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arxiv: 1907.06996 · v1 · pith:W2FABR5Nnew · submitted 2019-07-16 · 💻 cs.CV · cs.NE· q-bio.NC

Perception of visual numerosity in humans and machines

Alberto Testolin , Serena Dolfi , Mathijs Rochus , Marco Zorzi This is my paper

Pith reviewed 2026-05-24 20:58 UTC · model grok-4.3

classification 💻 cs.CV cs.NEq-bio.NC
keywords numerosity perceptiondeep neural networksvisual psychophysicsdevelopmental changesrepresentational similarity analysiscontinuous magnitudesnumerical cognition
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The pith

Deep networks replicate human numerosity perception with number as the main driver but continuous magnitudes exerting early influence

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines whether numerosity perception requires a specialized system or can arise from estimating continuous visual magnitudes such as area and density. Researchers applied the identical comparison task used with human participants to deep networks, employing stimuli designed to isolate the roles of numerical and non-numerical features. The networks reproduced human discrimination performance and its developmental trajectory, with numerosity dominating but non-numerical cues contributing more strongly at younger stages. Representational similarity analysis revealed spontaneous encoding of both numerosity and continuous magnitudes even in the absence of any task demand.

Core claim

Deep networks trained on numerosity comparison tasks accurately simulate human psychophysics and developmental changes, where discrimination relies primarily on numerosity information while non-numerical features exert significant impact especially early in development; representational similarity analysis shows that both numerosity and continuous magnitudes are spontaneously encoded even without task requirements.

What carries the argument

Deep neural network performing the numerosity comparison task, with representational similarity analysis measuring alignment of internal encodings to human data

If this is right

  • Numerosity discrimination improves as the relative weight of non-numerical features declines across development.
  • General visual processing networks can produce numerical behavior without requiring a dedicated number module.
  • Both numerical and non-numerical magnitudes arise as salient properties in visual representations even when unprompted by task goals.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same modeling approach could be extended to test whether other early mathematical concepts emerge from the statistics of natural visual input.
  • Varying the training distribution or architecture would reveal how experience modulates the balance between numerical and continuous cues.
  • If similar spontaneous encoding occurs in networks trained only on object recognition, it would strengthen the claim that numerosity is a byproduct of general vision.

Load-bearing premise

The chosen stimulus space and network training regime generate internal representations and decision behavior close enough to human vision that numerical versus non-numerical contributions can be compared directly.

What would settle it

Human performance data on a new stimulus set where numerosity is fully decorrelated from area and density would falsify the model if the network's predicted error patterns or developmental shifts deviate substantially from the observed human patterns.

Figures

Figures reproduced from arXiv: 1907.06996 by Alberto Testolin, Marco Zorzi, Mathijs Rochus, Serena Dolfi.

Figure 1
Figure 1. Figure 1: Stimulus space and model architecture. (A) The 3D stimulus space defined by the Numerosity, Size and Spacing orthogonal dimensions (adapted from (53)). Non-numerical features are represented as arrows to indicate the direction in which they increase, and each stimulus image can be represented as a point in this space. Example of stimuli pairs are shown below, where Numerosity can be fully congruent for Siz… view at source ↗
Figure 3
Figure 3. Figure 3: Maturation of number acuity in deep networks [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Representational similarity analysis. (A) Representational dissimilarity matrices for the best deep network architecture (distance measure: 1 – Pearson correlation) and the most relevant categorical models (distance measure: log distance between stimulus features). Each RDM was separately rank transformed and scaled into [0,1]. (B) Second-order correlation matrix showing the pairwise correlations between R… view at source ↗
read the original abstract

Numerosity perception is foundational to mathematical learning, but its computational bases are strongly debated. Some investigators argue that humans are endowed with a specialized system supporting numerical representation; 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 networks on the same numerosity comparison task that was administered to humans, using a stimulus space that allows to measure the contribution of non-numerical features. Our model accurately simulated the psychophysics of numerosity perception and the associated developmental changes: discrimination was driven by numerosity information, but non-numerical features had a significant impact, especially early during development. Representational similarity analysis further highlighted that both numerosity and continuous magnitudes were spontaneously encoded even when no task had to be carried out, demonstrating that numerosity is a major, salient property of our visual environment.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper trains deep convolutional networks on a numerosity comparison task using a stimulus set that decorrelates number from continuous magnitudes (area, density, etc.). It reports that the networks reproduce human psychophysical signatures and developmental trends, with numerosity as the dominant cue but non-numerical features exerting measurable influence (especially early in training). Representational similarity analysis on the networks' layers is used to demonstrate spontaneous encoding of both numerosity and continuous magnitudes even in the absence of an explicit task.

Significance. If the reported simulation holds, the work supplies a concrete computational demonstration that numerosity can emerge as a salient visual feature alongside continuous magnitudes, thereby offering a mechanistic reconciliation of the specialized-system versus magnitude-based accounts of numerical perception. The use of the identical task and stimulus space as the human experiments, together with the RSA analysis of spontaneous representations, strengthens the link between model behavior and developmental psychophysics.

minor comments (3)
  1. The abstract and introduction refer to 'developmental changes' but the methods section does not specify how the training schedule or data curriculum was aligned with human age groups; a brief clarification of this mapping would aid reproducibility.
  2. Figure 3 (RSA matrices) would benefit from explicit labeling of the layer indices corresponding to the reported correlation values, as the current caption leaves the mapping between network depth and the plotted layers implicit.
  3. The stimulus-generation procedure is described at a high level; adding a short supplementary table listing the exact ranges and sampling densities for each continuous magnitude would make the decorrelation claim easier to verify.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, recognition of the work's significance, and recommendation for minor revision. The report does not list any major comments.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper trains convolutional networks on a numerosity comparison task using a controlled stimulus space that decorrelates number from continuous magnitudes, then evaluates the resulting model behavior against independent human psychophysics data. No parameters are fitted to the target human discrimination thresholds or developmental trajectories; the comparison is purely external. Representational similarity analysis is performed on the network's internal activations without reference to human data. No self-citation chain, uniqueness theorem, or ansatz imported from prior author work is invoked to justify the central claims. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no identifiable free parameters, axioms, or invented entities; model details and any fitting choices are unavailable.

pith-pipeline@v0.9.0 · 5691 in / 1198 out tokens · 24255 ms · 2026-05-24T20:58:12.096245+00:00 · methodology

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Works this paper leans on

77 extracted references · 77 canonical work pages · 2 internal anchors

  1. [1]

    Dehaene S (2011) The number sense: How the mind creates mathematics (Oxford University Press)

    work page 2011

  2. [2]

    Trends Cogn Sci 14(12):542–551

    Piazza M (2010) Neurocognitive start-up tools for symbolic number representations. Trends Cogn Sci 14(12):542–551

    work page 2010

  3. [3]

    (Macmillan, London, UK)

    Butterworth B (1999) The mathematical brain. (Macmillan, London, UK)

    work page 1999

  4. [4]

    Nature 455

    Halberda J, Mazzocco MM, Feigenson L (2008) Individual differences in non-verbal number acuity correlate with maths achievement. Nature 455

    work page 2008

  5. [5]

    Trends Cogn Sci 7(4):145–147

    Dehaene S (2003) The neural basis of the Weber--Fechner law: a logarithmic mental number line. Trends Cogn Sci 7(4):145–147

    work page 2003

  6. [6]

    Neuron 44(3):547–555

    Piazza M, Izard V, Pinel P, Le Bihan D, Dehaene S (2004) Tuning curves for approximate numerosity in the human intraparietal sulcus. Neuron 44(3):547–555

    work page 2004

  7. [7]

    Anim Cogn 11(3):495–503

    Agrillo C, Dadda M, Serena G, Bisazza A (2008) Do fish count? Spontaneous discrimination of quantity in female mosquitofish. Anim Cogn 11(3):495–503. 16

    work page 2008

  8. [8]

    Psychol Sci 12(3):238–243

    Brannon EM, Wusthoff CJ, Gallistel CR, Gibbon J (2001) Numerical Subtraction in the Pigeon: Evidence for a Linear Subjective Number Scale. Psychol Sci 12(3):238–243

    work page 2001

  9. [9]

    (2009) Number-based visual generalisation in the honeybee

    Gross HJ, et al. (2009) Number-based visual generalisation in the honeybee. PLoS One 4(1). doi:10.1371/journal.pone.0004263

    work page doi:10.1371/journal.pone.0004263 2009

  10. [10]

    Psychol Sci 17(5):401–406

    Cantlon JF, Brannon EM (2006) Shared System for Ordering Small and Large Numbers in Monkeys and Humans. Psychol Sci 17(5):401–406

    work page 2006

  11. [11]

    Annu Rev Neurosci

    Nieder A, Dehaene S (2009) Representation of Number in the Brain. Annu Rev Neurosci

    work page 2009

  12. [12]

    Cereb Cortex 26(2):748–763

    Park J, Dewind NK, Woldorff MG, Brannon EM (2015) Rapid and Direct Encoding of Numerosity in the Visual Stream. Cereb Cortex 26(2):748–763

    work page 2015

  13. [13]

    Proc Natl Acad Sci 106(25):10382–5

    Izard V, Sann C, Spelke ES, Streri A (2009) Newborn infants perceive abstract numbers. Proc Natl Acad Sci 106(25):10382–5

    work page 2009

  14. [14]

    Dev Sci 8(1):88–101

    Xu F, Spelke ES, Goddard S (2005) Number sense in human infants. Dev Sci 8(1):88–101

    work page 2005

  15. [15]

    Number Sense

    Halberda J, Feigenson L (2008) Developmental change in the acuity of the “Number Sense”: The Approximate Number System in 3-, 4-, 5-, and 6-year-olds and adults. Dev Psychol 44(5):1457–65

    work page 2008

  16. [16]

    Proc Natl Acad Sci 109(28):11116–20

    Halberda J, Ly R, Wilmer JB, Naiman DQ, Germine L (2012) Number sense across the lifespan as revealed by a massive Internet-based sample. Proc Natl Acad Sci 109(28):11116–20

    work page 2012

  17. [17]

    Proc Natl Acad Sci 110(45):18116–18120

    Starr A, Libertus ME, Brannon EM (2013) Number sense in infancy predicts mathematical abilities in childhood. Proc Natl Acad Sci 110(45):18116–18120

    work page 2013

  18. [18]

    (2017) Associations of non-symbolic and symbolic numerical magnitude processing with mathematical competence: a meta-analysis

    Schneider M, et al. (2017) Associations of non-symbolic and symbolic numerical magnitude processing with mathematical competence: a meta-analysis. Dev Sci 20(3). doi:10.1111/desc.12372

    work page doi:10.1111/desc.12372 2017

  19. [19]

    (2010) Developmental Trajectory of Number Acuity Reveals a Severe Impairment in Developmental Dyscalculia

    Piazza M, et al. (2010) Developmental Trajectory of Number Acuity Reveals a Severe Impairment in Developmental Dyscalculia. Cognition 116(1):33–41

    work page 2010

  20. [20]

    Curr Biol 18(6):425–428

    Burr DC, Ross J (2008) A Visual Sense of Number. Curr Biol 18(6):425–428

    work page 2008

  21. [21]

    Nat Commun 8:13968

    Ferrigno S, Jara-Ettinger J, Piantadosi ST, Cantlon JF (2017) Universal and uniquely human factors in spontaneous number perception. Nat Commun 8:13968

    work page 2017

  22. [22]

    Nat Commun 7:12536

    Cicchini GM, Anobile G, Burr DC (2016) Spontaneous perception of numerosity in humans. Nat Commun 7:12536

    work page 2016

  23. [23]

    Trends Cogn Sci 8(7):307–314

    Feigenson L, Dehaene S, Spelke ES (2004) Core systems of number. Trends Cogn Sci 8(7):307–314

    work page 2004

  24. [24]

    Proc Natl Acad Sci 103(51):19599–19604

    Hurewitz F, Gelman R, Schnitzer B (2006) Sometimes area counts more than number. Proc Natl Acad Sci 103(51):19599–19604

    work page 2006

  25. [25]

    Continuous Extent

    Feigenson L, Carey S, Spelke E (2002) Infants’ Discrimination of Number vs. Continuous Extent. Cogn Psychol 44(1):33–66

    work page 2002

  26. [26]

    Psychol Sci 10(5):408–411

    Clearfield MW, Mix KS (1999) Number Versus Contour Length in Infants’ Discrimination of Small Visual Sets. Psychol Sci 10(5):408–411

    work page 1999

  27. [27]

    Cognition 121(2):248–252

    Gebuis T, Gevers W (2011) Numerosities and space; indeed a cognitive illusion! A reply to de Hevia and Spelke (2009). Cognition 121(2):248–252

    work page 2011

  28. [28]

    Dehaene S, Izard V, Piazza M (2005) Control over non-numerical parameters in numerosity experiments. 17

    work page 2005

  29. [29]

    Acta Psychol (Amst) 161:177–184

    Clayton S, Gilmore C, Inglis M (2015) Dot comparison stimuli are not all alike: The effect of different visual controls on ANS measurement. Acta Psychol (Amst) 161:177–184

    work page 2015

  30. [30]

    J Exp Psychol Gen 141(4):642–8

    Gebuis T, Reynvoet B (2012) The interplay between nonsymbolic number and its continuous visual properties. J Exp Psychol Gen 141(4):642–8

    work page 2012

  31. [31]

    (2013) Individual Differences in Inhibitory Control, Not Non-Verbal Number Acuity, Correlate with Mathematics Achievement

    Gilmore C, et al. (2013) Individual Differences in Inhibitory Control, Not Non-Verbal Number Acuity, Correlate with Mathematics Achievement. PLoS One 8(6):1–9

    work page 2013

  32. [32]

    Cogn Psychol 69:25–45

    Cappelletti M, Didino D, Stoianov I, Zorzi M (2014) Number skills are maintained in healthy ageing. Cogn Psychol 69:25–45

    work page 2014

  33. [33]

    Acta Psychol (Amst) 171:1–71

    Gebuis T, Cohen Kadosh R, Gevers W (2016) Sensory-integration system rather than approximate number system underlies numerosity processing : A critical review. Acta Psychol (Amst) 171:1–71

    work page 2016

  34. [34]

    sense of number

    Leibovich T, Katzin N, Harel M, Henik A (2017) From “sense of number” to “sense of magnitude” - The role of continuous magnitudes in numerical cognition. Behav Brain Sci 164. doi:10.3389/fpsyg.2017.00652

    work page doi:10.3389/fpsyg.2017.00652 2017

  35. [35]

    Cognition 168:222–233

    Starr A, DeWind NK, Brannon EM (2017) The contributions of numerical acuity and non- numerical stimulus features to the development of the number sense and symbolic math achievement. Cognition 168:222–233

    work page 2017

  36. [36]

    Cognition 181(June 2017):1–29

    Piazza M, Feo V De, Panzeri S, Dehaene S (2018) Learning to focus on number. Cognition 181(June 2017):1–29

    work page 2018

  37. [37]

    Dev Sci 19(5):817–833

    Bugden S, Ansari D (2016) Probing the nature of deficits in the ‘Approximate Number System’ in children with persistent Developmental Dyscalculia. Dev Sci 19(5):817–833

    work page 2016

  38. [38]

    Nature 521(7553):436–444

    LeCun Y, Bengio Y, Hinton GE (2015) Deep learning. Nature 521(7553):436–444

    work page 2015

  39. [39]

    Annu Rev Vis Sci 1(1):417–446

    Kriegeskorte N (2015) Deep Neural Networks: A New Framework for Modeling Biological Vision and Brain Information Processing. Annu Rev Vis Sci 1(1):417–446

    work page 2015

  40. [40]

    Nat Hum Behav 1(9):657–664

    Testolin A, Stoianov I, Zorzi M (2017) Letter perception emerges from unsupervised deep learning and recycling of natural image features. Nat Hum Behav 1(9):657–664

    work page 2017

  41. [41]

    Neuron 95(2):245–258

    Hassabis D, Kumaran D, Summerfield C, Botvinick M (2017) Neuroscience-Inspired Artificial Intelligence. Neuron 95(2):245–258

    work page 2017

  42. [42]

    Nat Neurosci 19(3):356–365

    Yamins DLK, DiCarlo JJ (2016) Using goal-driven deep learning models to understand sensory cortex. Nat Neurosci 19(3):356–365

    work page 2016

  43. [43]

    Front Comput Neurosci 10(73)

    Testolin A, Zorzi M (2016) Probabilistic Models and Generative Neural Networks: Towards an Unified Framework for Modeling Normal and Impaired Neurocognitive Functions. Front Comput Neurosci 10(73). doi:10.3389/fncom.2016.00073

    work page doi:10.3389/fncom.2016.00073 2016

  44. [44]

    Behav Brain Sci 40:1--72

    Lake BM, Ullman TD, Tenenbaum JB, Gershman SJ (2017) Building Machines That Learn and Think Like People. Behav Brain Sci 40:1--72

    work page 2017

  45. [45]

    Front Psychol 4(August):515

    Zorzi M, Testolin A, Stoianov IPIP (2013) Modeling language and cognition with deep unsupervised learning: a tutorial overview. Front Psychol 4(August):515

    work page 2013

  46. [46]

    Neural Comput 18(7):1527–1554

    Hinton GE, Osindero S, Teh Y (2006) A fast learning algorithm for deep belief nets. Neural Comput 18(7):1527–1554

    work page 2006

  47. [47]

    Trends Cogn Sci 14(3):119–30

    Fiser J, Berkes P, Orbán G, Lengyel M (2010) Statistically optimal perception and learning: from behavior to neural representations. Trends Cogn Sci 14(3):119–30

    work page 2010

  48. [48]

    Friston KJ (2010) The free-energy principle: a unified brain theory? Nat Rev Neurosci 11(2):127–38. 18

    work page 2010

  49. [49]

    sense of number

    Viswanathan P, Nieder A (2013) Neuronal correlates of a visual “sense of number” in primate parietal and prefrontal cortices. Proc Natl Acad Sci 110(27):11187–92

    work page 2013

  50. [50]

    PLoS Biol 6(2):0275–0285

    Izard V, Dehaene-Lambertz G, Dehaene S (2008) Distinct cerebral pathways for object identity and number in human infants. PLoS Biol 6(2):0275–0285

    work page 2008

  51. [51]

    visual number sense

    Stoianov I, Zorzi M (2012) Emergence of a “visual number sense” in hierarchical generative models. Nat Neurosci 15(2):194–6

    work page 2012

  52. [52]

    Philos Trans R Soc B Biol Sci 373(1740)

    Zorzi M, Testolin A (2018) An emergentist perspective on the origin of number sense. Philos Trans R Soc B Biol Sci 373(1740). doi:10.1098/rstb.2017.0043

    work page doi:10.1098/rstb.2017.0043 2018

  53. [53]

    Cognition 142:247–265

    DeWind NK, Adams GK, Platt ML, Brannon EM (2015) Modeling the approximate number system to quantify the contribution of visual stimulus features. Cognition 142:247–265

    work page 2015

  54. [54]

    Front Syst Neurosci 2(November):1–28

    Kriegeskorte N (2008) Representational similarity analysis – connecting the branches of systems neuroscience. Front Syst Neurosci 2(November):1–28

    work page 2008

  55. [55]

    J R Stat Soc B 57(1):289–300

    Benjamini Y, Hochberg Y (1995) Controlling the False Discovery Rate: A Practical and Powerful Approach to Multiple Testing. J R Stat Soc B 57(1):289–300

    work page 1995

  56. [56]

    J Mach Learn Res 9:2579– 2605

    Maaten L Van Der, Hinton GE (2008) Visualizing data using t-SNE. J Mach Learn Res 9:2579– 2605

    work page 2008

  57. [57]

    Cognition 128(3):331–352

    Cantrell L, Smith LB (2013) Open questions and a proposal: A critical review of the evidence on infant numerical abilities. Cognition 128(3):331–352

    work page 2013

  58. [58]

    Behav Brain Sci

    Stoianov I, Zorzi M (2017) Computational foundations of the visual number sense. Behav Brain Sci

    work page 2017

  59. [59]

    Nat Rev Neurosci 17(6):366–382

    Nieder A (2016) The neuronal code for number. Nat Rev Neurosci 17(6):366–382

    work page 2016

  60. [60]

    (2016) The limitations of deep learning in adversarial settings

    Papernot N, et al. (2016) The limitations of deep learning in adversarial settings. Proc - 2016 IEEE Eur Symp Secur Privacy, EURO S P 2016:372–387

    work page 2016

  61. [61]

    Psychol Sci:095679761882354

    Perry C, Zorzi M, Ziegler JC (2019) Understanding Dyslexia Through Personalized Large-Scale Computational Models. Psychol Sci:095679761882354

    work page 2019

  62. [62]

    Biotechniques 39(6):859–862

    Selinummi J, Seppälä J, Yli-Harja O, Puhakka JA (2005) Software for quantification of labeled bacteria from digital microscope images by automated image analysis. Biotechniques 39(6):859–862

    work page 2005

  63. [63]

    Understanding Traffic Density from Large-Scale Web Camera Data

    Zhang S, Wu G, Costeira JP, Moura JMF (2017) Understanding Traffic Density from Large- Scale Web Camera Data. Available at: http://arxiv.org/abs/1703.05868

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  64. [64]

    Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pp 833–841

    Zhang C, Li H, Wang X, Yang X (2015) Cross-scene crowd counting via deep convolutional neural networks. Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pp 833–841

    work page 2015

  65. [65]

    Adv Neural Inf Process Syst:1324–1332

    Lempitsky V, Zisserman A (2010) Learning To Count Objects in Images. Adv Neural Inf Process Syst:1324–1332

    work page 2010

  66. [66]

    (2015) Towards AI-Complete Question Answering: A Set of Prerequisite Toy Tasks

    Weston J, et al. (2015) Towards AI-Complete Question Answering: A Set of Prerequisite Toy Tasks. International Conference on Representation Learning doi:10.1016/j.jpowsour.2014.09.131

    work page doi:10.1016/j.jpowsour.2014.09.131 2015

  67. [67]

    Neural Arithmetic Logic Units

    Trask A, et al. (2018) Neural Arithmetic Logic Units Available at: http://arxiv.org/abs/1808.00508

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  68. [68]

    Phys D Nonlinar Phenom 42(1–3):335–346

    Harnad S (1990) The symbol grounding problem. Phys D Nonlinar Phenom 42(1–3):335–346

    work page 1990

  69. [69]

    Behav Brain Sci 3(03):417–424

    Searle JR (1980) Minds, brains, and programs. Behav Brain Sci 3(03):417–424. 19

    work page 1980

  70. [70]

    Can J Exp Psychol 70(1):12–23

    Leibovich T, Ansari D (2016) The Symbol-Grounding Problem in Numerical Cognition: A Review of Theory, Evidence, and Outstanding Questions. Can J Exp Psychol 70(1):12–23

    work page 2016

  71. [71]

    (2014) A Toolbox for Representational Similarity Analysis

    Nili H, et al. (2014) A Toolbox for Representational Similarity Analysis. PLoS Comput Biol 10(4). doi:10.1371/journal.pcbi.1003553

    work page doi:10.1371/journal.pcbi.1003553 2014

  72. [72]

    Cogn Sci 9(1):147–169

    Ackley D, Hinton GE, Sejnowski TJ (1985) A learning algorithm for Boltzmann machines. Cogn Sci 9(1):147–169

    work page 1985

  73. [73]

    Neural Comput 14(8):1771–1800

    Hinton GE (2002) Training products of experts by minimizing contrastive divergence. Neural Comput 14(8):1771–1800

    work page 2002

  74. [74]

    Front Psychol 4(May):251

    Testolin A, Stoianov I, De Filippo De Grazia M, Zorzi M (2013) Deep unsupervised learning on a desktop PC : A primer for cognitive scientists. Front Psychol 4(May):251

    work page 2013

  75. [75]

    Hertz JA, Krogh AS, Palmer RG (1991) Introduction to the theory of neural computation (Addison-Weasley, Redwood City, CA)

    work page 1991

  76. [76]

    (2014) Generative Adversarial Nets

    Goodfellow I, et al. (2014) Generative Adversarial Nets. Adv Neural Inf Process Syst:2672– 2680

    work page 2014

  77. [77]

    blue-cyan-gray-red-yellow

    Zanetti A, Testolin A, Zorzi M, Wawrzynski P (2019) Numerosity Representation in InfoGAN: An Empirical Study. Advances in Computational Intelligence. IWANN, eds I. R, G. J, A. C (Springer International Publishing, Cham), pp 49–6 20 Supplementary Methods Stimulus space definition The relationship betwee n the three orthogonal dimensions Numerosity, Size an...

    work page 2019