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Instance Optimality

Module by: Ronald DeVore. E-mail the author

Now we consider another way (actually two related ways) to measure optimality of an encoder/decoder pair.

1. Instance optimality. Suppose we are in NNℝ^N with an n×Nn×Nn times N measurement matrix ΦΦΦ and a decoder ΔΔΔ . Recall that
σ k ( x ) X : = inf z : # z k x z X . σ k ( x ) X : = inf z : # z k x z X .
(1)
We say that the encoding/decoding strategy Φ,ΔΦ,ΔΦ ,Δ is instance optimal of order kkk with constant C0C0C_0 if
x Δ Φ ( x ) X C 0 σ k ( x ) X x Δ Φ ( x ) X C 0 σ k ( x ) X
(2)
for all xNxNx in ℝ^N . (Note that we are no longer restricting xxx to a class KKK .) Better ΦΦΦ ’s have larger kkk for which this holds. The name “instance optimal” indicates that the encoding/decoding performance depends on each instance of xxx .
2. Mixed-norm instance optimality (MNIO). Let q<pq<pq <p . The encoder/decoder pair Φ,ΔΦ,ΔΦ ,Δ is MNIO for p,q,kp,q,kp ,q ,k , and C0C0C_0 if
x Δ Φ ( x ) p N C 0 σ k ( x ) q N k 1 q 1 p . x Δ Φ ( x ) p N C 0 σ k ( x ) q N k 1 q 1 p .
(3)
Cases of interest include asking whether
x Δ Φ ( x ) 2 N C 0 σ k ( x ) 1 N . x Δ Φ ( x ) 2 N C 0 σ k ( x ) 1 N .
(4)
and whether
x Δ Φ ( x ) 2 N C 0 σ k ( x ) 1 N k . x Δ Φ ( x ) 2 N C 0 σ k ( x ) 1 N k .
(5)

Let’s focus on instance optimality. It would be interesting to know whether a given ΦΦΦ satisfies this property. To answer this question, we state an equivalent condition to instance optimality.

Theorem 1

Consider the statements

1. Φ,ΔΦ,ΔΦ ,Δ is instance optimal of order kkk on XXX .
2. ΦΦΦ has the following nullspace property (NSP): η X C 1 η T c X η N ( Φ ) , # T k . η X C 1 η T c X η N ( Φ ) , # T k .
3. η X C 1 σ k ( η ) X η N ( Φ ) . η X C 1 σ k ( η ) X η N ( Φ ) .
4. η T X C 1 η T c X η N ( Φ ) , # T k . η T X C 1 η T c X η N ( Φ ) , # T k .

Then (b) and (c) are equivalent with the same constant; (d) is equvalent to (b) and (c) but with a different constant. Also (a) with a value kkk implies (b) with the same kkk , and (b) with a value 2k2k2 k implies (a) with a value kkk .

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Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

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