Random Cycle Coding: Lossless Compression of Cluster Assignments via Bits-Back Coding (2412.00369v1)

Abstract: We present an optimal method for encoding cluster assignments of arbitrary data sets. Our method, Random Cycle Coding (RCC), encodes data sequentially and sends assignment information as cycles of the permutation defined by the order of encoded elements. RCC does not require any training and its worst-case complexity scales quasi-linearly with the size of the largest cluster. We characterize the achievable bit rates as a function of cluster sizes and number of elements, showing RCC consistently outperforms previous methods while requiring less compute and memory resources. Experiments show RCC can save up to 2 bytes per element when applied to vector databases, and removes the need for assigning integer ids to identify vectors, translating to savings of up to 70% in vector database systems for similarity search applications.

• The paper introduces Random Cycle Coding, achieving optimal lossless compression of cluster assignments via bits-back coding.
• It utilizes induced permutation cycles and Foata's Bijection to encode clusters efficiently without redundant labels.
• Experiments on datasets like SIFT1M validate up to 70% storage savings, demonstrating RCC's practical impact in vector databases.

Random Cycle Coding: Lossless Compression of Cluster Assignments via Bits-Back Coding

The paper "Random Cycle Coding: Lossless Compression of Cluster Assignments via Bits-Back Coding" explores an innovative approach to encoding cluster assignments using an optimal method known as Random Cycle Coding (RCC). This methodology is significant within the context of efficiently communicating clusters in data systems, with a particular focus on modern vector similarity databases like FAISS.

Summary of Methodology and Contributions

The central contribution of this work is the development of Random Cycle Coding (RCC) which innovatively uses cycles of permutations to encode cluster assignments without the need for external labels. This approach efficiently encodes the ordering information within data and compresses the cluster assignments by taking advantage of the inherent structure of permutations and cycles. RCC achieves the Shannon bound in bit-rate savings, ensuring it is optimally efficient.

The methodology can be deconstructed as follows:

• Induced Permutation: Each permutation of a dataset induces disjoint cycles that encode the clusters. These cycles signify the cluster memberships and allow the encoding directly without extraneous labels.
• Bits-Back Coding: RCC applies bits-back coding employing an exact posterior, thereby optimizing the encoding efficiency to achieve theoretical minimum bit rates characterizing the cluster structure.
• Foata's Bijection: Utilization of Foata's Bijection helps transform permutations into an efficient canonical form, preserving cycle structure amid permutations and ensuring consistent encoding and decoding.

A key observation made in this research is the savings potential when cluster assignments are communicated without redundant integer ID labels. It demonstrates that the method can achieve savings of up to 70% in typical vector database systems, explicitly shown in experiments on real-world databases like SIFT1M and BigANN.

Numerical and Experimental Insights

The authors provide highly quantitative insights by demonstrating the savings RCC brings over existing methodologies. For instance, when sequences of high-dimensional vectors are clustered, compared to Random Order Coding (ROC) variants, RCC showed superiority in both numerical savings and computational efficiency.

• Time Complexity: The RCC operates with a quasi-linear time complexity proportional to the largest cluster size, making it computationally feasible for large datasets.
• Savings Potential: The paper thoroughly explores both mathematically and experimentally the settings under which RCC achieves maximal compression gains, particularly when there are few larger clusters rather than numerous small ones.
• Performance in Real-World Applications: By testing on datasets commonly used in vector databases, RCC's applicability is confirmed with significant storage savings, marking it as a practical solution for cluster encoding in similarity search applications.

Implications and Future Directions

The implications of Random Cycle Coding extend beyond just cluster compression within vector databases. Given its optimal nature, this novel coding methodology could serve as a foundational basis for further innovations in compression algorithms across various domains that rely on clustering large datasets. The theoretical underpinning of RCC using information theory principles suggests that future developments could enhance or extend this approach to different types of data or more complex clustering scenarios.

Conclusion

In summary, "Random Cycle Coding: Lossless Compression of Cluster Assignments via Bits-Back Coding" delivers a precise and practically viable method for cluster assignment encoding. It integrates permutation theory and bits-back coding into a coherent system that optimally and effectively compresses cluster data structures. The research contributes a profound theoretical advancement and shows great promise for wide-ranging practical applications in data systems optimized for similarity searches.