The Theory and Applications of Hölder Widths
We introduce the Hölder width, which measures the best error performance of some recent nonlinear approximation methods, such as deep neural network approximation. Then, we investigate the relationship between Hölder widths and other widths, showing that some Hölder widths are essentially smaller th...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2024-12-01
|
| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/14/1/25 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1832589149774807040 |
|---|---|
| author | Man Lu Peixin Ye |
| author_facet | Man Lu Peixin Ye |
| author_sort | Man Lu |
| collection | DOAJ |
| description | We introduce the Hölder width, which measures the best error performance of some recent nonlinear approximation methods, such as deep neural network approximation. Then, we investigate the relationship between Hölder widths and other widths, showing that some Hölder widths are essentially smaller than <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi></mrow></semantics></math></inline-formula>-Kolmogorov widths and linear widths. We also prove that, as the Hölder constants grow with <i>n</i>, the Hölder widths are much smaller than the entropy numbers. The fact that Hölder widths are smaller than the known widths implies that the nonlinear approximation represented by deep neural networks can provide a better approximation order than other existing approximation methods, such as adaptive finite elements and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi></mrow></semantics></math></inline-formula>-term wavelet approximation. In particular, we show that Hölder widths for Sobolev and Besov classes, induced by deep neural networks, are <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">O</mi><mo>(</mo><msup><mi>n</mi><mrow><mo>−</mo><mn>2</mn><mi>s</mi><mo>/</mo><mi>d</mi></mrow></msup><mo>)</mo></mrow></semantics></math></inline-formula> and are much smaller than other known widths and entropy numbers, which are <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">O</mi><mo>(</mo><msup><mi>n</mi><mrow><mo>−</mo><mi>s</mi><mo>/</mo><mi>d</mi></mrow></msup><mo>)</mo></mrow></semantics></math></inline-formula>. |
| format | Article |
| id | doaj-art-e0e242b559dc4c11900f5fdd1cdfdf52 |
| institution | Kabale University |
| issn | 2075-1680 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-e0e242b559dc4c11900f5fdd1cdfdf522025-01-24T13:22:11ZengMDPI AGAxioms2075-16802024-12-011412510.3390/axioms14010025The Theory and Applications of Hölder WidthsMan Lu0Peixin Ye1Department of Applied Mathematics, School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, ChinaDepartment of Applied Mathematics, School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, ChinaWe introduce the Hölder width, which measures the best error performance of some recent nonlinear approximation methods, such as deep neural network approximation. Then, we investigate the relationship between Hölder widths and other widths, showing that some Hölder widths are essentially smaller than <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi></mrow></semantics></math></inline-formula>-Kolmogorov widths and linear widths. We also prove that, as the Hölder constants grow with <i>n</i>, the Hölder widths are much smaller than the entropy numbers. The fact that Hölder widths are smaller than the known widths implies that the nonlinear approximation represented by deep neural networks can provide a better approximation order than other existing approximation methods, such as adaptive finite elements and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi></mrow></semantics></math></inline-formula>-term wavelet approximation. In particular, we show that Hölder widths for Sobolev and Besov classes, induced by deep neural networks, are <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">O</mi><mo>(</mo><msup><mi>n</mi><mrow><mo>−</mo><mn>2</mn><mi>s</mi><mo>/</mo><mi>d</mi></mrow></msup><mo>)</mo></mrow></semantics></math></inline-formula> and are much smaller than other known widths and entropy numbers, which are <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">O</mi><mo>(</mo><msup><mi>n</mi><mrow><mo>−</mo><mi>s</mi><mo>/</mo><mi>d</mi></mrow></msup><mo>)</mo></mrow></semantics></math></inline-formula>.https://www.mdpi.com/2075-1680/14/1/25Hölder widthsdeep neural networksentropy numbersnonlinear approximation<i>n</i>-Kolmogorov widthsnonlinear (<i>n</i>, <i>N</i>)-widths |
| spellingShingle | Man Lu Peixin Ye The Theory and Applications of Hölder Widths Axioms Hölder widths deep neural networks entropy numbers nonlinear approximation <i>n</i>-Kolmogorov widths nonlinear (<i>n</i>, <i>N</i>)-widths |
| title | The Theory and Applications of Hölder Widths |
| title_full | The Theory and Applications of Hölder Widths |
| title_fullStr | The Theory and Applications of Hölder Widths |
| title_full_unstemmed | The Theory and Applications of Hölder Widths |
| title_short | The Theory and Applications of Hölder Widths |
| title_sort | theory and applications of holder widths |
| topic | Hölder widths deep neural networks entropy numbers nonlinear approximation <i>n</i>-Kolmogorov widths nonlinear (<i>n</i>, <i>N</i>)-widths |
| url | https://www.mdpi.com/2075-1680/14/1/25 |
| work_keys_str_mv | AT manlu thetheoryandapplicationsofholderwidths AT peixinye thetheoryandapplicationsofholderwidths AT manlu theoryandapplicationsofholderwidths AT peixinye theoryandapplicationsofholderwidths |