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ADD TO TALK

- RON STUFF

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

Goals

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

networkinfrastructure

humanintelligence

Networked Sensing

Challenges

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

networkinfrastructure

humanintelligence

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

- 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 Exploit geometry of these models

- Unifying theme: low-dimensional signal structure
- Sparse signal models
- Graphical models
- 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

- Goal: preserve information from x in y
- One avenue: stable embedding
- Key question: how small can M be?

signalfromsparse,graphical,manifoldmodel

measurements

Sparse Models

K-dim subspaces

Sparse Models

K-dim subspaces

Sparse Models

K-dim subspaces

- Stable embedding <> Restricted isometry property (RIP) from compressive sensing
- Stability whp if

Single-Pixel Camera

M randomizedmeasurements

N mirrors

target N=65536 pixels

M=1300 measurements (2%)

M=11000 measurements (16%)

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

- 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

- 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

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

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

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 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

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

Naïve approach

take M CS measurements,

recover N-pixel image from CS measurements (expensive)

conventional matched filter

Smashed Filter

Worldly approach

take M CS measurements,

matched filter directly on CS measurements(inexpensive)

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

- Sparse signal models
- multi-signal sparse models [Wakin, next talk]

- Manifold models
- joint manifold models [next]

- Graphical models

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

- Fusion: stack corresponding image vectors taken at the same time
- Fused images still lie on L-dim manifold in“joint manifold”

Joint Manifolds

- Given submanifolds
- L-dimensional
- homeomorphic (we can continuously map between any pair)

- Define joint manifoldas concatenation of

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

- Can take randomized measurements of stacked images and process or make inferences

w/ unfused RDR

w/ unfused and no RDR

Multisensor Fusion via JM+RDR

- Can compute randomized measurements in-place
- ex: as we transmit to collection/processing point

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 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

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

- 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

- 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

Summary

- Scalable distributed sensing requires a re-think of the entire sensing and data processing pipeline
- New data representation: random encoding
- preserves info in a wide range of data types
- acts as source/channel fountain codes
- supports efficient processing and inference algorithms
- supports efficient fusion from multiple sensors
- supports a range of actuation/navigation strategies
- scalable in resolution N and number of sensors J
- secure

dsp.rice.edu/cs

Manifold Learning

- Given training points in , learn the mapping to the underlying K-dimensional articulation manifold
- ISOMAP, LLE, HLLE, …
- Example
- images of rotating teapot (L=1)
- articulation space = circle

Compressive Manifold Learning

- ISOMAP algorithm based on geodesic distances between points
- Random measurements preserve these distances
- Theorem: If , then the ISOMAP residual variance in the projected domain is bounded by the additive error factor

translatingdisk manifold(L=2)

full data (N=4096)

M = 100

M = 50

M = 25

Manifold Learning via Joint Manifolds

- Goal: Learn embeddingof 2D translating ellipse(with noise)
N=45x45=2025 pixelsJ=20 views at different angles

Manifold Learning via Joint Manifolds

- Goal: Learn embeddingof 2D translating ellipse(with noise)
N=45x45=2025 pixelsJ=20 views

- Embeddingslearnedseparately

Manifold Learning via Joint Manifolds

- Goal: Learn embeddingof 2D translating ellipse(with noise)
N=45x45=2025 pixelsJ=20 views

- Embeddingslearnedseparately
- Embedding learned jointly

Manifold Learning via JM+RE

- Goal:Learn embeddingvia random compressivemeasurements
N=45x45=2025 pixels

J=20 views

- Embeddingslearnedseparately
- Embedding learned jointly
M=100 measurements per view

Scalable Communication

- RE is democratic
- each measurement carries the same amount of information
- robust to measurement loss and quantization simple encoding

- Ex: wireless streaming application with data loss
- conventional: complicated (unequal) error protection of compressed data
- DCT/wavelet low frequency coefficients

- RE: merely stream additional measurements and reconstruct using those that arrive safely (fountain-like)

- conventional: complicated (unequal) error protection of compressed data
- Joint manifold fusion supports in-network computation

Our Approach

- Re-think the sensing and data processing pipeline
- New data representation: random encoding
- preserves info in a wide range of data types
- acts as source/channel fountain code
- supports efficient processing and inference algorithms
- supports efficient fusion from multiple sensors
- supports a range of actuation/navigation strategies
- scalable in resolution N and number of sensors J
- secure

TO ADD for afosr

- Richb talk:
- Overview of the program
- Concise models: sparsity, manifolds, graphical models
- Analysis, processing, fusion, learning
- Goals
- Team members

- Our progress at rice
- JAM
- Stable manifold embedding
- App to JAM

- Overview of the program
- Mike talk:
- DCS
- Video
- Manifold lifting

Manifold Models

- IAM definition
- Can get from other talks
- Parametric models

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