Abstract
We argue that many optimization methods can be viewed as representatives of “forcing”, a methodological approach that attempts to bridge the gap between data and mathematics on the basis of an a priori trust in the power of a mathematical technique, even when detailed, credible models of a phenomenon are lacking or do not justify the use of this technique. In particular, we show that forcing is implied in particle swarms optimization methods, and in modeling image processing problems through optimization. From these considerations, we extrapolate a principle for general data analysis methods, what we call ‘Brandt’s principle’, namely the assumption that an algorithm that approaches a steady state in its output has found a solution to a problem, or needs to be replaced. We finally propose that biological systems, and other phenomena that respect general rules of morphogenesis, are a natural setting for the application of this principle.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Arfken, G. B., & Weber, H. J. (2005). Mathematical methods for physicists. Boston: Elsevier Academic Press.
Bailey, K. D. (2006). Living systems theory and social entropy theory. Systems Research and Behavioral Science, 23, 291–300.
Berger, J. O. (2010). Statistical decision theory and Bayesian analysis. New York: Springer.
Bhutada, G. G., Anand, R. S., & Saxena, S. C. (2012). PSO-based learning of sub-band adaptive thresholding function for image denoising. Signal, Image and Video Processing, 6(1), 1–7.
Boyd, S., & Lieven, V. (2004). Convex optimization. Cambridge/New York: Cambridge University Press.
Brandt, A. (2001). Multiscale scientific computation: Review 2001. In J. B. Timothy, T. F. Chan & R. Haimes (Eds.), Multiscale and multiresolution methods: Theory and applications. Berlin/New York: Springer.
Brandt, A., & Livne, O. E. (2011). Multigrid techniques: 1984 guide with applications to fluid dynamics. Philadelphia: Society for Industrial and Applied Mathematics.
Bresson, X., Esedoḡlu, S., Vandergheynst, P., Thiran, J.-P., & Osher, S. (2007). Fast global minimization of the active contour/snake model. Journal of Mathematical Imaging and Vision, 28(2), 151–167.
Chen, S. S., Donoho, D. L., & Saunders, M. A. (2001). Atomic decomposition by basis pursuit. SIAM Review, 43(1), 129–159.
Delimata, P., & Suraj, Z. (2013). Hybrid methods in data classification and reduction. In A. Skowron & Z. Suraj (Eds.), Rough sets and intelligent systems – Professor Zdzisław Pawlak in memoriam. Intelligent Systems Reference Library, 43, 263–291.
Donoho, D. L., Elad, M., & Temlyakov, V. N. (2006). Stable recovery of sparse overcomplete representations in the presence of noise. IEEE Transactions on Information Theory, 52(1), 6–18.
Donoho, D. L., & Johnstone, I. M. (1994). Ideal spatial adaptation by wavelet shrinkage. Biometrika, 81(3), 425–455.
Donoho, D. L., & Johnstone, I. M. (1995). Adapting to unknown smoothness via wavelet shrinkage. Journal of the American Statistical Association, 90(432), 1200–1224.
Dorigo, M., et al. (2008). Particle swarm optimization. Scholarpedia, 3(11), 1486.
Epstein, J. M. (2001). Learning to be thoughtless: Social norms and individual computation. Computational Economics, 18, 9–24.
Epstein, J. M. (2006). Generative social science: Studies in agent-based computational modeling. Princeton: Princeton University Press.
Field, D. J. (1999). Wavelets, vision and the statistics of natural scenes. Philosophical Transactions of the Royal Society A, 357, 2527–2542.
Grenander, U., & Miller, M. (2007). Pattern theory: From representation to inference. Oxford: Oxford University Press.
Hastie, T., Tibshirani, R., & Friedman, J. (2009). The Elements of statistical learning. New York: Springer.
Izenman, A. J. (2008). Modern multivariate statistical techniques: Regression, classification, and manifold learning. New York: Springer.
Kass, M., Witkin, A., & Terzopoulos, D. (1987). Snakes: Active contour models. International Journal of Computer Vision, 1(4), 321–331.
Kennedy, J., & Eberhart, R. C. (1995). Particle swarm optimization. In Proceedings of the IEEE 978 International Conference on Neural Networks, Perth, WA (Vol. 4, pp. 1942–1948). New York: IEEE.
Mallat, S. (2008). A wavelet tour of signal processing. San Diego: Academic Press.
Miller, J. G. (1978). Living systems. New York: McGraw Hill.
Minelli, A. (2011). A principle of Developmental Inertia. In B. Hallgrimsson & B. K. Hall (Eds.), Epigenetics: Linking genotype and phenotype in development and evolution. Berkeley, CA: University of California Press.
Napoletani, D., Panza, M., & Struppa, D. C. (2011). Agnostic science. Towards a philosophy of data analysis. Foundations of Science, 16(19), 1–20.
Napoletani, D., Panza, M., & Struppa, D. C. (2013). Artificial diamonds are still diamonds. Foundations of Science, 18(3), 591–594.
Napoletani, D., Panza, M. & Struppa, D. C. (2013). Processes rather than descriptions? Foundations of Science, 18(3), 587–590.
Napoletani, D., Panza, M. & Struppa, D. C. (2014). Is big data enough? A reflection on the changing role of mathematics in applications. Notices of the American Mathematical Society, 61(5), 485–490.
Nasri, M., & Pour, H. N. (2009). Image denoising in the wavelet domain using a new adaptive thresholding function. Journal of Neurocomputing, 72, 1012–1025.
Olshausen, B. A., & Field, D. J. (1996). Emergence of simple-cell receptive field properties by learning a sparse code for natural images. Nature, 381, 607–609.
Pedersen, M. E. H., & Chipperfield, A. J. (2010). Simplifying particle swarm optimization. Applied Soft Computing, 10(2), 618–628.
Poli, R. (2008). Analysis of the publications on the applications of particle swarm optimisation. Journal of Artificial Evolution and Applications. doi: 10.1155/2008/685175.
Reynolds, C. W. (1987). Flocks, herds, and schools: A distributed behavioral model. Computer Graphics, 21(4), 25–34.
Wegener, I. (2005). Complexity theory: Exploring the limits of efficient algorithms. Berlin; New York: Springer.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Napoletani, D., Panza, M., Struppa, D.C. (2017). Forcing Optimality and Brandt’s Principle. In: Lenhard, J., Carrier, M. (eds) Mathematics as a Tool. Boston Studies in the Philosophy and History of Science, vol 327. Springer, Cham. https://doi.org/10.1007/978-3-319-54469-4_13
Download citation
DOI: https://doi.org/10.1007/978-3-319-54469-4_13
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-54468-7
Online ISBN: 978-3-319-54469-4
eBook Packages: Earth and Environmental ScienceEarth and Environmental Science (R0)