{"id":6790,"date":"2019-08-06T17:04:45","date_gmt":"2019-08-06T15:04:45","guid":{"rendered":"https:\/\/cmc.deusto.eus\/enzuazua\/?p=6790"},"modified":"2022-03-21T07:28:22","modified_gmt":"2022-03-21T06:28:22","slug":"the-curse-of-dimensionality","status":"publish","type":"post","link":"https:\/\/cmc.deusto.eus\/enzuazua\/the-curse-of-dimensionality\/","title":{"rendered":"The Curse of Dimensionality"},"content":{"rendered":"\n
\"\"
The Earth from Apollo 8. Foto: NASA<\/figcaption><\/figure>\n\n\n\n

The universe is immense and the Earth is huge but we\nstill live confined in a world that often feels too small. We are conditioned\nby our own human nature and live hemmed in by four dimensions, just as our\nrooms are constrained by four walls. The world is infinitely complex but in\nspite of that (or perhaps because of it) we live like tigers in a zoo: wearing\na useless furrow into the ground as we pace to and fro in the same, small area\ndelimited by our cages. <\/p>\n\n\n\n

But there only appear to be just four walls and four\ndimensions: the world actually has many more. This is evidenced in many areas,\nthough it is not always obvious and we sometimes fail to realise it.<\/p>\n\n\n\n

In economics, for instance, there are many relevant\ndimensions and variables and their number is increasing day by day in this increasingly\ninterconnected world.  How many\ndimensions and how many parameters are involved in the economic future of a\ncountry? There are stock market indices, employment and unemployment levels, inflation,\nthe balance of trade, GDP, taxes, fraud, transparency, infrastructures,\ncommunications, political status, etc. <\/p>\n\n\n\n

The setting which underlies the current economy is\ndiabolically complicated and involves higher dimensions. It is becoming harder\nand harder to shield oneself from the risk of upsets and lead the simple but\nsecure life that people were once able to enjoy, with steady jobs for life,\nbuying their bread from the same bakery every day. Businesses have been forced\nto internationalise and to innovate continually in order to survive. The same\ngoes for the world\u2019s most powerful universities and research organisations. Those\nin charge are finding it more and more difficult to manage situations marked by\never-greater numbers of links and correlations, where the idea of a \u201ccaptive\nmarket\u201d is increasingly a thing of the past and competition is ever fiercer and\nmore changeable. The shipbuilding industry here in the Basque Country is a case\nin point.<\/p>\n\n\n\n

\"\"
Richard Bellman<\/figcaption><\/figure>\n\n\n\n

Mathematician Richard Bellman<\/a><\/strong>\n(1920-1984) coined the expression \u201cthe curse of dimensionality\u201d as the ultimate\ndescription of the enormous complexity of the world in which we live. He observed\nthat spaces with increasingly large dimensions have geometrical properties\nwhich are hard to guess at and which differ from the space in which we actually\nlive (or believe we live), but that their consequences are important.<\/p>\n\n\n\n

As the size of a cube increases (a line in one\ndimension, a square in two, a cube in three and a hypercube in higher\ndimensions) there are fewer and fewer points close to the centre but more and\nmore on the vertices and edges, i.e. on the periphery. Just as the sea washes\nflotsam to shore, so the interior mass of a cube accumulates at its edges as\nthe dimensions increase. <\/p>\n\n\n\n

The same happens with the political map. The Basque\nCountry is small but in it we enact a scale model of the scenario described by\nthe curse of dimensionality, a context in which size does not matter because a\nsmall scale such as ours is still compatible with high dimensionality. Ours is\na multilingual society with languages of very different kinds and origins. This\nmakes it a society of distinctions, where integration is not always an easy\nmatter. Geographically, the Basque Country<\/a><\/strong> comprises a\nnumber of valleys; politically, it is divided into provinces governed by strong\nprovincial councils with their own assemblies, etc. Institutionally, our small\nsociety is so complex that it provides a perfect test bed for checking whether\nthe curse of dimensionality is applicable. <\/p>\n\n\n\n

And the predictions of the curse actually do come\ntrue. Parliamentary majorities are becoming harder and harder to secure and the\npolitical centre is slipping away. Where has it gone? Does it still exist? Moreover,\nthe situation seems set to be long-lasting, as the political spectrum here is\norganised around four main vertices which do not generally mix but which seem\nstable and durable. But of course, although our society is small it contains\nover two million people, each with their own concerns; that means that we are\nactually moving in a very high dimension of reality. The prediction that it\nwill become more and more difficult to bring people together around the centre\nis thus borne out. <\/p>\n\n\n\n

As a result, it is almost five years now since the\ngovernment of the Autonomous Community of the Basque Country was last able to\nobtain a natural, stable, absolute majority in parliament that enabled it to\ngovern comfortably. <\/p>\n\n\n\n

Bellman also made seminal contributions to dynamic\noptimisation, a field broadly applicable to a number of areas such as resource\nallocation and the design of infrastructures (rail and metro networks, roads,\nhospitals, air travel routes, etc.). It was he who first stated (and worked out\nthe equations to demonstrate) that an optimal strategy must always be optimal,\ni.e. if a better decision could have been made at any time then things could\nhave been done better overall. But this is hard to achieve if one acts merely\non the basis of one\u2019s own intuition: hence the usefulness of Bellman\u2019s\nequations, which enable the best strategy at each moment in time to be\ncalculated.<\/p>\n\n\n\n

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San Mam\u00e9s Barria football stadium<\/a><\/strong> <\/figcaption><\/figure>\n\n\n\n

There are pragmatists who give in to the curse of dimensionality, see the intrinsic difficulties of the problems at hand, realise their own limitations and seek to do things reasonably by resorting to mathematical methods. For instance Athletic Club de Bilbao called in Alvin E. Roth<\/a><\/strong>, winner of the 1994 Nobel Prize for Economics, to help them allocate the seats in their new San Mam\u00e9s Barria football stadium<\/a><\/strong>. Satisfying the demands of all the approximately 40,000 season-ticket holders when allocating the seating in the new stadium, with its improved but different layout, was no easy task. With the curse looming, it was decided that it was worth taking the risk and spending the money to hire the best, in the shape of the team which had organised the American network of organ donors for transplants and the distribution of pupils across the network of publicly-run schools in New York.<\/p>\n\n\n\n

This is a fine example of how science can help in\ndecision-making and management. It may perhaps mark a path that can be followed\nin other areas where even more is at stake to escape the curse of\ndimensionality.<\/p>\n","protected":false},"excerpt":{"rendered":"

In economics, there are many relevant dimensions and variables and their number is increasing day by day in this increasingly interconnected world.<\/p>\n","protected":false},"author":3,"featured_media":5727,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[52,502,501],"class_list":["post-6790","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-math-in-motion","tag-basque-country","tag-dimensions","tag-economics"],"_links":{"self":[{"href":"https:\/\/cmc.deusto.eus\/enzuazua\/wp-json\/wp\/v2\/posts\/6790","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/cmc.deusto.eus\/enzuazua\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/cmc.deusto.eus\/enzuazua\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/cmc.deusto.eus\/enzuazua\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/cmc.deusto.eus\/enzuazua\/wp-json\/wp\/v2\/comments?post=6790"}],"version-history":[{"count":2,"href":"https:\/\/cmc.deusto.eus\/enzuazua\/wp-json\/wp\/v2\/posts\/6790\/revisions"}],"predecessor-version":[{"id":6795,"href":"https:\/\/cmc.deusto.eus\/enzuazua\/wp-json\/wp\/v2\/posts\/6790\/revisions\/6795"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/cmc.deusto.eus\/enzuazua\/wp-json\/wp\/v2\/media\/5727"}],"wp:attachment":[{"href":"https:\/\/cmc.deusto.eus\/enzuazua\/wp-json\/wp\/v2\/media?parent=6790"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/cmc.deusto.eus\/enzuazua\/wp-json\/wp\/v2\/categories?post=6790"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/cmc.deusto.eus\/enzuazua\/wp-json\/wp\/v2\/tags?post=6790"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}