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Point masses, Jensen–Shannon saturation, and weight clipping

Example

Let $X = [0, 1]$ with distance $d (x, y) = ∣ x - y ∣$ , let $P = δ_{0}$ , and for $t \in [0, 1]$ let $Q_{t} = δ_{t}$ . For each $t > 0$ , let $F_{t} = {0, t}$ with the induced finite metric and ordering $(0, t)$ , and use probability vectors $p = (1, 0)$ and $q = (0, 1)$ ; for $t = 0$ , take $F_{0} = {0}$ and $p = q = (1)$ . For finite-support probabilities $R, S$ , define $KL (R ∥ S) = + \infty$ if $R$ gives positive mass where $S$ gives zero mass, and otherwise define $KL (R ∥ S) = \sum_{x : R (x) > 0} R (x) \log (R (x) / S (x))$ , with natural logarithms. Define $JS (R, S) = \frac{1}{2} KL (R ∥ (R + S) / 2) + \frac{1}{2} KL (S ∥ (R + S) / 2)$ . For every $t > 0$ , the finite-space transport cost $W_{1} (p, q)$ on $F_{t}$ , the compact-space coupling cost $W_{1} (P, Q_{t})$ , and the integral probability metric over all real functions satisfying $∣ f (x) - f (y) ∣ \leq ∣ x - y ∣$ on $X$ are all $t$ , whereas $JS (P, Q_{t}) = \log 2$ ; at $t = 0$ these three transport/IPM values and the JS value are zero. Thus the compact-space $W_{1} (P, Q_{t}) \to 0$ as $t ↓ 0$ while $JS (P, Q_{t})$ stays equal to $\log 2$ for every $t > 0$ .

For the same data space, consider the two-layer scalar affine critic $f_{a, b} (x) = a (b x)$ whose two scalar weights are constrained by $∣ a ∣, ∣ b ∣ \leq c$ for a fixed $c > 0$ . When $c > 1$ , the clipped family contains $f (x) = c^{2} x$ , which is not $1$ -Lipschitz; when $0 < c < 1$ , it cannot contain $f (x) = x$ ; when $c = 1$ , it cannot contain the nonlinear $1$ -Lipschitz function $f (x) = x^{2} / 2$ . Hence this fixed clipped affine family is not exactly the class of $1$ -Lipschitz critics for any $c > 0$ .

Example references

Dirac measure

Dirac measure is a probability measure

Integral probability metric

The coupling definition of the 1-Wasserstein cost on a compact metric space: Compact-space coupling definition of $W_{1}$

Finite-metric primal transport cost

Facts & Assumptions

Given: $X = [0, 1]$ with its usual metric, $P = δ_{0}$ , $Q_{t} = δ_{t}$ for $0 \leq t \leq 1$ , the finite-support definitions of $KL$ and $JS$ in the Example, and the critic family $f_{a, b} (x) = a b x$ with $∣ a ∣, ∣ b ∣ \leq c$ .

[F1]

A Dirac set function assigns mass $1$ exactly to measurable sets containing its point, and it is a probability measure. The Dirac set function at a point A Dirac set function is a probability measure

[F2]

For a nonnegative measurable function $g$ , $\int g d δ_{x} = g (x)$ . Integration against a Dirac measure evaluates at its point

[F3]

For finite probability vectors $p, q$ , $W_{1} (p, q)$ is the minimum of $\sum_{i, j} d (x_{i}, x_{j}) π_{i j}$ over nonnegative coupling matrices with row sums $p$ and column sums $q$ . The finite-metric 1-Wasserstein transport cost

[F4]

$W_{1}$ on a compact metric space is the infimum of $\int d (x, y) d γ (x, y)$ over Borel probability couplings with the prescribed marginals. The coupling definition of the 1-Wasserstein cost on a compact metric space

[F5]

The IPM is the supremum of the absolute differences of the test-function integrals over its specified class. Integral probability metrics

[F6]

A measure is countably additive on pairwise disjoint measurable sets; finite unions of null sets are null. Measures on sigma-algebras

Verification

1.1givenF3algebra

For $t > 0$ , the second row sum $0$ and nonnegative entries force $π_{21} = π_{22} = 0$ ; the first column sum $0$ then forces $π_{11} = 0$ , and the first row sum $1$ forces $π_{12} = 1$ , which also gives the required second column sum $1$ . This is the unique feasible matrix, and [F3] gives finite-space cost $d (0, t) = t$ . For $t = 0$ , the singleton probability vector has unique coupling $(1)$ and cost $0$ .

1.2givenF1F2F4F6algebra

By [F1], $P$ and $Q_{t}$ are Borel probability measures. If $γ$ is any coupling of $P$ and $Q_{t}$ , then $γ ({0} \times X) = 1$ and $γ (X \times {t}) = 1$ . The complement of their intersection is contained in the union of their two null complements, so finite subadditivity from [F6] gives $γ ({(0, t)}) = 1$ . A probability measure concentrated on the singleton $(0, t)$ equals its Dirac measure there: sets containing $(0, t)$ have mass $1$ , and sets omitting it have mass $0$ . The Dirac measure at $(0, t)$ has the required marginals, so it is the unique coupling. By [F2] and [F4], its cost is $d (0, t) = t$ , so $W_{1} (P, Q_{t}) = t$ , including $t = 0$ .

1.3givenF5algebra

Every function in the specified $1$ -Lipschitz class satisfies $∣ f (0) - f (t) ∣ \leq t$ , and $f (x) = x$ belongs to the class and has $∣ f (0) - f (t) ∣ = t$ . By [F5], the IPM is exactly $t$ .

1.4givenalgebra

If $t > 0$ , the mixture $(P + Q_{t}) / 2$ assigns mass $1 / 2$ to each of $0$ and $t$ . The finite sum definition gives $KL (P ∥ (P + Q_{t}) / 2) = \log 2$ and $KL (Q_{t} ∥ (P + Q_{t}) / 2) = \log 2$ , so $JS (P, Q_{t}) = \log 2$ . If $t = 0$ , the two laws agree, each KL term is $0$ , and the JS value is $0$ .

2.1givenalgebra∎

Under the weight bounds, the slope of $f_{a, b} (x) = a b x$ has absolute value at most $c^{2}$ . If $c > 1$ , the allowed choice $a = b = c$ gives slope $c^{2} > 1$ . If $0 < c < 1$ , every allowed slope has magnitude less than $1$ , so the function $x$ cannot occur. If $c = 1$ , every member is linear, while $x^{2} / 2$ is $1$ -Lipschitz because $∣ x^{2} - y^{2} ∣ / 2 = ∣ x - y ∣ (x + y) / 2 \leq ∣ x - y ∣$ on $[0, 1]$ ; it is not linear, since its value at $1$ is $1 / 2$ and at $1 / 2$ is $1 / 8$ . These three exhaustive cases prove the stated failure of exactness for every $c > 0$ .

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Sources

Martin Arjovsky, Soumith Chintala and Léon Bottou, Wasserstein GAN, §§2–3, PDF pp. 2–7 (standard reference, not scraped)
Filippo Santambrogio, Introduction to Optimal Transport Theory, §§1.1–1.3, PDF pp. 3–7 (standard reference, not scraped)