LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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Product and Householder QR errors bound the represented joint covariance

Statement

Let $G \in R^{n \times q}$ , with $q \geq n \geq 1$ , be an exact prediction or measurement prearray. Work in the operationwise no-underflow model with $0 < u < 1$ and $q u < 1$ . Let $\hat{G}$ be its computed version and assume $∥ \hat{G} - G ∥_{F} \leq e_{f o r m} + e_{i n},$ where $e_{f o r m}, e_{i n} \geq 0$ , $e_{f o r m}$ bounds rounded products used to form the array, and $e_{i n}$ bounds separately specified input-factor uncertainty. If the successful scaled-Householder QR of ${\hat{G}}^{T} \in R^{q \times n}$ has reduced upper factor $R$ and $L = R^{T}$ , set $κ_{q} = κ_{q} (u), β_{q, n} = (1 + κ_{q})^{n} - 1, ϵ = e_{f o r m} + e_{i n} + β_{q, n} ∥ \hat{G} ∥_{F},$ where $κ_{q}$ is the explicit local coefficient in The scaled computed Householder step satisfies the local backward error premise. Then $∥ L L^{T} - G G^{T} ∥_{F} \leq 2 ∥ G ∥_{F} ϵ + ϵ^{2} .$ If $200 q n u < 1$ , the rational envelope lemma An explicit rational envelope for the scaled Householder QR error gives $β_{q, n} \leq η_{q, n} : = 200 q n u / (1 - 200 q n u)$ . In that regime the same conclusion holds with $ϵ = e_{f o r m} + e_{i n} + η_{q, n} ∥ \hat{G} ∥_{F}$ . For one left-to-right rounded product $\hat{U V}$ with positive inner dimension $r \geq 1$ and $r u < 1$ , a valid choice is $e_{f o r m} = γ_{r} ∥ U ∥_{F} ∥ V ∥_{F}$ . A product with zero inner dimension is exactly zero and contributes zero to $e_{f o r m}$ ; for several disjoint product blocks, sum the nonempty-product bounds. This conclusion allows rank loss and does not imply that the nearby joint covariance has the original independent-noise block structure.

Facts & Assumptions

Given: The exact and computed arrays, operationwise round-to-nearest arithmetic with all successful nonzero intermediates normal, $r u < 1$ for each nonempty product, and successful scaled-Householder QR with $q u < 1$ .

[F1]

A left-to-right computed dot product of length $r$ has componentwise relative error at most $γ_{r} = r u / (1 - r u)$ , and its absolute error is at most $γ_{r} \sum_{j} ∣ x_{j} y_{j} ∣$ . The computed floating-point dot product obeys componentwise and normwise error bounds

[F2]

Successful scaled-Householder QR of an $m \times n$ real matrix, $m \geq n$ , gives $A + Δ A = \hat{Q} \hat{R}$ with $∥ Δ A ∥_{F} \leq ((1 + κ_{m})^{n} - 1) ∥ A ∥_{F}$ ; the coefficient comes from the local computed-reflector bound. The scaled Householder QR implementation is backward stable

[F3]

For any factor perturbation $E$ , the Gram-matrix error is at most $2 ∥ G ∥_{F} ∥ E ∥_{F} + ∥ E ∥_{F}^{2}$ . A factor perturbation gives an explicit Gram-matrix error

[F4]

If $200 q n u < 1$ , then $β_{q, n} \leq 200 q n u / (1 - 200 q n u)$ . An explicit rational envelope for the scaled Householder QR error

[F5]

For real vectors $a, b$ , if $b \neq 0$ then expanding $0 \leq ∥ a - t b ∥_{2}^{2}$ at $t = (a^{T} b) / ∥ b ∥_{2}^{2}$ gives $∣ a^{T} b ∣ \leq ∥ a ∥_{2} ∥ b ∥_{2}$ ; if $b = 0$ the inequality is immediate.

Proof

Given: The hypotheses in the Statement.

1.1F1F5givenalgebra

Exact-zero operations satisfy the relative model with error factor zero, so the rounded-factor expansion underlying [F1] also covers zero products and partial sums in the present model. An empty inner product is exact zero. For each entry of a nonempty product $U V$ , [F1] therefore gives an error at most $γ_{r} \sum_{j} ∣ U_{i j} V_{j k} ∣$ . Apply [F5] to the vectors $(∣ U_{i j} ∣)_{j}$ and $(∣ V_{j k} ∣)_{j}$ , then square and sum over entries: $\sum_{i, k} {(\sum_{j} ∣ U_{i j} ∣ ∣ V_{j k} ∣)}^{2} \leq \sum_{i, k} (\sum_{j} ∣ U_{i j} ∣^{2}) (\sum_{j} ∣ V_{j k} ∣^{2}) = ∥ U ∥_{F}^{2} ∥ V ∥_{F}^{2} .$ Therefore $∥ \hat{U V} - U V ∥_{F} \leq γ_{r} ∥ U ∥_{F} ∥ V ∥_{F}$ . Disjoint product blocks add by the Frobenius norm definition, and separately bounded input-factor errors add by the triangle inequality.

1.2F2givenalgebra

Apply [F2] to ${\hat{G}}^{T}$ , which has $q \geq n$ rows and $n$ columns. Transposing its factorization gives $\hat{G} + Δ^{T} = R^{T} {\hat{Q}}^{T} = L {\hat{Q}}^{T}, ∥ Δ^{T} ∥_{F} \leq β_{q, n} ∥ \hat{G} ∥_{F} .$ Set $E = (\hat{G} - G) + Δ^{T}$ . Then $∥ E ∥_{F} \leq ϵ$ . Since ${\hat{Q}}^{T} \hat{Q} = I$ , the transposed factorization also gives $L L^{T} = L {\hat{Q}}^{T} \hat{Q} L^{T} = (G + E) (G + E)^{T} .$

2.1F3F4step 1.2algebra∎

Apply [F3] to the factor identity and perturbation $E$ from step 1.2. It gives $∥ L L^{T} - G G^{T} ∥_{F} \leq 2 ∥ G ∥_{F} ϵ + ϵ^{2}$ for the stated $β_{q, n}$ choice. Under $200 q n u < 1$ , [F4] makes this epsilon no larger than the version using $η_{q, n}$ . Since $t \mapsto 2 ∥ G ∥_{F} t + t^{2}$ is increasing for $t \geq 0$ , the same covariance bound follows with the rational-envelope epsilon. The argument permits zero singular values and places no structural restriction on the perturbation of $G G^{T}$ .

Depends on

Used by

The square-root filter has an explicit conditioned finite-horizon error bound Theorem

Dependency tree · two levels

20 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Kevin S. Tracy, A Square-Root Kalman Filter Using Only QR Decompositions (standard reference, not scraped)
M. A. Salzmann, Some Aspects of Kalman Filtering, Technical Report 140 (standard reference, not scraped)
Simo Särkkä, Bayesian Filtering and Smoothing (standard reference, not scraped)