TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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Exact finite-data Lloyd descent and finite termination

Statement

For $N$ indexed points in $R^{d}$ with positive normalized weights, $k \geq 1$ centers, and any initial center tuple, the exact algorithm of Exact finite-data Lloyd quantizer terminates after at most $2^{N k}$ assignment/update passes. Every pass that changes a center strictly decreases the empirical squared-error distortion; at termination, the returned assignment is nearest-center and each nonempty cluster center is its weighted centroid. Each pass uses $O (N k d)$ exact arithmetic and comparisons, so the crude total operation bound is $O (2^{N k} N k d)$ . The conclusion is a fixed point, not a global-optimality claim.

Facts & Assumptions

Given: $N, d, k \geq 1$ , points $x_{1}, \dots, x_{N} \in R^{d}$ , weights $w_{i} > 0$ with sum one, and initial centers $c^{0}$ ; all comparisons and arithmetic are exact.

[F1]

Exact finite-data Lloyd quantizer: each pass assigns each point to its least-index nearest center, replaces every nonempty cluster center by its weighted centroid, leaves every empty-cluster center unchanged, and stops when the center tuple is unchanged.

[F2]

The weighted centroid minimizes finite-cluster squared error: for a nonempty cluster $S$ of weight $W_{S}$ and centroid $μ_{S}$ , its cost at center $c$ is its cost at $μ_{S}$ plus $W_{S} ∥ c - μ_{S} ∥_{2}^{2}$ .

[F3]

Put $X_{N} : = {x_{i} : 1 \leq i \leq N}$ , a finite Borel subset of $R^{d}$ , and let $μ_{N} : = \sum_{i = 1}^{N} w_{i} δ_{x_{i}}$ on $X_{N}$ , where $δ_{x_{i}}$ is unit point mass at $x_{i}$ . This is a probability measure on the relative Borel sigma algebra of $X_{N}$ with finite second moment $\int_{X_{N}} ∥ x ∥_{2}^{2} d μ_{N} (x) = \sum_{i = 1}^{N} w_{i} ∥ x_{i} ∥_{2}^{2} < \infty$ . For an ordered center tuple $c$ , let $Q_{c} : X_{N} \to {1, \dots, k}$ be its least-index nearest-center rule. Each fiber is Borel because $X_{N}$ is finite, so Finite measurable squared-error quantizer and nearest-center rule gives

$D_{μ_{N}} (Q_{c}; c_{1}, \dots, c_{k}) = \sum_{i = 1}^{N} w_{i} ∥ x_{i} - c_{Q_{c} (x_{i})} ∥_{2}^{2} = \sum_{i = 1}^{N} w_{i} \min_{1 \leq j \leq k} ∥ x_{i} - c_{j} ∥_{2}^{2} = : J (c) .$

Proof

1.1givenF1F3

At pass $t$ , the assignments in [F1] minimize each point's squared distance for the fixed tuple $c^{t}$ , so their assigned cost is exactly $J (c^{t})$ from [F3].

1.2givenF1F2F3algebra

Replacing the centers by cluster centroids changes the assigned cost by subtracting $\sum_{j : S_{j}^{t} \neq \emptyset} W_{S_{j}^{t}} ∥ c_{j}^{t} - μ_{S_{j}^{t}} ∥_{2}^{2}$ , by [F2]; empty clusters contribute no cost and their centers are unchanged by [F1]. If any center changes, one nonempty-cluster term is positive. Reassigning to nearest centers cannot increase cost, so $J (c^{t + 1}) < J (c^{t})$ .

1.3givenF1construct

Every center value occurring in the iteration is either its own initial value or the centroid of one of the at most $2^{N} - 1$ nonempty indexed subsets of the data; thus each center has at most $2^{N}$ possible values and there are at most $2^{N k}$ possible ordered center tuples.

2.1step 1.2step 1.3algebra

A changed pass cannot revisit a previous center tuple, because $J$ is a function of that tuple and strictly decreases at every changed pass by step 1.2. Hence at most $2^{N k} - 1$ passes change the tuple; the next pass leaves it unchanged and the algorithm stops, for at most $2^{N k}$ passes.

3.1F1step 2.1

At the stopping pass, the tuple is unchanged. The deterministic nearest-center rule therefore gives the same assignments on the next evaluation, and the update formula in [F1] already made every nonempty center its assigned centroid. Thus the returned pair is an assignment/centroid fixed point.

4.1givenstep 2.1algebra∎

One pass evaluates $N k$ squared distances in $d$ coordinates and performs at most $O (N d + k d)$ accumulation/update work, which is $O (N k d)$ since $N, d, k \geq 1$ . Multiplying by the pass bound gives $O (2^{N k} N k d)$ exact operations; this exponential bound makes no polynomial-time claim.

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Sources

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)
Robert M. Gray, Source Coding Theory, first edition (standard reference, not scraped)