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


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

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Richard baraniuk volkan cevher rice university ron devore texas a m university martin wainwright

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


    2 graphical models

    2. Graphical Models


    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 models2

    Graphical Models


    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


    Hierarchical graphical models

    Hierarchical Graphical Models


    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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