{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T11:31:25Z","timestamp":1776771085497,"version":"3.51.2"},"reference-count":71,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2020,1,9]],"date-time":"2020-01-09T00:00:00Z","timestamp":1578528000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100012190","name":"Ministry of Science and Higher Education of the Russian Federation","doi-asserted-by":"publisher","award":["14.Y26.31.0022"],"award-info":[{"award-number":["14.Y26.31.0022"]}],"id":[{"id":"10.13039\/501100012190","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100006041","name":"Innovate UK","doi-asserted-by":"publisher","award":["KTP010522"],"award-info":[{"award-number":["KTP010522"]}],"id":[{"id":"10.13039\/501100006041","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100006041","name":"Innovate UK","doi-asserted-by":"publisher","award":["KTP009890"],"award-info":[{"award-number":["KTP009890"]}],"id":[{"id":"10.13039\/501100006041","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100010198","name":"Ministerio de Econom\u00eda, Industria y Competitividad, Gobierno de Espa\u00f1a","doi-asserted-by":"publisher","award":["FIS2017-82900-P"],"award-info":[{"award-number":["FIS2017-82900-P"]}],"id":[{"id":"10.13039\/501100010198","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>High-dimensional data and high-dimensional representations of reality are inherent features of modern Artificial Intelligence systems and applications of machine learning. The well-known phenomenon of the \u201ccurse of dimensionality\u201d states: many problems become exponentially difficult in high dimensions. Recently, the other side of the coin, the \u201cblessing of dimensionality\u201d, has attracted much attention. It turns out that generic high-dimensional datasets exhibit fairly simple geometric properties. Thus, there is a fundamental tradeoff between complexity and simplicity in high dimensional spaces. Here we present a brief explanatory review of recent ideas, results and hypotheses about the blessing of dimensionality and related simplifying effects relevant to machine learning and neuroscience.<\/jats:p>","DOI":"10.3390\/e22010082","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T04:06:51Z","timestamp":1578629211000},"page":"82","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":47,"title":["High-Dimensional Brain in a High-Dimensional World: Blessing of Dimensionality"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6224-1430","authenticated-orcid":false,"given":"Alexander N.","family":"Gorban","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Leicester, Leicester LE1 7RH, UK"},{"name":"Laboratory of Advanced Methods for High-Dimensional Data Analysis, Lobachevsky University, 603022 Nizhny Novgorod, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8789-7532","authenticated-orcid":false,"given":"Valery A.","family":"Makarov","sequence":"additional","affiliation":[{"name":"Laboratory of Advanced Methods for High-Dimensional Data Analysis, Lobachevsky University, 603022 Nizhny Novgorod, Russia"},{"name":"Instituto de Matem\u00e1tica Interdisciplinar, Faculty of Mathematics, Universidad Complutense de Madrid, Avda Complutense s\/n, 28040 Madrid, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7359-7966","authenticated-orcid":false,"given":"Ivan Y.","family":"Tyukin","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Leicester, Leicester LE1 7RH, UK"},{"name":"Laboratory of Advanced Methods for High-Dimensional Data Analysis, Lobachevsky University, 603022 Nizhny Novgorod, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,1,9]]},"reference":[{"key":"ref_1","unstructured":"Bellman, R. (1957). Dynamic Programming, Princeton University Press."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"503","DOI":"10.1090\/S0002-9904-1954-09848-8","article-title":"The theory of dynamic programming","volume":"60","author":"Bellman","year":"1954","journal-title":"Bull. Am. Math. Soc."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1115\/1.3658896","article-title":"Reduction of dimensionality, dynamic programming, and control processes","volume":"83","author":"Bellman","year":"1961","journal-title":"J. Basic Eng."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Gorban, A.N., Kazantzis, N., Kevrekidis, I.G., \u00d6ttinger, H.C., and Theodoropoulos, C. (2006). Model Reduction and Coarse\u2013Graining Approaches for Multiscale Phenomena, Springer.","DOI":"10.1007\/3-540-35888-9"},{"key":"ref_5","unstructured":"Jolliffe, I. (1993). Principal Component Analysis, Springer."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Gorban, A.N., K\u00e9gl, B., Wunsch, D., and Zinovyev, A. (2008). Principal Manifolds for Data Visualisation and Dimension Reduction, Springer.","DOI":"10.1007\/978-3-540-73750-6"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1142\/S0129065710002383","article-title":"Principal manifolds and graphs in practice: from molecular biology to dynamical systems","volume":"20","author":"Gorban","year":"2010","journal-title":"Int. J. Neural Syst."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1561\/2200000059","article-title":"Tensor networks for dimensionality reduction and large-scale optimization: Part 1 low-rank tensor decompositions","volume":"9","author":"Cichocki","year":"2016","journal-title":"Found. Trends\u00ae Mach. Learn."},{"key":"ref_9","first-page":"431","article-title":"Tensor networks for dimensionality reduction and large-scale optimization: Part 2 applications and future perspectives","volume":"9","author":"Cichocki","year":"2017","journal-title":"Found. Trends\u00ae Mach. Learn."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Beyer, K., Goldstein, J., Ramakrishnan, R., and Shaft, U. (1999, January 10\u201312). When is \u201cnearest neighbor\u201d meaningful?. Proceedings of the 7th International Conference on Database Theory (ICDT), Jerusalem, Israel.","DOI":"10.1007\/3-540-49257-7_15"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1427","DOI":"10.1016\/j.camwa.2012.09.011","article-title":"Is the k-NN classifier in high dimensions affected by the curse of dimensionality?","volume":"65","author":"Pestov","year":"2013","journal-title":"Comput. Math. Appl."},{"key":"ref_12","unstructured":"Warwick, K., and K\u00e1rn\u00fd, M. (1997). Utilizing geometric anomalies of high dimension: when complexity makes computation easier. Computer-Intensive Methods in Control and Signal Processing: The Curse of Dimensionality, Springer."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"88","DOI":"10.1006\/jmva.1997.1691","article-title":"On the effect of inliers on the spatial median","volume":"63","author":"Brown","year":"1997","journal-title":"J. Multivar. Anal."},{"key":"ref_14","unstructured":"Donoho, D.L. (2000, January 6\u201312). High-Dimensional Data Analysis: The Curses and Blessings of Dimensionality. Invited Lecture at Mathematical Challenges of the 21st Century. Proceedings of the AMS National Meeting, Los Angeles, CA, USA. Available online: http:\/\/citeseerx.ist.psu.edu\/viewdoc\/summary?doi=10.1.1.329.3392."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Chen, D., Cao, X., Wen, F., and Sun, J. (2013, January 23\u201328). Blessing of dimensionality: High-dimensional feature and its efficient compression for face verification. Proceedings of the 2013 IEEE Conference on Computer Vision and Pattern Recognition, Portland, OR, USA.","DOI":"10.1109\/CVPR.2013.389"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1109\/TPAMI.2016.2539946","article-title":"Blessing of dimensionality: Recovering mixture data via dictionary pursuit","volume":"39","author":"Liu","year":"2016","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1007\/s00357-009-9037-9","article-title":"The remarkable simplicity of very high dimensional data: Application of model-based clustering","volume":"26","author":"Murtagh","year":"2009","journal-title":"J. Classif."},{"key":"ref_18","first-page":"1135","article-title":"The More, the Merrier: The Blessing of Dimensionality for Learning Large Gaussian Mixtures","volume":"Volume 35","author":"Balcan","year":"2014","journal-title":"Proceedings of the 27th Conference on Learning Theory"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1016\/j.ifacol.2016.10.755","article-title":"The blessing of dimensionality: Separation theorems in the thermodynamic limit","volume":"49","author":"Gorban","year":"2016","journal-title":"IFAC-PapersOnLine"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"380","DOI":"10.1080\/01621459.2016.1256815","article-title":"Embracing the blessing of dimensionality in factor models","volume":"113","author":"Li","year":"2018","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Landgraf, A.J., and Lee, Y. (2019). Generalized principal component analysis: Projection of saturated model parameters. Technometrics.","DOI":"10.1080\/00401706.2019.1668854"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1289","DOI":"10.1109\/TIT.2006.871582","article-title":"Compressed sensing","volume":"52","author":"Donoho","year":"2006","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"4273","DOI":"10.1098\/rsta.2009.0152","article-title":"Observed universality of phase transitions in high-dimensional geometry, with implications for modern data analysis and signal processing","volume":"367","author":"Donoho","year":"2009","journal-title":"Phil. Trans. R. Soc. A"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Candes, E., Rudelson, M., Tao, T., and Vershynin, R. (2005, January 23\u201325). Error correction via linear programming. Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS\u201905), Pittsburgh, PA, USA.","DOI":"10.1109\/SFCS.2005.5464411"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Pereda, E., Garc\u00eda-Torres, M., Meli\u00e1n-Batista, B., Ma\u00f1as, S., M\u00e9ndez, L., and Gonz\u00e1lez, J.J. (2018). The blessing of Dimensionality: Feature Selection outperforms functional connectivity-based feature transformation to classify ADHD subjects from EEG patterns of phase synchronisation. PLoS ONE, 13.","DOI":"10.1371\/journal.pone.0201660"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1016\/0893-9659(93)90023-G","article-title":"Quasiorthogonal dimension of Euclidian spaces","volume":"6","author":"Kainen","year":"1993","journal-title":"Appl. Math. Lett."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"427","DOI":"10.1111\/j.1467-9868.2005.00510.x","article-title":"Geometric representation of high dimension, low sample size data","volume":"67","author":"Hall","year":"2005","journal-title":"J. R. Stat. Soc. B"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1016\/j.ins.2015.09.021","article-title":"Approximation with random bases: Pro et contra","volume":"364\u2013365","author":"Gorban","year":"2016","journal-title":"Inf. Sci."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"60","DOI":"10.1002\/rsa.10073","article-title":"An elementary proof of a theorem of Johnson and Lindenstrauss","volume":"22","author":"Dasgupta","year":"2003","journal-title":"Random Sruct. Algorithms"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"20170237","DOI":"10.1098\/rsta.2017.0237","article-title":"Blessing of dimensionality: Mathematical foundations of the statistical physics of data","volume":"376","author":"Gorban","year":"2018","journal-title":"Philos. Trans. R. Soc. A"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Vershynin, R. (2018). High-Dimensional Probability: An Introduction with Applications in Data Science, Cambridge University Press.","DOI":"10.1017\/9781108231596"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1006\/aima.2000.1949","article-title":"Concentration property on probability spaces","volume":"156","author":"Giannopoulos","year":"2000","journal-title":"Adv. Math."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Ledoux, M. (2005). The Concentration of Measure Phenomenon, AMS. Number 89 in Mathematical Surveys & Monographs.","DOI":"10.1090\/surv\/089"},{"key":"ref_34","unstructured":"Gibbs, J.W. (1960). Elementary Principles in Statistical Mechanics, Developed with Especial Reference to the Rational Foundation of Thermodynamics, Dover Publications."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"178","DOI":"10.1007\/s000390300004","article-title":"Isoperimetry of waists and concentration of maps","volume":"13","author":"Gromov","year":"2003","journal-title":"Geom. Funct. Anal."},{"key":"ref_36","unstructured":"L\u00e9vy, P. (1951). Probl\u00e8mes Concrets D\u2019analyse Fonctionnelle, Gauthier-Villars."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Dubhashi, D.P., and Panconesi, A. (2009). Concentration of Measure for the Analysis of Randomized Algorithms, Cambridge University Press.","DOI":"10.1017\/CBO9780511581274"},{"key":"ref_38","unstructured":"Ball, K. (1997). An Elementary Introduction to Modern Convex Geometry. Flavors of Geometry, Cambridge University Press."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1016\/j.ins.2018.07.040","article-title":"Correction of AI systems by linear discriminants: Probabilistic foundations","volume":"466","author":"Gorban","year":"2018","journal-title":"Inf. Sci."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1016\/j.plrev.2018.09.005","article-title":"The unreasonable effectiveness of small neural ensembles in high-dimensional brain","volume":"29","author":"Gorban","year":"2019","journal-title":"Phys. Life Rev."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1007\/BF00334039","article-title":"On the shape of the convex hull of random points","volume":"77","year":"1988","journal-title":"Probab. Theory Relat. Fields"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1016\/j.neunet.2017.07.014","article-title":"Stochastic separation theorems","volume":"94","author":"Gorban","year":"2017","journal-title":"Neural Netw."},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Tyukin, I.Y., Gorban, A.N., McEwan, A.A., and Meshkinfamfard, S. (2019). Blessing of dimensionality at the edge. arXiv.","DOI":"10.1109\/ICIAI.2019.8850825"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1016\/j.ins.2019.02.001","article-title":"One-trial correction of legacy AI systems and stochastic separation theorems","volume":"484","author":"Gorban","year":"2019","journal-title":"Inf. Sci."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1111\/j.1469-1809.1936.tb02137.x","article-title":"The Use of Multiple Measurements in Taxonomic Problems","volume":"7","author":"Fisher","year":"1936","journal-title":"Ann. Eugenics"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"98","DOI":"10.1016\/j.plrev.2019.03.014","article-title":"Some insights from high-dimensional spheres: Comment on \u201cThe unreasonable effectiveness of small neural ensembles in high-dimensional brain\u201d by Alexander N. Gorban et al.","volume":"29","year":"2019","journal-title":"Phys. Life Rev."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1016\/j.plrev.2019.06.003","article-title":"Symphony of high-dimensional brain. Reply to comments on \u201cThe unreasonable effectiveness of small neural ensembles in high-dimensional brain\u201d","volume":"29","author":"Gorban","year":"2019","journal-title":"Phys. Life Rev."},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Grechuk, B. (2019, January 14\u201319). Practical stochastic separation theorems for product distributions. Proceedings of the IEEE IJCNN 2019\u2014International Joint Conference on Neural Networks, Budapest, Hungary.","DOI":"10.1109\/IJCNN.2019.8851817"},{"key":"ref_49","first-page":"309","article-title":"Probabilistic Bounds for Binary Classification of Large Data Sets","volume":"Volume 1","author":"Oneto","year":"2019","journal-title":"Proceedings of the International Neural Networks Society"},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Tyukin, I.Y., Gorban, A.N., and Grechuk, B. (2019, January 14\u201319). Kernel Stochastic Separation Theorems and Separability Characterizations of Kernel Classifiers. Proceedings of the IEEE IJCNN 2019\u2014International Joint Conference on Neural Networks, Budapest, Hungary.","DOI":"10.1109\/IJCNN.2019.8852278"},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Meshkinfamfard, S., Gorban, A.N., and Tyukin, I.V. Tackling Rare False-Positives in Face Recognition: A Case Study. Proceedings of the 2018 IEEE 20th International Conference on High Performance Computing and Communications; IEEE 16th International Conference on Smart City; IEEE 4th International Conference on Data Science and Systems (HPCC\/SmartCity\/DSS).","DOI":"10.1109\/HPCC\/SmartCity\/DSS.2018.00260"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"230","DOI":"10.1016\/j.ins.2018.11.057","article-title":"Fast construction of correcting ensembles for legacy artificial intelligence systems: Algorithms and a case study","volume":"485","author":"Tyukin","year":"2019","journal-title":"Inf. Sci."},{"key":"ref_53","doi-asserted-by":"crossref","unstructured":"Tyukin, I.Y., Gorban, A.N., Sofeikov, K., and Romanenko, I. (2018). Knowledge transfer between artificial intelligence systems. Front. Neurorobot., 12.","DOI":"10.3389\/fnbot.2018.00049"},{"key":"ref_54","unstructured":"Allison, P.M., Sofeikov, K., Levesley, J., Gorban, A.N., Tyukin, I., and Cooper, N.J. (2018). Exploring automated pottery identification [Arch-I-Scan]. Internet Archaeol., 50."},{"key":"ref_55","unstructured":"Romanenko, I., Gorban, A., and Tyukin, I. (2019). Image Processing. (10,489,634 B2), U.S. Patent, Available online: https:\/\/patents.google.com\/patent\/US10489634B2\/en."},{"key":"ref_56","doi-asserted-by":"crossref","unstructured":"Xu, R., and Wunsch, D. (2008). Clustering, Wiley.","DOI":"10.1002\/9780470382776"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"33922","DOI":"10.1038\/srep33922","article-title":"Fluorescence-based assay as a new screening tool for toxic chemicals","volume":"6","author":"Moczko","year":"2016","journal-title":"Sci. Rep."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1111\/j.1600-0587.2012.07348.x","article-title":"Collinearity: A review of methods to deal with it and a simulation study evaluating their performance","volume":"36","author":"Dormann","year":"2013","journal-title":"Ecography"},{"key":"ref_59","doi-asserted-by":"crossref","unstructured":"Albergante, L., Bac, J., and Zinovyev, A. (2019, January 14\u201319). Estimating the effective dimension of large biological datasets using Fisher separability analysis. Proceedings of the IEEE IJCNN 2019\u2014International Joint Conference on Neural Networks, Budapest, Hungary.","DOI":"10.1109\/IJCNN.2019.8852450"},{"key":"ref_60","unstructured":"Artin, E. (2015). The Gamma Function, Courier Dover Publications."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1016\/j.plrev.2019.04.006","article-title":"The heresy of unheard-of simplicity: Comment on \u201cThe unreasonable effectiveness of small neural ensembles in high-dimensional brain\u201d by A.N. Gorban, V.A. Makarov, and I.Y. Tyukin","volume":"29","author":"Kreinovich","year":"2019","journal-title":"Phys. Life Rev."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"1102","DOI":"10.1038\/nature03687","article-title":"Invariant visual representation by single neurons in the human brain","volume":"435","author":"Reddy","year":"2005","journal-title":"Nature"},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1068\/p010371","article-title":"Single units and sensation: a neuron doctrine for perceptual psychology?","volume":"1","author":"Barlow","year":"1972","journal-title":"Perception"},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1016\/j.plrev.2019.02.014","article-title":"Akakhievitch revisited: Comment on \u201cThe unreasonable effectiveness of small neural ensembles in high-dimensional brain\u201d by Alexander N. Gorban et al.","volume":"29","year":"2019","journal-title":"Phys. Life Rev."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"527","DOI":"10.1016\/S0306-4522(00)00496-6","article-title":"Total number and distribution of inhibitory and excitatory synapses on hippocampal CA1 pyramidal cells","volume":"102","author":"Emri","year":"2001","journal-title":"Neuroscience"},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"629","DOI":"10.1016\/j.neuron.2013.11.026","article-title":"Structured synaptic connectivity between hippocampal regions","volume":"81","author":"Druckmann","year":"2014","journal-title":"Neuron"},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"2560","DOI":"10.1073\/pnas.0308577100","article-title":"Topographic specificity of functional connections from hippocampal CA3 to CA1","volume":"101","author":"Brivanlou","year":"2004","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"4856","DOI":"10.1007\/s11538-018-0415-5","article-title":"High-dimensional brain: A tool for encoding and rapid learning of memories by single neurons","volume":"81","author":"Tyukin","year":"2019","journal-title":"Bull. Math. Biol."},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1016\/j.plrev.2019.02.008","article-title":"High and low dimensionality in neuroscience and artificial intelligence: Comment on \u201cThe unreasonable effectiveness of small neural ensembles in high-dimensional brain\u201d by A.N. Gorban et al.","volume":"29","author":"Varona","year":"2019","journal-title":"Phys. Life Rev."},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"108","DOI":"10.1016\/j.plrev.2019.02.010","article-title":"\u201cBrainland\u201d vs. \u201cflatland\u201d: how many dimensions do we need in brain dynamics? Comment on the paper \u201cThe unreasonable effectiveness of small neural ensembles in high-dimensional brain\u201d by Alexander N. Gorban et al.","volume":"29","author":"Barrio","year":"2019","journal-title":"Phys. Life Rev."},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"104","DOI":"10.1016\/j.plrev.2019.03.005","article-title":"The reasonable ineffectiveness of biological brains in applying the principles of high-dimensional cybernetics: Comment on \u201cThe unreasonable effectiveness of small neural ensembles in high-dimensional brain\u201d by Alexander N. Gorban et al.","volume":"29","year":"2019","journal-title":"Phys. Life Rev."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/1\/82\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T13:42:34Z","timestamp":1760362954000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/1\/82"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,1,9]]},"references-count":71,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2020,1]]}},"alternative-id":["e22010082"],"URL":"https:\/\/doi.org\/10.3390\/e22010082","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,1,9]]}}}