DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

DPLR structured state-space parameterization

Definition

An $N$ -state single-input, single-output complex linear state-space model has $B, C \in C^{N}$ , state equation $x^{'} (t) = A x (t) + B u (t)$ , and output $y (t) = C^{*} x (t)$ ; its direct term is set to zero in the S4 convolution convention. Here $X^{*}$ denotes conjugate transpose; a square complex matrix $H$ is normal when $H^{*} H = H H^{*}$ ; and a square matrix $V$ is unitary when $V^{*} V = V V^{*} = I$ . A matrix is in diagonal-plus-low-rank (DPLR) form of rank $r$ if

A = Λ - P Q^{*},

where $Λ \in C^{N \times N}$ is diagonal and $P, Q \in C^{N \times r}$ . It is in normal-plus-low-rank (NPLR) form of rank $r$ if

A = V Λ V^{*} - P Q^{*},

where $V \in C^{N \times N}$ is unitary, $Λ$ is diagonal, and $P, Q \in C^{N \times r}$ . The unitary state change $\tilde{x} = V^{*} x$ converts this NPLR representation to DPLR form and changes $B, C$ to $V^{*} B, V^{*} C$ , respectively.

For a step size $Δ > 0$ such that $I - Δ A / 2$ is invertible, the bilinear recurrent matrices are $\overline{A} = (I - Δ A / 2)^{- 1} (I + Δ A / 2), \overline{B} = (I - Δ A / 2)^{- 1} Δ B, \overline{C} = C .$ With initial state $x_{- 1} = 0$ , the discrete recurrence is $x_{t} = \overline{A} x_{t - 1} + \overline{B} u_{t}$ , $y_{t} = {\overline{C}}^{*} x_{t}$ . Its length- $L$ impulse kernel is the vector $K_{L} = ({\overline{C}}^{*} {\overline{A}}^{j} \overline{B})_{j = 0}^{L - 1}$ . For a nonzero direct term $D$ , the output has the additional separate skip term $D u_{t}$ .

For a filter evaluation of transform length $M \geq L$ , the output vector for the length- $M$ truncated generating function is ${\tilde{C}}_{M} = (I - {\overline{A}}^{M})^{*} C$ . A filter procedure may receive ${\tilde{C}}_{M}$ as given data; an operation count for evaluating the filter from this transformed vector does not include the separate work of forming it from $C$ .

Definition references

Continuous-time linear state-space model

Gu, Goel, and Ré, “Efficiently Modeling Long Sequences with Structured State Spaces,” §§2.1, 2.3, 3.2, Appendix C.3

Depends on

Continuous-time linear state-space model

Used by

Dependency tree · two levels

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Sources

Gu, Goel, and Ré, Efficiently Modeling Long Sequences with Structured State Spaces, §§2.1, 2.3, 3.2, Appendix C.3 (standard reference, not scraped)