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Optimal point density within the compact compander class

Statement

Fix $a < b$ and a positive normalized $C^{1}$ source density $f$ on $[a, b]$ . Among all positive normalized $C^{1}$ point densities $g$ used in Positive C1 compact one-dimensional compander, the asymptotic coefficient from Exact high-rate limit for a positive compact one-dimensional compander is uniquely minimized by

$g_{*} (x) = \frac{f (x)^{1 / 3}}{\int_{a}^{b} f (t)^{1 / 3} d t},$

and the minimum limit is

$\frac{1}{12} {(\int_{a}^{b} f (x)^{1 / 3} d x)}^{3} .$

This is optimality within the stated compact compander class, not unrestricted scalar or vector quantizer optimality.

Facts & Assumptions

Given: $a < b$ and a positive normalized $C^{1}$ density $f$ on $[a, b]$ .

[F1]

Positive C1 compact one-dimensional compander: an admissible point density $g$ is positive, normalized, and $C^{1}$ on the same compact interval.

[F2]

Exact high-rate limit for a positive compact one-dimensional compander: the asymptotic coefficient for $g$ is $\frac{1}{12} \int_{a}^{b} f / g^{2}$ .

[F3]

Real powers for positive bases, with the zero-base positive-exponent convention and Continuity and derivatives of positive-base real powers: for positive input, $f^{1 / 3}$ is defined as the positive real power and is $C^{1}$ with derivative $f^{'} / (3 f^{2 / 3})$ .

[F4]

Every continuous function on an interval has a primitive; two primitives differ by a constant; and $\int_{a}^{b} f = G (b) - G (a)$ for any primitive $G$ : every continuous integrand on $[a, b]$ has a finite Riemann integral.

[F5]

The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ : the derivative of a composition of differentiable Euclidean maps is the composition of their derivatives.

[F6]

A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value: a continuous real function on compact $[a, b]$ is bounded and attains its extrema.

Proof

1.1givenF1F3F4F6algebra

Set $A : = \int_{a}^{b} f (x)^{1 / 3} d x$ , $B (g) : = \int_{a}^{b} f (x) / g (x)^{2} d x$ , and $z (x) : = f (x) / (B (g) g (x)^{3})$ . Positivity and continuity on compact $[a, b]$ make $A, B (g)$ finite and strictly positive by [F3], [F4] and [F6]. Using $\int g = 1$ , direct cancellation gives $\int_{a}^{b} z (x) g (x) d x = 1$ and $\int_{a}^{b} z (x)^{1 / 3} g (x) d x = A / B (g)^{1 / 3}$ .

2.1F3step 1.1algebra

For each $x$ , $z (x) + 2 - 3 z (x)^{1 / 3} = (z (x)^{1 / 3} - 1)^{2} (z (x)^{1 / 3} + 2) \geq 0$ by the positive real-power convention [F3]. Multiply by $g (x) > 0$ and integrate; step 1.1 and $\int g = 1$ give $3 A / B (g)^{1 / 3} \leq 3$ , hence $B (g) \geq A^{3}$ .

3.1F3F6step 1.1step 2.1algebra

Equality $B (g) = A^{3}$ makes the integral of $(z^{1 / 3} - 1)^{2} (z^{1 / 3} + 2)$ against $g (x) d x$ equal $1 + 2 - 3 A / B (g)^{1 / 3} = 0$ , by step 1.1. By [F3] the root is positive; by [F6], $g$ has a positive minimum. If the continuous integrand were positive at any point, it would be bounded below by a positive number on a one-sided or two-sided interval of positive length, so that integral would be positive. Therefore the integrand vanishes everywhere, and its factorization gives $z = 1$ and $f / g^{3} = B (g)$ on $[a, b]$ .

4.1F1F2F3F5step 1.1step 3.1algebra∎

Step 3.1 gives $g = f^{1 / 3} / B (g)^{1 / 3}$ ; integrating and using $\int g = 1$ gives $B (g)^{1 / 3} = A$ , hence $g = g_{*}$ . Conversely, [F3] and [F5] show $f^{1 / 3}$ is positive and $C^{1}$ , so $g_{*}$ is an admissible positive normalized $C^{1}$ density; direct substitution gives $B (g_{*}) = A^{3}$ . Thus $g_{*}$ is the unique minimizer of the coefficient in [F2], with minimum $A^{3} / 12$ .

Depends on

Used by

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Sources

Robert M. Gray, Source Coding Theory, first edition (standard reference, not scraped)
Robert Gallager, 6.450 Principles of Digital Communications I, Chapter 3: Quantization (standard reference, not scraped)
Robert M. Gray and David L. Neuhoff, Quantization, IEEE Transactions on Information Theory 44(6) (1998) (standard reference, not scraped)