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Numerical Computation of Non-Comm VoI Metrics & Spectra for Random Graphs

This research program focuses on information-driven learning to quantify VoI measures and analyze random graphs using Eigen-analysis methods. The project aims to exploit quantified uncertainty for sensor fusion and learning roles using non-commutative information theory approaches. The study addresses the challenges of Eigen-VoI quantification, phase transitions, and rates of convergence with impact on information fusion and network analytics. The research also explores predicting graph spectra based on expected degree sequences, aiding in community structure discovery in networks.

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Numerical Computation of Non-Comm VoI Metrics & Spectra for Random Graphs

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  1. Numerical computation of Non-Comm. VoI Metrics & Spectra of Random Graphs Co-PI Raj RaoNadakuditi University of Michigan

  2. 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

  3. 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

  4. 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

  5. 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

  6. 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”

  7. 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

  8. Numerically computing D-transform • Desired: • Allow continuous and discrete valued inputs • O(n log n) where n is number of singular values • Numerically stable

  9. Empirical VoI quantification • Based on an eigen-gap based segment, compute non-commVoI subspaces

  10. 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)

  11. 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

  12. 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

  13. 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)

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

  15. Phase transition in comm. detection • Unidentifable: If cin – cout < 2 • cin = Avg. degree “within”; cout= Avg. degree “without”

  16. Year 2 plans • 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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