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Category learning from equivalence constraints

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Abstract

Information for category learning may be provided as positive or negative equivalence constraints (PEC/NEC)—indicating that some exemplars belong to the same or different categories. To investigate categorization strategies, we studied category learning from each type of constraint separately, using a simple rule-based task. We found that participants use PECs differently than NECs, even when these provide the same amount of information. With informative PECs, categorization was rapid, reasonably accurate and uniform across participants. With informative NECs, performance was rapid and highly accurate for only some participants. When given directions, all participants reached high-performance levels with NECs, but the use of PECs remained unchanged. These results suggest that people may use PECs intuitively, but not perfectly. In contrast, using informative NECs enables a potentially more accurate categorization strategy, but a less natural, one which many participants initially fail to implement—even in this simplified setting.

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Notes

  1. For clarity of presentation and simplicity of experimentation, this example—as our experimental paradigm—uses binary feature values and categories defined by rules. Nevertheless, the conclusions of the analysis—as the results of the study—extend to other categorization scenarios.

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Acknowledgments

This study was supported by a “Center of Excellence” grant from the Israel Science Foundation, a grant from the US-Israel Binational Science Foundation, and a grant by the EU under the DIRAC integrated project IST-027787. Preliminary results of this study were presented in the annual meeting of the Cognitive Science Society, Stresa, Italy, July 2005. We would like to thank Lee Brooks and Gil Diesendruck for their comments. We also thank Michael Ziessler and an anonymous reviewer for their useful comments.

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Authors and Affiliations

  1. Interdisciplinary Center for Neural Computation, Hebrew University, Edmond Safra Campus, 91904, Jerusalem, Israel

    Rubi Hammer, Tomer Hertz, Shaul Hochstein & Daphna Weinshall

  2. Neurobiology Department, Institute of Life Sciences, Hebrew University, Jerusalem, Israel

    Rubi Hammer & Shaul Hochstein

  3. School of Computer Sciences and Engineering, Hebrew University, Jerusalem, Israel

    Tomer Hertz & Daphna Weinshall

Authors
  1. Rubi Hammer
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  2. Tomer Hertz
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  3. Shaul Hochstein
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Correspondence to Rubi Hammer.

Appendix 1

Appendix 1

We analyze the dependence of the number of possible PECs, NECs, and highNECs on the number of objects and categories. Note that all PECs are informative for identifying relevant dimensions while in the case of NECs, only the highNECs (negative constraints made up of two objects from two different categories that differ in their value on only a single dimension) are adequately informative for such a task. To simplify the discussion, we assume that the number of objects in each category is identical.

Specifically, let

  • c = the number of categories.

  • n = the number of objects in each category.

  • d = the number of relevant dimensions, assuming binary dimension, d = log2 c

It follows that

$$ {\text{number of PEC}} = \frac{nc(n - 1)}{2};\quad {\text{number of NEC}} = \frac{{n^{2} c(c - 1)}}{2};\quad {\text{number of highNEC}} = \frac{\text{ncd}}{2}. $$

This calculation shows that the total number of PECs is much smaller than the total number of NECs specifically when the number of categories, c, is large. In addition, highNECs (NECs which provide 1 Bit of information) are a small subset of NECs when the number of category members, n, is large. Specifically:

$$ \frac{\text{PEC}}{\text{NEC}} = \frac{nc(n - 1)}{{n^{2} c(c - 1)}} \approx \frac{(n - 1)}{n(c - 1)} \approx \frac{1}{c} \ll 1 $$
$$ \frac{\text{highNEC}}{\text{NEC}} = \frac{\text{ncd}}{{n^{2} c(c - 1)}} \approx \frac{{\log_{2} c}}{nc} \ll \frac{1}{c} $$

In the current experiment, nc = 32. When d = 2, c = 4 and n = 8. Then, there are 112 PECs and 384 NECs, of which 32 are highNEC. When d = 3, c = 8 and n = 4. Then, there are 48 PECs and 448 NECs, of which only 48 are highNEC.

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Hammer, R., Hertz, T., Hochstein, S. et al. Category learning from equivalence constraints. Cogn Process 10, 211–232 (2009). https://doi.org/10.1007/s10339-008-0243-x

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