Research map / Learning theory and algorithms / Optimal transport

Sliced Inner Product Gromov–Wasserstein Distances

NeurIPS 2026 · Learning theory and algorithms › Optimal transport · approach: Sliced Wasserstein variants · 0 citations (OpenAlex)

Read the paperCheck your idea against 108,967 papers

AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.

Derives a closed-form solution for univariate inner-product Gromov–Wasserstein (IGW) and builds a rotationally invariant sliced IGW distance with Monte Carlo and subgradient optimization guarantees, applied to LLM representation comparison and user text clustering.

Problem

Gromov–Wasserstein alignment with inner product cost is statistically and computationally hard in high dimensions, and unlike Wasserstein it has no closed-form one-dimensional solution, which blocks slicing-based scalable variants.

Approach

Gives a fully general closed-form for univariate IGW (reducing to sorting for uniform empirical measures), then defines a sliced IGW distance with an outer minimization over Stiefel manifold to restore orthogonal invariance, with MC and constraint-dissolving subgradient optimization guarantees.

Key results

• Monte Carlo error of the sliced IGW estimate follows the predicted O(sqrt(log(m)/m)) rate on a Gaussian setting with d_x=5, d_y=10.
• In the heterogeneous user clustering experiment, sliced IGW achieves ARI (0.61, 0.44, 0.52) and purity (0.83, 0.75, 0.79), versus ARI (0.41, 0.17, 0.17) for Gaussian IGW.
• In the homogeneous clustering setting both methods recover true clusters exactly (ARI 1, purity 1).

Surprising findings

• Sliced IGW is not a metric in general: there exist measures with zero sliced IGW but nonzero IGW, and analyticity of characteristic functions is needed for equivalence.
• A naive average-sliced IGW is not orthogonally invariant, so an outer minimization over the Stiefel manifold is required.

Limitations

• Equivalence of sliced IGW zero and IGW zero requires analytic characteristic functions.
• Stiefel optimization uses a subgradient method with only asymptotic stationarity guarantees (no non-asymptotic rate).
• Gaussian-based clustering baseline and LLM comparison are small-scale experiments (800 passages, 24 synthetic users).

Experimental setup

Datasets: amazon-polarity ag-news arXiv-summarization IMDB

Benchmarks: amazon-polarity ag-news arXiv-summarization IMDB

Metrics: sliced IGW distance Monte Carlo error adjusted Rand index purity 1-CKA

Baselines: Gaussian IGW centered kernel alignment (CKA)

Builds on: sliced Wasserstein distance sliced quadratic Gromov-Wasserstein constraint dissolving approach for Stiefel manifold optimization

Closest papers by meaning

• A Novel Sliced Fused Gromov-Wasserstein Distance (AAAI 2026 · similarity 0.88)
• Sliced Gromov-Wasserstein (NeurIPS 2019 · similarity 0.88)
• Tree-Sliced Wasserstein Distance with Nonlinear Projection (ICML 2025 · similarity 0.85)
• Tree-Wasserstein Distance for High Dimensional Data with a Latent Feature Hierarchy (ICLR 2025 · similarity 0.84)
• Statistical and Computational Guarantees of Kernel Max-Sliced Wasserstein Distances (ICML 2025 · similarity 0.84)
• Partial Gromov-Wasserstein Metric (ICLR 2025 · similarity 0.84)