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

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

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Non-comm. info. measures in high-dim.-relatively-small-sample regime

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

Co-PI Raj RaoNadakuditi

University of Michigan

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

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

- 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.

- e.g. f-divergence, Shannon mutual info., Sanov’sthm.

- Non-commutative analogs of above

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

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”

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

Numerically computing D-transform

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

Empirical VoI quantification

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

- 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)

- 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

- 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

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)

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

Phase transition in comm. detection

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

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

- Information fusion

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