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Hierarchical low rank matrices and error propagation — Examples

1 · Prerequisites

2 · Summary

The examples compute a rank-one separated Cauchy block, show why an inadmissible identity block is kept dense when a subunit spectral error is required, and exhibit two accepted local low-rank update truncations whose accumulated error destroys positive definiteness.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passOpen item page →

A nearfield identity block requires full rank for subunit spectral error

Example

Let $τ = {1, 2}$ have coordinates $0$ and $1$ , choose leaf threshold $b = 2$ , and use the block $τ \times τ$ . Its distance from itself is zero, so it is inadmissible and terminates as a dense near block. For $I_{2}$ , no rank-one approximation has operator-norm error below one.

Facts & Assumptions

Given: The cluster $τ$ , its two coordinates, the threshold $b = 2$ , and the identity matrix $I_{2}$ .

[F1]

Finite cluster trees and geometric admissibility: a pair is admissible only at positive distance, and the block recursion stops when both clusters are leaves.

[F2]

H-matrix block representation: near block leaves are stored as full dense matrices.

[F3]

The best rank-at-most-k approximation in operator norm is the rank-k truncation of a singular value decomposition: every rank-at-most-one matrix has operator-norm error at least the second singular value of the identity matrix.

[F4]

The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||: the operator norm is the maximum vector stretch over unit vectors.

Verification

1.1givenF1F2algebra

Since $dist (τ, τ) = 0$ , $(τ, τ)$ is not admissible. The cluster has size $b$ and is a leaf, so its block pair is terminal and near by [F1]; [F2] therefore stores this block densely.

2.1givenF3F4algebra∎

The singular values of $I_{2}$ are $1, 1$ . Applying [F3] with rank cap one shows $∥ I_{2} - B ∥_{2} \geq 1$ for every rank-at-most-one $B$ , where [F4] identifies this norm as the maximum unit-vector stretch. Thus replacing this dense near block by a rank-at-most-one matrix cannot meet any spectral error tolerance strictly below one.

ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-6-luna)Open item page →

Two accepted local truncations can destroy positive definiteness after accumulation

Example

Let $e_{2} = (0, 1)^{T}$ , $M = diag (1, - 1 / 50)$ , and $D_{1} = D_{2} = (3 / 50) e_{2} e_{2}^{*}$ . The exact sum $A = M + D_{1} + D_{2} = diag (1, 1 / 10)$ is positive definite. If each rank-one update is separately recompressed to rank zero with tolerance $δ = 3 / 50$ before assembly, both local truncations are accepted and the assembled approximation is $M$ , which is indefinite. Each local error $3 / 50$ is below the smaller diagonal coefficient $1 / 10$ of $A$ . The example concerns truncation of low-rank updates before their sum is assembled; it does not claim that every SVD truncation of a completed positive-definite H-matrix destroys positivity.

Facts & Assumptions

Given: The matrices $M, D_{1}, D_{2}$ , tolerance $δ = 3 / 50$ , and rank cap $k = 0$ for the two update compressions.

[F1]

Core-SVD recompression with rank and error tolerances: the procedure accepts a rank-zero output when the discarded singular-value tail is at most $δ$ .

[F2]

Core-SVD recompression is best at the declared Frobenius tolerance: the returned rank-zero approximation has Frobenius error exactly its discarded singular-value tail.

[F3]

Finite tensor arrays and the Frobenius inner product: the Frobenius norm is computed from the squared moduli of all matrix entries.

Verification

1.1givenalgebra

The exact sum is $A = diag (1, - 1 / 50 + 3 / 50 + 3 / 50) = diag (1, 1 / 10)$ . For every nonzero $z = (z_{1}, z_{2})^{T}$ , $z^{*} A z = ∣ z_{1} ∣^{2} + (1 / 10) ∣ z_{2} ∣^{2} > 0$ , so $A$ is positive definite.

1.2givenF1F2F3algebra

Each $D_{i}$ has rank one and its only positive singular value is $3 / 50$ , so its rank-zero discarded tail and Frobenius error are both $3 / 50 = δ$ by [F3]. Thus [F1] accepts each update as zero, and [F2] gives its exact local error.

2.1givenstep 1.1step 1.2algebra∎

After both updates are replaced by zero, the sum is $M$ . Its quadratic form at $e_{2}$ is $e_{2}^{*} M e_{2} = - 1 / 50 < 0$ , so it is not positive definite. The accumulated perturbation is $D_{1} + D_{2} = (3 / 25) e_{2} e_{2}^{*}$ , whose action in this direction exceeds either local error; hence the two separate local tolerances do not preserve definiteness of the assembled sum.

ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-6-luna)Open item page →

A rank-one approximation of a separated Cauchy block

Example

Take $x_{1} = 0, x_{2} = 1$ in $[0, 1]$ and $y_{1} = 2, y_{2} = 3$ in $[2, 3]$ . For $K_{i j} = 1 / (y_{j} - x_{i})$ and rank order $r = 1$ , the first geometric term is ${\tilde{K}}_{i j} = 1 / y_{j}$ . It gives a rank-one approximation with an exactly computable nonzero residual.

Facts & Assumptions

Given: The four points and the order $r = 1$ .

[F1]

A separated Cauchy block has a geometric rank bound: for $a = 1, b = 2, c = 1$ , the rank-one approximation has entries $1 / y_{j}$ and each residual entry has absolute value at most $1 / 2$ .

[F2]

Finite tensor arrays and the Frobenius inner product: the Frobenius norm is the square root of the sum of squared entry moduli.

Verification

1.1givenF1algebra

The exact block is $K = (\begin{array}{cc} 1 / 2 & 1 / 3 \\ 1 & 1 / 2 \end{array})$ . The rank-one approximation is $\tilde{K} = (\begin{array}{cc} 1 / 2 & 1 / 3 \\ 1 / 2 & 1 / 3 \end{array}) = (1, 1)^{T} (1 / 2, 1 / 3)$ , so its rank is one.

2.1F1F2step 1.1algebra∎

The residual is $K - \tilde{K} = (\begin{array}{cc} 0 & 0 \\ 1 / 2 & 1 / 6 \end{array})$ , hence $∥ K - \tilde{K} ∥_{F} = \sqrt{1 / 4 + 1 / 36} = \sqrt{10} / 6 \leq 1 = \sqrt{2 \cdot 2} (1 / 2) / (2 - 1)$ by [F2]. This attains the stated finite block bound at the entry $(2, 1)$ .

Hierarchical low rank matrices and error propagation — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

5 · Examples, counterexamples and false statements

A nearfield identity block requires full rank for subunit spectral error

Example

Facts & Assumptions

Verification

Two accepted local truncations can destroy positive definiteness after accumulation

Example

Facts & Assumptions

Verification

A rank-one approximation of a separated Cauchy block

Example

Facts & Assumptions

Verification

Sources