Adaptive sensing performance lower bounds for sparse signal detection and support estimation

Bernoulli

Volumn 20, Issue 4, 2014, Pages 2217-2246

  Castro, Rui M ^a  

^a EINDHOVEN UNIVERSITY OF TECHNOLOGY   (Netherlands)

Author keywords

Adaptive sensing; Minimax lower bounds; Sequential experimental design; Sparsity based models

Indexed keywords

EID: 84907437332     PISSN: 13507265     EISSN: None     Source Type: Journal    
DOI: 10.3150/13-BEJ555     Document Type: Article

References (36)

1. Addario-Berry, L., Broutin, N., Devroye, L. and Lugosi, G. (2010). On combinatorial testing problems. Ann. Statist. 38 3063-3092. MR2722464
2. Arias-Castro, E., Candès, E.J. and Davenport, M.A. (2013). On the fundamental limits of adaptive sensing. IEEE Trans. Inform. Theory 59 472-481. MR3008159
3. Arias-Castro, E., Candès, E.J., Helgason, H. and Zeitouni, O. (2008). Searching for a trail of evidence in a maze. Ann. Statist. 36 1726-1757. MR2435454
4. Balcan, N., Beygelzimer, A. and Langford, J. (2006). Agostic active learning. In 23rd International Conference on Machine Learning 65-72.
5. Bessler, S.A. (1960). Theory and applications of the sequential design of experiments, k-actions and infinitely many experiments: Part I - Theory. Technical Report 55, Stanford Univ., Applied Mathematics and Statistics Laboratories.
6. Blanchard, G. and Geman, D. (2005). Hierarchical testing designs for pattern recognition. Ann. Statist. 33 1155-1202. MR2195632
7. Butucea, C. and Ingster, Y. (2013). Detection of a sparse submatrix of a high-dimensional noisy matrix. Bernoulli 19 2652-2688.
8. Cai, T.T., Jin, J. and Low, M.G. (2007). Estimation and confidence sets for sparse normal mixtures. Ann. Statist. 35 2421-2449. MR2382653
9. Castro, R., Willett, R. and Nowak, R. (2005). Faster rates in regression via active learning. In Advances in Neural Information Processing Systems 18 179-186.
10. Castro, R.M. and Nowak, R.D. (2008). Minimax bounds for active learning. IEEE Trans. Inform. Theory 54 2339-2353. MR2450865
11. Chernoff, H. (1959). Sequential design of experiments. Ann. Math. Statist. 30 755-770. MR0108874
12. Cohn, D., Ghahramani, Z. and Jordan, M. (1996). Active learning with statistical models. J. Artificial Intelligence Res. 4 129-145.
13. Dasgupta, S. (2004). Analysis of a greedy active learning strategy. In Advances in Neural Information Processing Systems 17 337-344.
14. Dasgupta, S. (2005). Coarse sample complexity bounds for active learning. In Advances in Neural Information Processing Systems 18 235-242.
15. Dasgupta, S., Kalai, A. and Monteleoni, C. (2005). Analysis of perceptron-based active learning. In Eighteenth Annual Conference on Learning Theory (COLT) 249-263.
16. Donoho, D. and Jin, J. (2004). Higher criticism for detecting sparse heterogeneous mixtures. Ann. Statist. 32 962-994. MR2065195
17. Donoho, D.L. (2006). Compressed sensing. IEEE Trans. Inform. Theory 52 1289-1306. MR2241189
18. El-Gamal, M.A. (1991). The role of priors in active Bayesian learning in the sequential statistical decision framework. In Maximum Entropy and Bayesian Methods (Laramie, WY, 1990). Fund. Theories Phys. 43 (W.T. Grandy and L.H. Schich, eds.) 33-38. Dordrecht: Kluwer Academic. MR1173460
19. Fedorov, V.V. (1972). Theory of Optimal Experiments. New York: Academic Press. MR0403103
20. Freund, Y., Seung, H.S., Shamir, E. and Tishby, N. (1997). Selective sampling using the query by committee algorithm. Machine Learning 28 133-168.
21. Hall, P. and Molchanov, I. (2003). Sequential methods for design-adaptive estimation of discontinuities in regression curves and surfaces. Ann. Statist. 31 921-941. MR1994735
22. Hanneke, S. (2011). Rates of convergence in active learning. Ann. Statist. 39 333-361. MR2797849
23. Haupt, J., Baraniuk, R., Castro, R. and Nowak, R. (2012). Sequentially designed compressed sensing. In IEEE Statistical Signal Processing Workshop (IEEE SSP) Proceedings 401-404. Available at http://www.win.tue.nl/~rmcastro/publications/SCS.pdf.
24. Haupt, J., Castro, R.M. and Nowak, R. (2011). Distilled sensing: Adaptive sampling for sparse detection and estimation. IEEE Trans. Inform. Theory 57 6222-6235. MR2857969
25. Ingster, Y.I. (1997). Some problems of hypothesis testing leading to infinitely divisible distributions. Math. Methods Statist. 6 47-69. MR1456646
26. Ingster, Y.I. and Suslina, I.A. (2003). Nonparametric Goodness-of-fit Testing Under Gaussian Models. Lecture Notes in Statistics 169. New York: Springer. MR1991446
27. Kim, J.-C. and Korostelev, A. (2000). Rates of convergence for the sup-norm risk in image models under sequential designs. Statist. Probab. Lett. 46 391-399. MR1743998
28. Koltchinskii, V. (2010). Rademacher complexities and bounding the excess risk in active learning. J. Mach. Learn. Res. 11 2457-2485. MR2727771
29. Lai, T.L. and Robbins, H. (1985). Asymptotically efficient adaptive allocation rules. Adv. in Appl. Math. 6 4-22. MR0776826
30. Malloy, M. and Nowak, R. (2011). On the limits of sequential testing in high dimensions. In Asilomar Conference on Signals, Systems and Computers 1245-1249. Available at http://arxiv.org/abs/1105.4540.
31. Malloy, M. and Nowak, R. (2011). Sequential analysis in high-dimensional multiple testing and sparse recovery. In The IEEE International Symposium on Information Theory 2661-2665. Available at http://arxiv.org/abs/1103.5991v1.
32. Meinshausen, N. and Rice, J. (2006). Estimating the proportion of false null hypotheses among a large number of independently tested hypotheses. Ann. Statist. 34 373-393. MR2275246
33. Novak, E. (1996). On the power of adaption. J. Complexity 12 199-237. MR1408328
34. Tsybakov, A.B. (2009). Introduction to Nonparametric Estimation. New York: Springer. MR2724359
35. Wald, A. (1947). Sequential Analysis. New York: Wiley. MR0020764
36. Wasserman, L. (2006). All of Nonparametric Statistics. Springer Texts in Statistics. New York: Springer. MR2172729