Computational Complexity of Statistics: New Insights from Low-Degree Polynomials
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909647048278016 |
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| author | Wein, Alexander S. |
| author_facet | Wein, Alexander S. |
| contents | This is a survey on the use of low-degree polynomials to predict and explain the apparent statistical-computational tradeoffs in a variety of average-case computational problems. In a nutshell, this framework measures the complexity of a statistical task by the minimum degree that a polynomial function must have in order to solve it. The main goals of this survey are to (1) describe the types of problems where the low-degree framework can be applied, encompassing questions of detection (hypothesis testing), recovery (estimation), and more; (2) discuss some philosophical questions surrounding the interpretation of low-degree lower bounds, and notably the extent to which they should be treated as evidence for inherent computational hardness; (3) explore the known connections between low-degree polynomials and other related approaches such as the sum-of-squares hierarchy and statistical query model; and (4) give an overview of the mathematical tools used to prove low-degree lower bounds. A list of open problems is also included. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_10748 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Computational Complexity of Statistics: New Insights from Low-Degree Polynomials Wein, Alexander S. Statistics Theory Computational Complexity Machine Learning This is a survey on the use of low-degree polynomials to predict and explain the apparent statistical-computational tradeoffs in a variety of average-case computational problems. In a nutshell, this framework measures the complexity of a statistical task by the minimum degree that a polynomial function must have in order to solve it. The main goals of this survey are to (1) describe the types of problems where the low-degree framework can be applied, encompassing questions of detection (hypothesis testing), recovery (estimation), and more; (2) discuss some philosophical questions surrounding the interpretation of low-degree lower bounds, and notably the extent to which they should be treated as evidence for inherent computational hardness; (3) explore the known connections between low-degree polynomials and other related approaches such as the sum-of-squares hierarchy and statistical query model; and (4) give an overview of the mathematical tools used to prove low-degree lower bounds. A list of open problems is also included. |
| title | Computational Complexity of Statistics: New Insights from Low-Degree Polynomials |
| topic | Statistics Theory Computational Complexity Machine Learning |
| url | https://arxiv.org/abs/2506.10748 |