1 sparse models
1. Sparse Models

pixels

largewaveletcoefficients

(blue = 0)

K-dim subspaces

3 manifold models
3. Manifold Models
  • Image articulation manifold (IAM)
  • Manifold dimensionL= # imaging parameters
  • If images are smooththen manifold is smooth

articulation parameter space

thrust 2 randomized dimensionality reduction
Thrust 2: Randomized Dimensionality Reduction
  • Goal: preserve information from x in y
  • One avenue: stable embedding
  • Key question: how small can M be?

signalfromsparse,graphical,manifoldmodel

measurements

sparse models
Sparse Models

K-dim subspaces

sparse models1
Sparse Models

K-dim subspaces

  • Stable embedding <> Restricted isometry property (RIP) from compressive sensing
  • Stability whp if
single pixel camera
Single-Pixel Camera

M randomizedmeasurements

N mirrors

target N=65536 pixels

M=1300 measurements (2%)

M=11000 measurements (16%)

graphical models
Graphical Models

K-dim subspaces

  • Example: K-sparse signals
graphical models1
Graphical Models
  • Example: K-sparse signals with correlations
  • Rules out some/many subspaces
  • Stability whp with as low as

K-dim subspaces

ex clustered signals
Ex: Clustered Signals
  • Model clustering of significant pixelsin space domain using Ising Markov Random Field
  • Example: Recovery of background subtracted video from randomized measurements

target

Ising-modelrecovery

CoSaMPrecovery

LP (FPC)recovery

manifold models
Manifold Models
  • Can stably embed a compact, smooth L-dimensional manifold whp if
  • Recall that manifold dimension L is very small for many apps (# imaging parameters)
  • Constants scale with manifold’s
    • condition number (curvature)
    • volume
thrust 3 scalable inference
Thrust 3: Scalable Inference

Many applications involve signal inferenceand not reconstructiondetection < classification < estimation < reconstruction

Good news: RDR supports efficient learning, inference, processing directly on compressive measurements

Random projections ~ sufficient statisticsfor signals with concise geometrical structure

classification
Classification

Simple object classification problem

AWGN: nearest neighbor classifier

Common issue:

L unknown articulation parameters

Common solution: matched filter

find nearest neighbor under all articulations

matched filter geometry
Matched Filter Geometry

Classification with L unknown articulation parameters

Images are points in

Classify by finding closesttarget template to datafor each class

distance or inner product

data

target templatesfromgenerative modelor training data (points)

matched filter geometry1
Matched Filter Geometry

Detection/classification with L unknown articulation parameters

Images are points in

Classify by finding closesttarget template to data

As template articulationparameter changes, points map out a L-dimnonlinear manifold

Matched filter classification = closest manifold search

data

articulation parameter space

smashed filter
Smashed Filter

Recall stable manifoldembedding whp using

random measurements

Enables parameter estimation and MFdetection/classificationdirectly on randomizedmeasurements

recall L very small in many applications (# articulations)

example matched filter
Example: Matched Filter

Naïve approach

take M CS measurements,

recover N-pixel image from CS measurements (expensive)

conventional matched filter

smashed filter1
Smashed Filter

Worldly approach

take M CS measurements,

matched filter directly on CS measurements(inexpensive)

smashed filter2
Smashed Filter

Random shift and rotation (L=3 dim. manifold)

WG noise added to measurements

Goals: identify most likely shift/rotation parameters identify most likely class

more noise

classification rate (%)

avg. shift estimate error

more noise

number of measurements M

number of measurements M

thrust 4 scalable data fusion
Thrust 4: Scalable Data Fusion
  • Sparse signal models
    • multi-signal sparse models [Wakin, next talk]
  • Manifold models
    • joint manifold models [next]
  • Graphical models
manifold based fusion
Manifold-based Fusion
  • Example: Network of J cameras observing an articulating object
  • Each camera’s images lie on L-dim manifold in
  • How to efficiently fuse imagery from J cameras to solve an inference problem while minimizing network communication?
multisensor fusion
Multisensor Fusion
  • Fusion: stack corresponding image vectors taken at the same time
  • Fused images still lie on L-dim manifold in“joint manifold”
joint manifolds
Joint Manifolds
  • Given submanifolds
    • L-dimensional
    • homeomorphic (we can continuously map between any pair)
  • Define joint manifoldas concatenation of
joint manifolds properties
Joint Manifolds: Properties
  • Joint manifold inherits properties from component manifolds
    • compactness
    • smoothness
    • volume:
    • condition number ( ):
  • Translate into algorithm performance gains
  • Bounds are often loose in practice (good news)
multisensor fusion via jm rdr
Multisensor Fusion via JM+RDR
  • Can take randomized measurements of stacked images and process or make inferences

w/ unfused RDR

w/ unfused and no RDR

multisensor fusion via jm rdr1
Multisensor Fusion via JM+RDR
  • Can compute randomized measurements in-place
    • ex: as we transmit to collection/processing point
simulation results
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
  • Goal: classify a test image

class 1

class 2

simulation results1
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
    • joint manifold

Joint Manifold

real world experiment
“Real World” Experiment

manifold learnedfrom data

manifold learnedfrom RDR

real world experiment1
“Real World” Experiment

joint manifold learned from data

joint manifold learned from RDR

thrust 5 scalable learning
Thrust 5: Scalable Learning
  • Sparse signal models
    • learning new sparse dictionaries
  • Manifold models
    • Manifold lifting [Wakin, next talk]
    • Manifold learning as high-dimensional function estimation [DeVore]
  • Graphical model learning
graphical model learning
Graphical Model Learning
  • Learn Gaussian graphical model by learning inverse covariance matrix [Wainwright]
  • Learn best fitting sparse model (in term of number of edges) via L1 optimization
  • Provably consistent
summary
Summary
  • Re-think data acquisition/processing pipeline
  • Exploit low-dimensional geometrical structure of
    • sparse signal models
    • graphical signal models
    • manifold signal models
  • Scalable algorithms via randomized dim. reduction
  • Progress to date:
    • multi-signal sparse models
    • smashed filter for inference
    • joint manifold model for fusion
    • manifold lifting
    • graphical model learning

dsp.rice.edu

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