Abstract
A theory of patterns analysis has to suggest criteria how patterns in data can be defined in a meaningful way and how they should be compared. Similarity-based Pattern Analysis and Recognition is expected to adhere to fundamental principles of the scientific process that are expressiveness of models and reproducibility of their inference. Patterns are assumed to be elements of a pattern space or hypothesis class and data provide “information” which of these patterns should be used to interpret the data. The mapping between data and patterns is constructed by an inference algorithm, in particular by a cost minimization process. Fluctuations in the data usually limit the precision that we can achieve to uniquely identify a single pattern as interpretation of the data. We advocate an information-theoretic perspective on pattern analysis to resolve this dilemma where the tradeoff between informativeness of statistical inference and their stability is mirrored in the information-theoretic optimum of high information rate and zero communication error. The inference algorithm is considered as a noisy channel which naturally limits the resolution of the pattern space given the uncertainty of the data.
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Notes
- 1.
In the following, we restrict hypotheses to map an object to a pattern. The more general situation of object configurations can be analyzed in an analogous way but involves a more complex notation.
- 2.
For binary weights, Z β (X (n)) corresponds the microcanonical partition function by assuming that almost all solutions cost close to R(c ⊥,X (n))+1/β.
- 3.
The reader should keep in mind that we are not interested in deriving a new principle for coding, but we exploit the communication metaphor to derive a quantitative criterion of how precisely we can approximate the global minimizer of a cost function by an approximation set.
- 4.
Superscript (n) dropped for readability.
- 5.
Please note that AEP has to be proved for a selected cost function R.
- 6.
W.l.o.g. we use the symmetric encoding {−1,1} rather than {0,1} to simplify the calculations.
References
Alon, N., Ben-David, S., Cesa-Bianchi, N., Haussler, D.: Scale-sensitive dimensions, uniform convergence, and learnability. J. ACM 44(4), 615–631 (1997)
Buhmann, J.M.: Information theoretic model validation for clustering. In: International Symposium on Information Theory, Austin Texas. IEEE Press, New York (2010). http://arxiv.org/abs/1006.0375
Buhmann, J.M., Kühnel, H.: Vector quantization with complexity costs. IEEE Trans. Inf. Theory 39(4), 1133–1145 (1993)
Buhmann, J.M., Chehreghani, M.H., Frank, M., Streich, A.P.: Information theoretic model selection for pattern analysis. In: Guyon, I., Dror, G., Lemaire, V., Taylor, G., Silver, D. (eds.) ICML 2011 Workshop on “Unsupervised and Transfer Learning”, Bellevue, Washington, vol. 27, pp. 51–65 (2012). Clearwater Beach, Florida, JMLR: W&CP 5
Buhmann, J.M., Mihalák, M., Srámek, R., Widmayer, P.: Robust optimization in the presence of uncertainty. In: Inventions in Theoretical Computer Science 2013, Berkeley. ACM 2013, pp. 505–514 (2012). doi:10.1145/2422436.2422491
Busse, L.: Information in orderings (learning to order). Ph.D. thesis, # 20600, ETH Zurich, CH-8092 Zurich, Rämistrasse (2012)
Busse, L.M., Chehreghani, M.H., Buhmann, J.M.: The information content in sorting algorithms. In: International Symposium on Information Theory, pp. 2746–2750. IEEE Press, Cambridge (2012)
Chehreghani, M.H., Giovanni Busetto, A., Buhmann, J.M.: Information theoretic model validation for spectral clustering. In: AISTATS 2012, La Palma. J. Mach. Learn. Res. (W&CP), vol. 22, pp. 495–503 (2012)
Cover, T.M., Thomas, J.A.: Elements of Information Theory, 2nd edn. Wiley, New York (1991)
Csiczár, I., Körner, J.: Information Theory: Coding Theorems for Discrete Memoryless Systems. Academic Press, New York (1981)
Frank, M., Buhmann, J.M.: Selecting the rank of SVD by maximum approximation capacity. In: International Symposium on Information Theory, St. Petersburg, pp. 1036–1040. IEEE Press, New York (2011)
Grenander, U.: General Pattern Theory: a Mathematical Study of Regular Structures. Oxford University Press, Oxford (1994)
Grenander, U., Miller, M.I.: Pattern Theory: from Representation to Inference. Oxford University Press, Oxford (2007)
Han, L., Rossi, L., Torsello, A., Wilson, R.C., Hancock, E.R.: Information theoretic prototype selection for unattributed graphs. In: Gimel’farb, G.L., Hancock, E.R., Imiya, A., Kuijper, A., Kudo, M., Omachi, S., Windeatt, T., Yamada, K. (eds.) Structural, Syntactic, and Statistical Pattern Recognition. Lecture Notes in Computer Science, vol. 7626, pp. 33–41. Springer, Berlin (2012)
Hofmann, T., Buhmann, J.M.: Pairwise data clustering by deterministic annealing. IEEE Trans. Pattern Anal. Mach. Intell. 19(1), 1–14 (1997)
Lingamneni, A., Krishna Muntimadugu, K., Enz, C., Karp, R.M., Palem, K.V., Piguet, C.: Algorithmic methodologies for ultra-efficient inexact architectures for sustaining technology scaling. In: Proceedings of the 9th Conference on Computing Frontiers, CF’12, pp. 3–12. ACM, New York (2012)
Rose, K., Gurewitz, E., Fox, G.: Vector quantization by deterministic annealing. IEEE Trans. Inf. Theory 38(4), 1249–1257 (1992)
Vapnik, V.N.: Estimation of Dependencies Based on Empirical Data. Springer, New York (1982)
Vapnik, V.N., Chervonenkis, A.Ya.: On the uniform convergence of relative frequencies of events to their probabilities. Theory Probab. Appl. 16, 264–280 (1971)
Acknowledgement
This work has been partially supported by the FP7 EU project SIMBAD and by the SNF project 200021_138117. JB acknowledges very stimulating discussions with A. Busetto, L. Busse, M.H. Chehreghani, M. Frank, M. Mihalák, V. Roth, R. Srámek, W. Szpankowski and P. Widmayer.
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Buhmann, J.M. (2013). SIMBAD: Emergence of Pattern Similarity. In: Pelillo, M. (eds) Similarity-Based Pattern Analysis and Recognition. Advances in Computer Vision and Pattern Recognition. Springer, London. https://doi.org/10.1007/978-1-4471-5628-4_3
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