Asymptotic separation conjecture for mixtures of two uniform distributions

From papers

Let $P_1$ and $P_2$ be uniform distributions on closed intervals $[a,b]$ and $[c,d]$ , respectively, where $a<b<c<d$ . Let $P:=pP_1+(1-p)P_2$ be the mixture associated with the probability vector $(p,1-p)$ , where $0<p<1$ . For each $n$ , let $\alpha_n$ be an optimal $n$ -point set for $P$ , and let $V_n(P)$ denote its quantization error.

Asymptotic separation conjecture. For every probability vector $(p,1-p)$ , there exists a positive integer $N$ such that, for all $n eg N$ , the optimal set $\alpha_n$ contains points from both $[a,b]$ and $[c,d]$ , contains no point from $(b,c)$ , and, if $k:=\operatorname{card}(\alpha_n\cap[a,b])$ , then

\alpha_n(P)=\alpha_k(P_1)\cup\alpha_{n-k}(P_2),\qquad V_n(P)=\frac{1}{108}\left(\frac{p}{k^2}+\frac{1-p}{(n-k)^2}\right).

The conjecture predicts that sufficiently many quantization points eventually lie in the two supporting intervals and split into optimal sets for the two component distributions. It would settle the corresponding asymptotic behavior for mixtures of uniform distributions on separated intervals, while the source presents the related low-rate placement questions as open.

Progress summary

Open

The conjecture predicts that, at high resolution, optimal points split between the two intervals without entering the gap, but no general proof or counterexample has been found.

Gomez, Lopez, and Roychowdhury formulated the conjecture in work from $2022$ . It asserts that for all sufficiently large $n$ , every optimal quantizer separates between the two supporting intervals, has no gap points, and decomposes into optimal quantizers for the two components.

Known results

The related paper records proved special cases for $p=\frac{1}{100}$ , $p=\frac{2}{5}$ , and $p=\frac{1}{1000}$ , but not the general conjecture.
The same literature leaves related low-rate placement questions open, including whether one mixture can have both gap points at low rates and separated support points at a higher rate.

Current status (as of August 2026): The general asymptotic separation conjecture remains open; only selected weight cases and related partial results are recorded, with no retrieved proof, counterexample, or verification.

Sources

Sources & referencesView supporting material

Primary source

Asha Barua, Gustavo Fernandez, Ashley Gomez, Ogla Lopez and Mrinal Kanti Roychowdhury, “Quantization for the mixtures of uniform distributions on connected and disconnected line segments”, arXiv:2203.12664 (2025).

1 week ago

Conjecture 3.16, as stated for arbitrary supporting intervals, is false: its displayed quantization error is missing both interval lengths. Nevertheless, the geometric separation assertion holds in full generality, with the following corrected exact formula.

Put ℓ₁=b−a, ℓ₂=d−c, g=c−b>0, and h₀=min{p/ℓ₁,(1−p)/ℓ₂}>0. There exists N such that for every n≥N, every optimal n-point quantizer has points in both supporting intervals, no points in their gap, and

Vₙ(P)=min_{1≤j<n}{pℓ₁²/(12j²)+(1−p)ℓ₂²/[12(n−j)²]}.

Indeed, choosing ⌊n/2⌋ equally spaced midpoint centers in [a,b] and the remaining centers in [c,d] gives

Vₙ(P)≤C/n², C=3[pℓ₁²+(1−p)ℓ₂²]/4.

For any optimal codebook Γₙ, let Rₙ=max_{x∈[a,b]∪[c,d]}dist(x,Γₙ). If Rₙ≥ε, the 1-Lipschitz property of the distance function and the density lower bound h₀ give

Vₙ(P)≥h₀ε² min{ε/2,ℓ₁,ℓ₂}/4.

Consequently Rₙ→0 uniformly over optimal codebooks. Choose N so that Rₙ<g/2 for n≥N. Every optimal center has a positive-mass Voronoi cell and equals the conditional mean of that cell. If the Voronoi cell of a center z met both components, there would be x∈[a,b] and y∈[c,d] with

g≤y−x≤|y−z|+|z−x|≤2Rₙ<g,

a contradiction. Each cell therefore meets just one support component, and its centroid belongs to that component. Both components contain centers, and none lies in the gap.

If j centers belong to [a,b], the two component optimization problems separate. A uniform distribution on an interval of length ℓ has unique optimal j-point midpoint grid and exact quadratic distortion ℓ²/(12j²). Minimizing over j gives the corrected formula above and proves the entire intended separation statement.

To refute the published displayed formula, take [a,b]=[0,1], [c,d]=[2,3], and p=1/2. For every sufficiently large even n=2k, strict convexity and symmetry give the unique allocation j=k. The actual error is

V₂ₖ(P)=1/(12k²),

whereas the conjecture claims

V₂ₖ(P)=1/(108k²).

These differ by a factor of nine for infinitely many n. The constant 1/108 is valid only in the earlier special case ℓ₁=ℓ₂=1/3, not for the arbitrary intervals quantified in Conjecture 3.16.

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Shivam Patel · 21 hours ago