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Common-covariance LDA scores are affine and QDA scores are quadratic

Statement

Assume the Axiom of Choice. Let $d \geq 1$ and $K \geq 2$ be finite, let $π_{k} > 0$ with $\sum_{k = 1}^{K} π_{k} = 1$ , let $μ_{k} \in R^{d}$ , and let each real symmetric matrix $Σ_{k} \in R^{d \times d}$ be positive definite, meaning $v^{T} Σ_{k} v > 0$ for every nonzero $v$ . Define

f_{k} (x) = (2 π)^{- d / 2} \det (Σ_{k})^{- 1 / 2} \exp ⁣ (- \frac{1}{2} (x - μ_{k})^{T} Σ_{k}^{- 1} (x - μ_{k})), p_{k} (x) = \frac{π_{k} f_{k} (x)}{\sum_{j = 1}^{K} π_{j} f_{j} (x)} .

For LDA, assume $Σ_{k} = Σ$ for all $k$ ; then $\log p_{k} (x)$ differs by a term independent of $k$ from the affine score

δ_{k}^{L} (x) = x^{T} Σ^{- 1} μ_{k} - \frac{1}{2} μ_{k}^{T} Σ^{- 1} μ_{k} + \log π_{k} .

For QDA, allow the $Σ_{k}$ to vary with $k$ ; then $\log p_{k} (x)$ differs by a term independent of $k$ from the quadratic score

δ_{k}^{Q} (x) = - \frac{1}{2} \log \det (Σ_{k}) - \frac{1}{2} (x - μ_{k})^{T} Σ_{k}^{- 1} (x - μ_{k}) + \log π_{k} .

Thus maximizing the posterior probabilities uses affine scores in LDA and scores of degree at most two in QDA; ties are resolved by the smallest class index.

Statement references

Gaussian LDA and QDA model contracts

Finite linear-score classifiers with a deterministic tie rule

A positive-definite multivariate normal law has its Gaussian density

The Axiom of Choice

Hastie, Tibshirani, and Friedman, ESL, §4.3, equations (4.7)–(4.12)

Stanford CS229 Spring 2022 lecture notes, §4.1

Facts & Assumptions

Given: AC, finite $d \geq 1$ and $K \geq 2$ , positive priors summing to one, means $μ_{k}$ , symmetric positive-definite covariance matrices, and the LDA common-covariance or QDA class-specific-covariance premise in the Statement.

[A1]

AC is the assumption used by the multivariate-normal law and its positive-definite density theorem. The Axiom of Choice

[F1]

The generative model has priors $π_{k}$ and conditional laws $N_{d} (μ_{k}, Σ_{k})$ ; LDA shares $Σ$ and QDA permits distinct positive-definite $Σ_{k}$ . Gaussian LDA and QDA model contracts

[F2]

Under AC, a finite-dimensional multivariate normal law with positive-definite covariance has the displayed Lebesgue density. A positive-definite multivariate normal law has its Gaussian density

[F3]

A finite ordered family of affine class scores predicts the smallest label index attaining the maximum. Finite linear-score classifiers with a deterministic tie rule

Proof

1.1A1F1F2givenalgebra

For each $x$ , [F2] gives $f_{k} (x) > 0$ and finite because $Σ_{k}$ is positive definite; since $π_{k} > 0$ , the denominator $m (x) = \sum_{j} π_{j} f_{j} (x)$ is positive. The density Bayes formula $p_{k} (x) = π_{k} f_{k} (x) / m (x)$ is therefore well-defined, positive, and sums to one.

2.1step 1.1F2algebra

Taking logarithms gives $\log p_{k} (x) = \log π_{k} - \frac{d}{2} \log (2 π) - \frac{1}{2} \log \det (Σ_{k}) - \frac{1}{2} (x - μ_{k})^{T} Σ_{k}^{- 1} (x - μ_{k}) - \log m (x)$ . The terms $- \frac{d}{2} \log (2 π)$ and $- \log m (x)$ do not depend on $k$ , so the remaining class-dependent expression is exactly $δ_{k}^{Q} (x)$ .

3.1step 2.1F1F2algebra

Under the LDA premise $Σ_{k} = Σ$ , the determinant term in step 2.1 is also independent of $k$ . Symmetry of $Σ^{- 1}$ gives $(x - μ_{k})^{T} Σ^{- 1} (x - μ_{k}) = x^{T} Σ^{- 1} x - 2 x^{T} Σ^{- 1} μ_{k} + μ_{k}^{T} Σ^{- 1} μ_{k}$ . Removing the terms independent of $k$ leaves $δ_{k}^{L} (x)$ . The LDA score is affine in $x$ , whereas the QDA score in step 2.1 has degree at most two.

4.1step 1.1step 2.1step 3.1F3algebra∎

For LDA, the affine scores in step 3.1 have a maximizing index and use the smallest-index tie rule by [F3]. For QDA, each $δ_{k}^{Q} (x)$ is finite because $Σ_{k}$ is positive definite and $π_{k} > 0$ ; since $K$ is finite, the maximizing set is nonempty and finite, so it has a unique smallest index. In either model, adding the same finite quantity to every score preserves the maximizing indices, and the log-posterior scores in steps 2.1 and 3.1 differ from the displayed scores by such a quantity. Thus the posterior argmax has the stated deterministic tie rule for both LDA and QDA.

Depends on

Used by

A Gaussian LDA model and a conditional logistic model can share a decision boundary Example

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Sources

Hastie, Tibshirani, and Friedman, The Elements of Statistical Learning, 2nd ed., corrected 12th printing (standard reference, not scraped)
Stanford CS229 Spring 2022 lecture notes (standard reference, not scraped)