{"title":"Subsampling under distributional constraints","authors":"Florian Combes, Ricardo Fraiman, Badih Ghattas","doi":"10.1002/sam.11661","DOIUrl":null,"url":null,"abstract":"Some complex models are frequently employed to describe physical and mechanical phenomena. In this setting, we have an input <mjx-container aria-label=\"upper X\" ctxtmenu_counter=\"0\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper X\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/7d0e6166-8cd5-42ea-ba28-4c46d5e0ba78/sam11661-math-0001.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-role=\"latinletter\" data-semantic-speech=\"upper X\" data-semantic-type=\"identifier\">X</mi></mrow>$$ X $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> in a general space, and an output <mjx-container aria-label=\"upper Y equals f left parenthesis upper X right parenthesis\" ctxtmenu_counter=\"1\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow data-semantic-children=\"0,8\" data-semantic-content=\"1\" data-semantic- data-semantic-role=\"equality\" data-semantic-speech=\"upper Y equals f left parenthesis upper X right parenthesis\" data-semantic-type=\"relseq\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"9\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\"relseq,=\" data-semantic-parent=\"9\" data-semantic-role=\"equality\" data-semantic-type=\"relation\" rspace=\"5\" space=\"5\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"2,6\" data-semantic-content=\"7,2\" data-semantic- data-semantic-parent=\"9\" data-semantic-role=\"simple function\" data-semantic-type=\"appl\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"8\" data-semantic-role=\"simple function\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"8\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\"4\" data-semantic-content=\"3,5\" data-semantic- data-semantic-parent=\"8\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"6\" data-semantic-role=\"open\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"6\" data-semantic-role=\"close\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo></mjx-mrow></mjx-mrow></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/462084af-71bd-4d96-b7da-d144735fbe99/sam11661-math-0002.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-children=\"0,8\" data-semantic-content=\"1\" data-semantic-role=\"equality\" data-semantic-speech=\"upper Y equals f left parenthesis upper X right parenthesis\" data-semantic-type=\"relseq\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"9\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">Y</mi><mo data-semantic-=\"\" data-semantic-operator=\"relseq,=\" data-semantic-parent=\"9\" data-semantic-role=\"equality\" data-semantic-type=\"relation\">=</mo><mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"2,6\" data-semantic-content=\"7,2\" data-semantic-parent=\"9\" data-semantic-role=\"simple function\" data-semantic-type=\"appl\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-operator=\"appl\" data-semantic-parent=\"8\" data-semantic-role=\"simple function\" data-semantic-type=\"identifier\">f</mi><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"appl\" data-semantic-parent=\"8\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\"></mo><mrow data-semantic-=\"\" data-semantic-children=\"4\" data-semantic-content=\"3,5\" data-semantic-parent=\"8\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"6\" data-semantic-role=\"open\" data-semantic-type=\"fence\" stretchy=\"false\">(</mo><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"6\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">X</mi><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"6\" data-semantic-role=\"close\" data-semantic-type=\"fence\" stretchy=\"false\">)</mo></mrow></mrow></mrow>$$ Y=f(X) $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> where <mjx-container aria-label=\"f\" ctxtmenu_counter=\"2\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"f\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/c29317d8-616a-414b-b372-90c993754c09/sam11661-math-0003.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-role=\"latinletter\" data-semantic-speech=\"f\" data-semantic-type=\"identifier\">f</mi></mrow>$$ f $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> is a very complicated function, whose computational cost for every new input is very high, and may be also very expensive. We are given two sets of observations of <mjx-container aria-label=\"upper X\" ctxtmenu_counter=\"3\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper X\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/3f3db9eb-4570-4f8a-97ff-6809e982abe8/sam11661-math-0004.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-role=\"latinletter\" data-semantic-speech=\"upper X\" data-semantic-type=\"identifier\">X</mi></mrow>$$ X $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>, <mjx-container aria-label=\"upper S 1\" ctxtmenu_counter=\"4\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow><mjx-msub data-semantic-children=\"0,1\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper S 1\" data-semantic-type=\"subscript\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-script style=\"vertical-align: -0.15em; margin-left: -0.032em;\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"integer\" data-semantic-type=\"number\" size=\"s\"><mjx-c></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/4ec4ad24-2769-40bd-a111-0ff284af8d8b/sam11661-math-0005.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub data-semantic-=\"\" data-semantic-children=\"0,1\" data-semantic-role=\"latinletter\" data-semantic-speech=\"upper S 1\" data-semantic-type=\"subscript\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">S</mi><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"2\" data-semantic-role=\"integer\" data-semantic-type=\"number\">1</mn></msub></mrow>$$ {S}_1 $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> and <mjx-container aria-label=\"upper S 2\" ctxtmenu_counter=\"5\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow><mjx-msub data-semantic-children=\"0,1\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper S 2\" data-semantic-type=\"subscript\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-script style=\"vertical-align: -0.15em; margin-left: -0.032em;\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"integer\" data-semantic-type=\"number\" size=\"s\"><mjx-c></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/bb365000-a4df-473a-b8e4-dfcf2b501f67/sam11661-math-0006.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub data-semantic-=\"\" data-semantic-children=\"0,1\" data-semantic-role=\"latinletter\" data-semantic-speech=\"upper S 2\" data-semantic-type=\"subscript\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">S</mi><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"2\" data-semantic-role=\"integer\" data-semantic-type=\"number\">2</mn></msub></mrow>$$ {S}_2 $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> of different sizes such that only <mjx-container aria-label=\"f left parenthesis upper S 1 right parenthesis\" ctxtmenu_counter=\"6\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow data-semantic-children=\"0,6\" data-semantic-content=\"7,0\" data-semantic- data-semantic-role=\"simple function\" data-semantic-speech=\"f left parenthesis upper S 1 right parenthesis\" data-semantic-type=\"appl\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"8\" data-semantic-role=\"simple function\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"8\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\"3\" data-semantic-content=\"4,5\" data-semantic- data-semantic-parent=\"8\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"6\" data-semantic-role=\"open\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-msub data-semantic-children=\"1,2\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"latinletter\" data-semantic-type=\"subscript\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"3\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-script style=\"vertical-align: -0.15em; margin-left: -0.032em;\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"3\" data-semantic-role=\"integer\" data-semantic-type=\"number\" size=\"s\"><mjx-c></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"6\" data-semantic-role=\"close\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo></mjx-mrow></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/4f57f6c2-b903-46a0-84c4-8e5a71ed7666/sam11661-math-0007.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-children=\"0,6\" data-semantic-content=\"7,0\" data-semantic-role=\"simple function\" data-semantic-speech=\"f left parenthesis upper S 1 right parenthesis\" data-semantic-type=\"appl\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-operator=\"appl\" data-semantic-parent=\"8\" data-semantic-role=\"simple function\" data-semantic-type=\"identifier\">f</mi><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"appl\" data-semantic-parent=\"8\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\"></mo><mrow data-semantic-=\"\" data-semantic-children=\"3\" data-semantic-content=\"4,5\" data-semantic-parent=\"8\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"fenced\" data-semantic-parent=\"6\" data-semantic-role=\"open\" data-semantic-type=\"fence\">(</mo><msub data-semantic-=\"\" data-semantic-children=\"1,2\" data-semantic-parent=\"6\" data-semantic-role=\"latinletter\" data-semantic-type=\"subscript\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"3\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">S</mi><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"3\" data-semantic-role=\"integer\" data-semantic-type=\"number\">1</mn></msub><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"fenced\" data-semantic-parent=\"6\" data-semantic-role=\"close\" data-semantic-type=\"fence\">)</mo></mrow></mrow>$$ f\\left({S}_1\\right) $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> is available. We tackle the problem of selecting a subset <mjx-container aria-label=\"upper S 3 subset of upper S 2\" ctxtmenu_counter=\"7\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow data-semantic-children=\"2,6\" data-semantic-content=\"3\" data-semantic- data-semantic-role=\"set\" data-semantic-speech=\"upper S 3 subset of upper S 2\" data-semantic-type=\"relseq\"><mjx-msub data-semantic-children=\"0,1\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"latinletter\" data-semantic-type=\"subscript\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-script style=\"vertical-align: -0.15em; margin-left: -0.032em;\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"integer\" data-semantic-type=\"number\" size=\"s\"><mjx-c></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo data-semantic- data-semantic-operator=\"relseq,⊂\" data-semantic-parent=\"7\" data-semantic-role=\"set\" data-semantic-type=\"relation\" rspace=\"5\" space=\"5\"><mjx-c></mjx-c></mjx-mo><mjx-msub data-semantic-children=\"4,5\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"latinletter\" data-semantic-type=\"subscript\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-script style=\"vertical-align: -0.15em; margin-left: -0.032em;\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"integer\" data-semantic-type=\"number\" size=\"s\"><mjx-c></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/ff1fb648-0e13-4503-9d3e-0e7e51f4478d/sam11661-math-0008.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-children=\"2,6\" data-semantic-content=\"3\" data-semantic-role=\"set\" data-semantic-speech=\"upper S 3 subset of upper S 2\" data-semantic-type=\"relseq\"><msub data-semantic-=\"\" data-semantic-children=\"0,1\" data-semantic-parent=\"7\" data-semantic-role=\"latinletter\" data-semantic-type=\"subscript\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">S</mi><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"2\" data-semantic-role=\"integer\" data-semantic-type=\"number\">3</mn></msub><mo data-semantic-=\"\" data-semantic-operator=\"relseq,⊂\" data-semantic-parent=\"7\" data-semantic-role=\"set\" data-semantic-type=\"relation\">⊂</mo><msub data-semantic-=\"\" data-semantic-children=\"4,5\" data-semantic-parent=\"7\" data-semantic-role=\"latinletter\" data-semantic-type=\"subscript\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"6\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">S</mi><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"6\" data-semantic-role=\"integer\" data-semantic-type=\"number\">2</mn></msub></mrow>$$ {S}_3\\subset {S}_2 $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> of smaller size on which to run the complex model <mjx-container aria-label=\"f\" ctxtmenu_counter=\"8\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"f\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/b4c52b54-aedc-4b11-ae25-6085d8e94e16/sam11661-math-0009.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-role=\"latinletter\" data-semantic-speech=\"f\" data-semantic-type=\"identifier\">f</mi></mrow>$$ f $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>, and such that the empirical distribution of <mjx-container aria-label=\"f left parenthesis upper S 3 right parenthesis\" ctxtmenu_counter=\"9\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow data-semantic-children=\"0,6\" data-semantic-content=\"7,0\" data-semantic- data-semantic-role=\"simple function\" data-semantic-speech=\"f left parenthesis upper S 3 right parenthesis\" data-semantic-type=\"appl\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"8\" data-semantic-role=\"simple function\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"8\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\"3\" data-semantic-content=\"4,5\" data-semantic- data-semantic-parent=\"8\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"6\" data-semantic-role=\"open\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-msub data-semantic-children=\"1,2\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"latinletter\" data-semantic-type=\"subscript\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"3\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-script style=\"vertical-align: -0.15em; margin-left: -0.032em;\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"3\" data-semantic-role=\"integer\" data-semantic-type=\"number\" size=\"s\"><mjx-c></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"6\" data-semantic-role=\"close\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo></mjx-mrow></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/4cb7dec6-3f53-4721-977e-c8e0032eb454/sam11661-math-0010.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-children=\"0,6\" data-semantic-content=\"7,0\" data-semantic-role=\"simple function\" data-semantic-speech=\"f left parenthesis upper S 3 right parenthesis\" data-semantic-type=\"appl\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-operator=\"appl\" data-semantic-parent=\"8\" data-semantic-role=\"simple function\" data-semantic-type=\"identifier\">f</mi><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"appl\" data-semantic-parent=\"8\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\"></mo><mrow data-semantic-=\"\" data-semantic-children=\"3\" data-semantic-content=\"4,5\" data-semantic-parent=\"8\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"fenced\" data-semantic-parent=\"6\" data-semantic-role=\"open\" data-semantic-type=\"fence\">(</mo><msub data-semantic-=\"\" data-semantic-children=\"1,2\" data-semantic-parent=\"6\" data-semantic-role=\"latinletter\" data-semantic-type=\"subscript\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"3\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">S</mi><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"3\" data-semantic-role=\"integer\" data-semantic-type=\"number\">3</mn></msub><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"fenced\" data-semantic-parent=\"6\" data-semantic-role=\"close\" data-semantic-type=\"fence\">)</mo></mrow></mrow>$$ f\\left({S}_3\\right) $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> is close to that of <mjx-container aria-label=\"f left parenthesis upper S 1 right parenthesis\" ctxtmenu_counter=\"10\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow data-semantic-children=\"0,6\" data-semantic-content=\"7,0\" data-semantic- data-semantic-role=\"simple function\" data-semantic-speech=\"f left parenthesis upper S 1 right parenthesis\" data-semantic-type=\"appl\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"8\" data-semantic-role=\"simple function\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"8\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\"3\" data-semantic-content=\"4,5\" data-semantic- data-semantic-parent=\"8\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"6\" data-semantic-role=\"open\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-msub data-semantic-children=\"1,2\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"latinletter\" data-semantic-type=\"subscript\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"3\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-script style=\"vertical-align: -0.15em; margin-left: -0.032em;\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"3\" data-semantic-role=\"integer\" data-semantic-type=\"number\" size=\"s\"><mjx-c></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"6\" data-semantic-role=\"close\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo></mjx-mrow></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/dd7f9f4b-7fc5-4fc4-82e6-210e76febd4c/sam11661-math-0011.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-children=\"0,6\" data-semantic-content=\"7,0\" data-semantic-role=\"simple function\" data-semantic-speech=\"f left parenthesis upper S 1 right parenthesis\" data-semantic-type=\"appl\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-operator=\"appl\" data-semantic-parent=\"8\" data-semantic-role=\"simple function\" data-semantic-type=\"identifier\">f</mi><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"appl\" data-semantic-parent=\"8\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\"></mo><mrow data-semantic-=\"\" data-semantic-children=\"3\" data-semantic-content=\"4,5\" data-semantic-parent=\"8\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"fenced\" data-semantic-parent=\"6\" data-semantic-role=\"open\" data-semantic-type=\"fence\">(</mo><msub data-semantic-=\"\" data-semantic-children=\"1,2\" data-semantic-parent=\"6\" data-semantic-role=\"latinletter\" data-semantic-type=\"subscript\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"3\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">S</mi><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"3\" data-semantic-role=\"integer\" data-semantic-type=\"number\">1</mn></msub><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"fenced\" data-semantic-parent=\"6\" data-semantic-role=\"close\" data-semantic-type=\"fence\">)</mo></mrow></mrow>$$ f\\left({S}_1\\right) $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>. We suggest three algorithms to solve this problem and show their efficiency using simulated datasets and the Airfoil self-noise data set.","PeriodicalId":48684,"journal":{"name":"Statistical Analysis and Data Mining","volume":"36 1","pages":""},"PeriodicalIF":2.1000,"publicationDate":"2024-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Statistical Analysis and Data Mining","FirstCategoryId":"94","ListUrlMain":"https://doi.org/10.1002/sam.11661","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE","Score":null,"Total":0}
引用次数: 0
Abstract
Some complex models are frequently employed to describe physical and mechanical phenomena. In this setting, we have an input in a general space, and an output where is a very complicated function, whose computational cost for every new input is very high, and may be also very expensive. We are given two sets of observations of , and of different sizes such that only is available. We tackle the problem of selecting a subset of smaller size on which to run the complex model , and such that the empirical distribution of is close to that of . We suggest three algorithms to solve this problem and show their efficiency using simulated datasets and the Airfoil self-noise data set.
期刊介绍：
Statistical Analysis and Data Mining addresses the broad area of data analysis, including statistical approaches, machine learning, data mining, and applications. Topics include statistical and computational approaches for analyzing massive and complex datasets, novel statistical and/or machine learning methods and theory, and state-of-the-art applications with high impact. Of special interest are articles that describe innovative analytical techniques, and discuss their application to real problems, in such a way that they are accessible and beneficial to domain experts across science, engineering, and commerce.
The focus of the journal is on papers which satisfy one or more of the following criteria:
Solve data analysis problems associated with massive, complex datasets
Develop innovative statistical approaches, machine learning algorithms, or methods integrating ideas across disciplines, e.g., statistics, computer science, electrical engineering, operation research.
Formulate and solve high-impact real-world problems which challenge existing paradigms via new statistical and/or computational models
Provide survey to prominent research topics.
京公网安备 11010802042870号
京ICP备2023020795号-1
分享
求助内容：
应助结果提醒方式：
扫码关注我们
