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Numerical computation of Non-Comm. VoI Metrics & Spectra of Random Graphs . Co-PI Raj Rao Nadakuditi University of Michigan. Research program Info-driven learning. Mission Information and Objectives. Non-commutative Info Theory. Info theoretic surrogates. Consensus learning.

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slide1

Numerical computation of Non-Comm. VoI Metrics & Spectra of Random Graphs

Co-PI Raj RaoNadakuditi

University of Michigan

slide2

Research program

Info-driven learning

Mission Information

and

Objectives

Non-commutative

Info Theory

Info theoretic

surrogates

Consensus

learning

Info-geometric

learning

Information-driven Learning

. Jordan (Lead); Ertin, Fisher,

Hero, Nadakuditi

Scalable, Actionable

VoI measures

Bounds, models and

learning algorithms

slide3

Eigen-analysis methods & apps.

  • Principal component analysis
    • Direction-finding (e.g. sniper localization)
    • Pre-processing/Denoising to SVM-based classification
    • (e.g. pattern, gait & face recognition)
    • Regression, Matched subspace detectors
    • Community/Anomaly detection in networks/graphs
  • Canonical Correlation Analysis
    • PCA-extension for fusing multiple correlated sources
  • LDA, MDS, LSI, Kernel(.) ++, MissingData(.)++
  • Eigen-analysis  Spectral Dim. Red. Subspace methods
  • Technical challenge:
    • Quantify eigen-VoI (Thrust 1) and Exploit quantified uncertainty (Thrust 2) for eigen-analysis based sensor fusion and learning
slide4

Role of Non-Comm. Info theory

  • For noisy, estimated subspaces, quantify:
    • Fundamental limits and phase transitions
    • Estimates of accuracy possibly, data-driven
    • Rates of convergence, learning rates
    • P-values
    • Impact of adversarial noise models
  • “Classical” info. measures in low-dim.-large sample regime
    • e.g. f-divergence, Shannon mutual info., Sanov’sthm.
              • vs.
  • Non-comm. info. measures in high-dim.-relatively-small-sample regime
    • Non-commutative analogs of above
slide5

Analytical signal-plus-noise model

  • Low dimensional (= k) latent signal model
  • Xnis n x m noise-only Gaussian matrix
  • c = n/m = # Sensors / # Samples
  • Theta ~ SNR
slide6

Empirical subspaces are unequal

  • c = n/m = # Sensors / # Samples
  • Theta ~ SNR, X is Gaussian
  • Insight: Subspace estimates are biased!
    • “Large-n-large-m” versus “Small-n-large-m”
slide7

A non-commutative VoImetric (beyond Gaussians)

  • Xnis n x m unitarily-invariant noise-only random matrix
  • Theorem [N. and Benaych-Georges, 2011]:
  • μ = Spectral measure of noise singular values
  • D = D-transform of μ “log-Fourier” transform in NCI
slide8

Numerically computing D-transform

  • Desired:
    • Allow continuous and discrete valued inputs
    • O(n log n) where n is number of singular values
    • Numerically stable
slide9

Empirical VoI quantification

  • Based on an eigen-gap based segment, compute non-commVoI subspaces
slide10

Accomplishment - I

  • Uk are Chebyshev polynomials
  • Series coefficients computed via DCT in O(n log n)
  • Closed-form G transform (and hence D transform) series expansion!
  • “Numerical computation of convolutions in free probability theory” (with Sheehan Olver)
slide11

Broader Impact

  • For noisy, estimated subspaces, quantify:
    • Fundamental limits and phase transitions
    • Estimates of accuracy possibly, data-driven
    • Rates of convergence, learning rates
    • P-values
    • Impact of adversarial noise models
    • Impact of finite training data
  • Facilitate fast, accurate performance prediction for eigen-methods!
  • Transition: MATLAB toolbox
slide12

Spectra of Networks

  • Role of spectra of social and related networks:
    • Community structure discovery
    • Dynamics
    • Stability
  • Open problem: Predict graph spectra given degree sequence
  • Broader Impact: ARL CTA & ITA, ARO MURI
slide13

Non. Comm. Prob. for Network Science

  • Role of spectra of social and related networks:
    • Community structure discovery
    • Dynamics
    • Stability
  • OpenSolved problem: Predict spectra of a graph given expected degree sequence
  • Answer: Free multiplicative convolution of degree sequence with semi-circle
  • “Spectra of graphs with expected degree sequence” (with Mark Newman)
slide14

Accomplishment - II

  • Predicting spectra (numerical free convolution – Accomplishment I)
  • “When is a hub not a hub (spectrally)?”
  • New phenomena, new VoI analytics
slide15

Phase transition in comm. detection

  • Unidentifable: If cin – cout < 2
  • cin = Avg. degree “within”; cout= Avg. degree “without”
slide16

Relation to other research thrusts

  • Accomplishments
    • Numerical computation of Non-Commconvolutions
    • Predicting spectra of complicated networks
  • Impact
    • Information fusion
      • Numerical computation of Non-Comm. Metrics
      • Performance prediction
      • New VoI analytics for networks
      • Predicting graph spectra from degree sequence
    • Information exploitation
      • Selective fusion of subspace information
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