Linear Partial Gromov-Wasserstein Embedding
Yikun Bai, Abihith Kothapalli, Hengrong Du, Rocio Diaz Martin, Soheil, Kolouri
TL;DR
The paper introduces a linearized embedding for the partial Gromov-Wasserstein problem, significantly reducing computational complexity while maintaining effectiveness in tasks like shape retrieval and transport-based learning.
Contribution
It proposes the LPGW embedding, a linearization method that simplifies PGW computation from quadratic to linear in the number of metric spaces, and proves it defines a valid metric.
Findings
LPGW reduces pairwise computations from O(K^2) to O(K).
LPGW preserves partial matching advantages of PGW.
LPGW improves efficiency in shape retrieval and transport-based learning.
Abstract
The Gromov-Wasserstein (GW) problem, a variant of the classical optimal transport (OT) problem, has attracted growing interest in the machine learning and data science communities due to its ability to quantify similarity between measures in different metric spaces. However, like the classical OT problem, GW imposes an equal mass constraint between measures, which restricts its application in many machine learning tasks. To address this limitation, the partial Gromov-Wasserstein (PGW) problem has been introduced. It relaxes the equal mass constraint, allowing the comparison of general positive Radon measures. Despite this, both GW and PGW face significant computational challenges due to their non-convex nature. To overcome these challenges, we propose the linear partial Gromov-Wasserstein (LPGW) embedding, a linearized embedding technique for the PGW problem. For different metric…
Peer Reviews
Decision·ICLR 2025 Poster
1. The paper is well organized and easy to follow. 2. The paper contains many details, which is very helpful for understanding. 3. The experiment is promising, justifying theoretic part.
I am not an expert in this area, I don't find weaknesses from my perspective.
The paper is well motivated and the study of the proposed formulation is very thorough, covering both general and special cases under suitable assumptions. The proposed algorithm combines the advantages from both partial GW and linearized GW, and extensive experiments supports this extension.
1. The paper, though overall well motivated, is mostly combining existing methods and theoretical novelty is limited, with discussions following from the the original ideas and definitions of linearized/partial GW/OT. A further question: what if TV is replaced by a general divergence for unbalanced formulation as in [1]? 2. Discussion of related works: the term "linearization" is from the tangent structure of gauged GW space as is first pointed out in [2]. Mapping based unbalanced GW formulat
The paper is, in my opinion well-written, and the theoretical findings appear correct. In the experiments, the LPGW algorithm compares favorably to the other approaches considered.
I believe that the paper is of a limited novelty. Indeed, both the linearized GW problem and the partial GW problem have appeared in previous works. Combining these two approaches to obtain a linearized partial GW problem appears to be incremental progress on the important question of efficient computation for the GW problem. Furthermore, the theoretical properties of the LPGW problem appear quite limited. Many of the results require the existence of a Monge map for the PGW problem, but suffic
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TopicsDermatological and Skeletal Disorders · Bone health and treatments · Fibroblast Growth Factor Research