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A collision-free column coloring recovers every structurally sparse Jacobian entry

Statement

Let the objects and notation be those of The column-interference graph and collision-free coloring for a sparse Jacobian, and suppose $χ$ is a proper coloring of the graph of a safe structural row pattern. For every row $0 \leq i < m$ , every color $c$ , and every column $j \in S_{i}$ with $χ (j) = c$ ,

$J_{i j} = (J s^{c})_{i} .$

Entries with $j \notin S_{i}$ are zero by the safe-pattern hypothesis. Thus all entries are recovered from one compressed product per color. A color collision in a row does not give a general recovery guarantee: two distinct Jacobians supported on that colliding row can have the same compressed product.

Facts & Assumptions

Given: A differentiable map, its Jacobian, a safe row pattern and a proper coloring as in the statement.

[F1]

The column-interference graph and collision-free coloring for a sparse Jacobian: $J_{i j} = 0$ outside $S_{i}$ ; an edge joins columns that share some $S_{i}$ ; a proper coloring assigns different colors to adjacent columns; and $s^{c}$ is the indicator of color $c$ .

[F2]

The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case: the Jacobian entries are the partial derivatives $J_{i j} = \partial_{j} F_{i} (x)$ .

[F3]

Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes: for an $m \times n$ matrix $A$ and a vector $v \in R^{n}$ , $(A v)_{i} = \sum_{j < n} A_{i j} v_{j}$ .

[F4]

A finite sum in a commutative monoid indexed by an arbitrary finite set and Semigroup and monoid: a finite sum is independent of its enumeration, the empty sum is $0$ , and addition is associative with identity $0$ .

Proof

Given: The hypotheses listed above.

1.1givenF1F3F4algebra

For a fixed row $i$ and color $c$ , [F3] gives $(J s^{c})_{i} = \sum_{j < n} J_{i j} s_{j}^{c}$ . By [F1], a summand can be nonzero only when $j \in S_{i}$ and $χ (j) = c$ . [F4] makes this an order-independent finite sum with zero terms off that set.

2.1givenF1F4step 1.1algebra

If two distinct columns of $S_{i}$ had color $c$ , the graph definition in [F1] would make them adjacent, contradicting proper coloring. Thus at most one summand in step 1.1 is nonzero. If the specified $j \in S_{i}$ has color $c$ , it is that sole summand and equals $J_{i j}$ because $s_{j}^{c} = 1$ ; if no such column exists, every summand is zero and the row product is $0$ by [F4]. Entries outside the safe pattern are zero by [F1].

3.1F1F2F3constructalgebra∎

To verify the collision caveat, let $F (z_{0}, z_{1}) = z_{0} + 2 z_{1}$ and $G (z_{0}, z_{1}) = 3 z_{0}$ , and evaluate both Jacobians at $(0, 0)$ . By [F2], their one-row Jacobians are $(1, 2)$ and $(3, 0)$ , both supported in $S_{0} = {0, 1}$ . Assign columns $0$ and $1$ the same color, whose seed is $(1, 1)^{T}$ by [F1]. Both compressed products equal $3$ by [F3], so this color collision cannot determine the entries.

Depends on

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Sources

Thomas F. Coleman and Jorge J. Moré, Estimation of Sparse Jacobian Matrices and Graph Coloring Problems, Argonne National Laboratory Report ANL-81-39 (standard reference, not scraped)
Bradley M. Bell and Kasper Kristensen, Computing Sparse Jacobians and Hessians Using Algorithmic Differentiation (standard reference, not scraped)