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Projector operators in clustering. (English) Zbl 1353.62068

Summary: In a recent paper, the notion of quantum perceptron has been introduced in connection with projection operators. Here, we extend this idea, using these kind of operators to produce a clustering machine, that is, a framework that generates different clusters from a set of input data. Also, we consider what happens when the orthonormal bases first used in the definition of the projectors are replaced by frames and how these can be useful when trying to connect some noised signal to a given cluster.

MSC:

62H30 Classification and discrimination; cluster analysis (statistical aspects)
68T10 Pattern recognition, speech recognition
42C15 General harmonic expansions, frames

Software:

C4.5; WEKA
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References:

[1] Minsky, Perceptrons (1969)
[2] Siomau, A quantum model for autonomous learning automata, Quantum Information Processing 13 (5) pp 1211– (2014)
[3] Messiah, Quantum mechanics, North Holland Publishing Company, Amsterdam, 1961. E. Merzbacher, Quantum Mechanics (1998) · Zbl 0102.42602
[4] Reed, Academic Press, New York, 1980; G. K. Pedersen, Analysis now, in: Methods of Modern Mathematical Physics, I (1989)
[5] Jain, Data clustering: a review, ACM computing surveys (CSUR) 31 pp 264– (1999)
[6] Goyal, Quantized frame expansions with erasures Appl, Computational Harmonic Analysis 10 (3) pp 203– (2001) · Zbl 0992.94009
[7] Casazza, Equal-norm tight frames with erasures, advances in computational mathematics 18 pp 387– (2003) · Zbl 1035.42029
[8] Benedetto, Finite normalized tight frames, Advances in Computational Mathematics 18 pp 357– (2003) · Zbl 1028.42022
[9] Springer, Frame potential minimization for clustering short time series, Advances in Data Analysis and Classification pp 341– (2011) · Zbl 1274.62430
[10] Christensen, An Introduction to Frames and Riesz Bases (2002) · Zbl 1017.42022
[11] Daubechies, Ten Lectures on Wavelets (1992) · Zbl 0776.42018
[12] Heil, A Basis Theory Primer: Expanded Edition (2010)
[13] Goguen, L-Fuzzy sets, Journal of Mathematical Analysis and Applications 18 pp 145– (1967) · Zbl 0145.24404
[14] Finlayson, Color by correlation: a simple, unifying framework for colour constancy, IEEE Transactions on Pattern Analysis and Machine Intelligence 23 pp 1209– (2001)
[15] Li, Regularized color clustering in medical image database, IEEE Transactions on Medical Imaging 19 pp 1150– (2000)
[16] Lourde, A digital guitar tuner, IJCSIS pp 6– (2009)
[17] Quinlan, C4.5: Programs for Machine Learning (1993)
[18] Hall, The WEKA data mining software: an update, SIGKDD Explorations 11 (1) pp 10– (2009) · Zbl 05740105
[19] Grochenig, Foundations of Time-Frequency Analysis (2001)
[20] Chan, Tight frame: an efficient way for high-resolution image reconstruction, Applied and Computational Harmonic Analysis 17 pp 91– (2004) · Zbl 1046.42026
[21] Eldar, Optimal tight frames and quantum measurement, IEEE Transactions on Information Theory 48 pp 599– (2002) · Zbl 1071.94510
[22] Eldar, Sampling with arbitrary sampling and reconstruction spaces and oblique dual frame vectors, Journal of Applied Analysis 9 pp 77– (2003) · Zbl 1028.94020
[23] Cai, Approximation of frame based missing data recovery, Applied and Computational Harmonic Analysis 31 pp 185– (2011) · Zbl 1220.94005
[24] Kovacevic, Life beyond bases: the advent of frames (part 1), IEEE Signal Processing Magazine 24 pp 86– (2007)
[25] Holmes, Optimal frames for erasures, Linear Algebra and its Applications 377 pp 31– (2004) · Zbl 1042.46009
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