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<p>We give an explicit solution formula for the polynomial regression problem in terms of Schur polynomials and Vandermonde determinants. We thereby generalize the work of Chang, Deng, and Floater to the case of model functions of the form <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\sum _{i=1}^{n} a_{i} x^{d_{i}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </msubsup> <msub> <mi>a</mi> <mi>i</mi> </msub> <msup> <mi>x</mi> <msub> <mi>d</mi> <mi>i</mi> </msub> </msup> </mrow> </math></EquationSource> </InlineEquation> for some integer exponents <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(d_{1}&gt;d_{2}&gt;\dots c &gt;d_{n} \ge 0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>d</mi> <mn>1</mn> </msub> <mo>&gt;</mo> <msub> <mi>d</mi> <mn>2</mn> </msub> <mo>&gt;</mo> <mo>⋯</mo> <mi>c</mi> <mo>&gt;</mo> <msub> <mi>d</mi> <mi>n</mi> </msub> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and phrase the results using Schur polynomials. Even though the solution circumvents the well-known problems with the forward stability of the normal equation, it is only of practical value if <i>n</i> is small because the number of terms in the formula grows rapidly with the number <i>m</i> of data points. The formula can be evaluated essentially without rounding.</p>

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A symmetric function approach to polynomial regression

  • 期刊论文

  • 发表日期：

    2026年2月16日

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  • Hans-Christian Herbig,
  • Daniel Herden,
  • Christopher W. Seaton

摘要

We give an explicit solution formula for the polynomial regression problem in terms of Schur polynomials and Vandermonde determinants. We thereby generalize the work of Chang, Deng, and Floater to the case of model functions of the form \(\sum _{i=1}^{n} a_{i} x^{d_{i}}\) ∑ i = 1 n a i x d i for some integer exponents \(d_{1}>d_{2}>\dots c >d_{n} \ge 0\) d 1 > d 2 > ⋯ c > d n ≥ 0 and phrase the results using Schur polynomials. Even though the solution circumvents the well-known problems with the forward stability of the normal equation, it is only of practical value if n is small because the number of terms in the formula grows rapidly with the number m of data points. The formula can be evaluated essentially without rounding.