Numerosity discrimination in deep neural networks: Initial competence, developmental refinement and experience statistics

Testolin, Alberto; Zou, Will Y.; McClelland, James L.

doi:10.1111/desc.12940

Cited by 50 publications

(52 citation statements)

References 61 publications

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“…In our stimulus space Numerosity, Size and Spacing varied within the same range, in order to make sure that one dimension was not statistically more salient (i.e., of higher variance) than the others. However, an interesting research direction could be to investigate how the representational space might change under different distributional properties, for example by creating stimuli that match the statistical distribution of visual features in natural environments 50 . www.nature.com/scientificreports www.nature.com/scientificreports/ Our simulations show that numerosity processing can be carried out using generic low-level computations, such as those emerging in multi-layered neural networks that learn a hierarchical generative model of the sensory data.…”

Section: Discussionmentioning

confidence: 99%

Visual sense of number vs. sense of magnitude in humans and machines

Testolin

Dolfi

Rochus

et al. 2020

Sci Rep

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Numerosity perception is thought to be foundational to mathematical learning, but its computational bases are strongly debated. Some investigators argue that humans are endowed with a specialized system supporting numerical representations; others argue that visual numerosity is estimated using continuous magnitudes, such as density or area, which usually co-vary with number. Here we reconcile these contrasting perspectives by testing deep neural networks on the same numerosity comparison task that was administered to human participants, using a stimulus space that allows the precise measurement of the contribution of non-numerical features. Our model accurately simulates the psychophysics of numerosity perception and the associated developmental changes: discrimination is driven by numerosity, but non-numerical features also have a significant impact, especially early during development. Representational similarity analysis further highlights that both numerosity and continuous magnitudes are spontaneously encoded in deep networks even when no task has to be carried out, suggesting that numerosity is a major, salient property of our visual environment.

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Section: Discussionmentioning

confidence: 99%

Visual sense of number vs. sense of magnitude in humans and machines

Testolin

Dolfi

Rochus

et al. 2020

Sci Rep

Self Cite

View full text Add to dashboard Cite

show abstract

“…In conclusion, we believe that a deeper understanding of numerosity perception will require considering alternatives to the search for evidence of adherence to idealized, essential characteristics: We should also strive to define what could be the underlying mechanisms giving rise to the complex behavioral patterns observed in these studies. Promising results in this direction have been recently achieved by connectionist modelingfor example, by showing how approximate adherence to Weber's law can emerge in generic neural networks that learn the statistics of their visual environment (Stoianov & Zorzi, 2012;Zorzi & Testolin, 2018), or how developmental trajectories of numerical acuity in children can be simulated by progressive deep learning (Testolin, Zou, & McClelland, 2020). Further research is required to explore these issues more fully, keeping in mind that we must be prudent when characterizing the actual patterns observed in the empirical data.…”

Section: Discussionmentioning

confidence: 99%

Do estimates of numerosity really adhere to Weber’s law? A reexamination of two case studies

Testolin

McClelland

2020

Psychon Bull Rev

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Both humans and nonhuman animals can exhibit sensitivity to the approximate number of items in a visual array or events in a sequence, and across various paradigms, uncertainty in numerosity judgments increases with the number estimated or produced. The pattern of increase is usually described as exhibiting approximate adherence to Weber’s law, such that uncertainty increases proportionally to the mean estimate, resulting in a constant coefficient of variation. Such a pattern has been proposed to be a signature characteristic of an innate “number sense.” We reexamine published behavioral data from two studies that have been cited as prototypical evidence of adherence to Weber’s law and observe that in both cases variability increases less than this account would predict, as indicated by a decreasing coefficient of variation with an increase in number. We also consider evidence from numerosity discrimination studies that show deviations from the constant coefficient of variation pattern. Though behavioral data can sometimes exhibit approximate adherence to Weber’s law, our findings suggest that such adherence is not a fixed characteristic of the mechanisms whereby humans and animals estimate numerosity. We suggest instead that the observed pattern of increase in variability with number depends on the circumstances of the task and stimuli, and reflects an adaptive ensemble of mechanisms composed to optimize performance under these circumstances.

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“…Furthermore, basic visuospatial processing in a hierarchical structure together with statistical properties of the object may be sufficient for the emergence of numerical competence even in non-trained networks (Zorzi and Testolin, 2018); moreover, progressive experience can refine and improve this representation resembling developmental advances. It has been suggested that a dedicated system for numerosity also in animals might not be necessary since learning mechanisms of the neural system could be sufficient to extract that information from natural environments, as it is in the case of artificial networks (Testolin et al, 2020b).…”

Section: Neural Networkmentioning

confidence: 99%

Numerosities and Other Magnitudes in the Brains: A Comparative View

2021

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The ability to represent, discriminate, and perform arithmetic operations on discrete quantities (numerosities) has been documented in a variety of species of different taxonomic groups, both vertebrates and invertebrates. We do not know, however, to what extent similarity in behavioral data corresponds to basic similarity in underlying neural mechanisms. Here, we review evidence for magnitude representation, both discrete (countable) and continuous, following the sensory input path from primary sensory systems to associative pallial territories in the vertebrate brains. We also speculate on possible underlying mechanisms in invertebrate brains and on the role played by modeling with artificial neural networks. This may provide a general overview on the nervous system involvement in approximating quantity in different animal species, and a general theoretical framework to future comparative studies on the neurobiology of number cognition.

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Numerosity discrimination in deep neural networks: Initial competence, developmental refinement and experience statistics

Cited by 50 publications

References 61 publications

Visual sense of number vs. sense of magnitude in humans and machines

Visual sense of number vs. sense of magnitude in humans and machines

Do estimates of numerosity really adhere to Weber’s law? A reexamination of two case studies

Numerosities and Other Magnitudes in the Brains: A Comparative View

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