lemmaroutine

Atom margins: sparse rows with exposed classes

Main and preparatory rows have ≤

K-1

nonzeros inside an exposed class

sparsity input for ports and recipes

Lemma·lem:atom-margins·§5.5 Actual ports and estimated assembly

Statement (paper)

Suppose a bounded nonnegative main function

k(v)

vanishes for

v\le\vartheta

. Its product

k(v)R_{\vartheta}(\rho)

and the preparatory product

h_{\vartheta}(v)R_{\vartheta}(\rho)

have at most

K-1

nonzero real entries in a row. They admit a common exposed class

A

, of at most

K-1

real entries, containing their supports and satisfying

I_{\vartheta}=1

throughout

A

. These assertions hold for arbitrary real raw estimates, without a sign or accuracy hypothesis. The auxiliary score

I_{\vartheta}

also has at most

K-1

nonzero real entries and has an exposed row class with a bounded-support smooth witness equal to one on that class.

In plain English

For a bounded nonnegative main function

k(v)

vanishing for

v\le\vartheta

, the rows of

k(v)R_{\vartheta}(\rho)

and of the preparatory product

h_{\vartheta}(v)R_{\vartheta}(\rho)

have at most

K-1

nonzero real entries, and share an exposed class

A

of at most

K-1

entries containing their supports on which

I_{\vartheta}=1

. The auxiliary score

I_{\vartheta}

likewise has at most

K-1

nonzero entries per row and an exposed row class with a bounded-support smooth witness equal to one on it; all this holds for arbitrary real raw estimates.

Why it's here

This gives the sparsity and exposure guarantees that later port and recipe statements rely on: each row of the main and preparatory tables has few nonzero entries lying in a common identifiable class. Because it needs no sign or accuracy assumption on the raw estimates, it applies to the actual estimated assembly, not only to idealized estimates.

Classified routine:

A direct sparsity and support bookkeeping consequence of the definitions of the cutoff functions

R_\vartheta

h_\vartheta

and

I_\vartheta

: at most

K-1

entries can exceed the threshold

\vartheta

in a row. It is a technical lemma supporting the estimated assembly, not a main new idea.

Uses

Nothing else in the paper

Used by

lemAtom ports: bounded rows with smooth witnesseslemEstimator recipes have regenerable records and bounded controlclaimRanks, compression and list selectionclaimFresh presence tests before value or motion

Sources

PaperOpen

Section: §5.5 Actual ports and estimated assembly
Page: —
Label: lem:atom-margins
Line: sections/atoms.tex:442

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