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

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