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Instance Optimality

Module by: Ronald DeVore

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

  1. Instance optimality. Suppose we are in ℝNℝNℝ^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 x∈ℝNx∈ℝNx 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 ) ∥ ℓ p N ≤ C 0 σ k ( x ) ℓ q N k 1 ∕ q − 1 ∕ p . ∥ x − Δ Φ ( x ) ∥ ℓ p N ≤ C 0 σ k ( x ) ℓ q N k 1 ∕ q − 1 ∕ p . (3)
    Cases of interest include asking whether
    ∥ x − Δ Φ ( x ) ∥ ℓ 2 N ≤ C 0 σ k ( x ) ℓ 1 N . ∥ x − Δ Φ ( x ) ∥ ℓ 2 N ≤ C 0 σ k ( x ) ℓ 1 N . (4)
    and whether
    ∥ x − Δ Φ ( x ) ∥ ℓ 2 N ≤ C 0 σ k ( x ) ℓ 1 N k . ∥ x − Δ Φ ( x ) ∥ ℓ 2 N ≤ C 0 σ k ( x ) ℓ 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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