1. Value of Information 1 st year review. UCLA 2012 Kickoff ARO MURI on Value-centered Information Theory for Adaptive Learning, Inference, Tracking, and Exploitation Value of Information 1 st year review. UCLA 2012 Numerical computation of Non- Comm. VoI Metrics & Spectra of Random Graphs Co-PI Raj Rao Nadakuditi University of Michigan

2. Value of Information 1 st year review. UCLA 2012 Kickoff 1 st year review. UCLA 2012 Value of Information Mission Information and Objectives Non-commutative Info Theory Info-geometric learning Consensus learning Info theoretic surrogates Information-driven Learning. Jordan (Lead); Ertin, Fisher, Hero, Nadakuditi Bounds, models and learning algorithms Scalable, Actionable VoI measures Research program Info-driven learning

3. Value of Information 1 st year review. UCLA 2012 Kickoff 1 st year review. UCLA 2012 Value of Information 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 Eigen-analysis methods & apps.

4. Value of Information 1 st year review. UCLA 2012 Kickoff 1 st year review. UCLA 2012 Value of Information 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’s thm. vs. Non-comm. info. measures in high-dim.-relatively-small- sample regime Non-commutative analogs of above

5. Value of Information 1 st year review. UCLA 2012 Kickoff 1 st year review. UCLA 2012 Value of Information Analytical signal-plus-noise model Low dimensional (= k) latent signal model X n is n x m noise-only Gaussian matrix c = n/m = # Sensors / # Samples Theta ~ SNR

6. Value of Information 1 st year review. UCLA 2012 Kickoff 1 st year review. UCLA 2012 Value of Information 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. Value of Information 1 st year review. UCLA 2012 Kickoff 1 st year review. UCLA 2012 Value of Information A non-commutative VoI metric (beyond Gaussians) X n is 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. Value of Information 1 st year review. UCLA 2012 Kickoff 1 st year review. UCLA 2012 Value of Information 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. Value of Information 1 st year review. UCLA 2012 Kickoff 1 st year review. UCLA 2012 Value of Information Empirical VoI quantification Based on an eigen-gap based segment, compute non- comm VoI subspaces

10. Value of Information 1 st year review. UCLA 2012 Kickoff 1 st year review. UCLA 2012 Value of Information 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. Value of Information 1 st year review. UCLA 2012 Kickoff 1 st year review. UCLA 2012 Value of Information 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 Broader Impact

12. Value of Information 1 st year review. UCLA 2012 Kickoff 1 st year review. UCLA 2012 Value of Information 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. Value of Information 1 st year review. UCLA 2012 Kickoff 1 st year review. UCLA 2012 Value of Information Non. Comm. Prob. for Network Science Role of spectra of social and related networks: Community structure discovery Dynamics Stability Open Solved 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. Value of Information 1 st year review. UCLA 2012 Kickoff 1 st year review. UCLA 2012 Value of Information Accomplishment - II Predicting spectra (numerical free convolution – Accomplishment I) “When is a hub not a hub (spectrally)?” New phenomena, new VoI analytics

15. Value of Information 1 st year review. UCLA 2012 Kickoff 1 st year review. UCLA 2012 Value of Information Phase transition in comm. detection

16. Value of Information 1 st year review. UCLA 2012 Kickoff 1 st year review. UCLA 2012 Value of Information Year 2 plans Accomplishments – Numerical computation of Non-Comm convolutions – 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