Random Permutation Codes: Lossless Source Coding of Non-Sequential Data (2411.14879v1)

Abstract: This thesis deals with the problem of communicating and storing non-sequential data. We investigate this problem through the lens of lossless source coding, also sometimes referred to as lossless compression, from both an algorithmic and information-theoretic perspective. Lossless compression algorithms typically preserve the ordering in which data points are compressed. However, there are data types where order is not meaningful, such as collections of files, rows in a database, nodes in a graph, and, notably, datasets in machine learning applications. Compressing with traditional algorithms is possible if we pick an order for the elements and communicate the corresponding ordered sequence. However, unless the order information is somehow removed during the encoding process, this procedure will be sub-optimal, because the order contains information and therefore more bits are used to represent the source than are truly necessary. In this work we give a formal definition for non-sequential objects as random sets of equivalent sequences, which we refer to as Combinatorial Random Variables (CRVs). The definition of equivalence, formalized as an equivalence relation, establishes the non-sequential data type represented by the CRV. The achievable rates of CRVs is fully characterized as a function of the equivalence relation as well as the data distribution. The optimal rates of CRVs are achieved within the family of Random Permutation Codes (RPCs) developed in later chapters. RPCs randomly select one-of-many possible sequences that can represent the instance of the CRV. Specialized RPCs are given for the case of multisets, graphs, and partitions/clusterings, providing new algorithms for compression of databases, social networks, and web data in the JSON file format.

• The paper introduces a novel method, Random Permutation Codes (RPCs), that efficiently compress non-sequential data via a bits-back coding strategy.
• It defines Combinatorial Random Variables (CRVs) to model diverse data structures, including multisets, partitions, and graphs.
• Empirical results show that RPCs achieve optimal compression rates and outperform traditional methods on large, sparse datasets.

Overview of "Random Permutation Codes: Lossless Source Coding of Non-Sequential Data"

The paper "Random Permutation Codes: Lossless Source Coding of Non-Sequential Data" by Daniel Severo introduces a novel framework for compressing non-sequential data through a family of algorithms known as Random Permutation Codes (RPCs). These algorithms aim to efficiently encode data types that inherently lack a sequential order, such as multisets, partitions, and graphs. The paper makes significant contributions by formally defining a new concept called Combinatorial Random Variables (CRVs) and demonstrating the potential of RPCs to achieve optimal compression rates under specific conditions.

Key Contributions

1. Combinatorial Random Variables (CRVs): The paper introduces CRVs as a generalization of data types that can be expressed as equivalence classes of sequences. A CRV is defined with respect to an arbitrary symbol alphabet, a sequence length, and an equivalence relation, where the equivalence relation is finer than permutation-equivalence. This concept allows for modeling complex non-sequential data structures such as multisets, partitions, and both directed and undirected graphs.
2. Optimal Compression via RPCs: The primary contribution lies in the development of RPCs, which can efficiently compress CRVs by using a bits-back coding technique. RPCs exploit the fact that the order in which data is stored can be manipulated to enable recovery of information without introducing redundancy. This is achieved through a sampling mechanism that uses asymmetric numeral systems (ANS) to optimally encode data points.
3. Sets, Multisets, and Clusters: Different combinatorial objects are encoded by varying the equivalence relation defining them. For example, a multiset is a CRV where the equivalence relation groups sequences with the same elements, regardless of their order. The paper extends this to more complex objects like partitions, clusters, and graphs.
4. Theoretical Insights and Practical Algorithms: The paper provides a detailed theoretical analysis of the achievable rates using RPCs, showing that the proposed methods can achieve rates equivalent to the Negative Evidence Lower Bound (NELBO) known in variational inference. Furthermore, the algorithms presented are computationally attractive, with complexity that scales based on the specific problem instance, making them practical for real-world applications.
5. Empirical Evaluation: Through experiments, the paper demonstrates that RPCs can outperform traditional methods in settings with large and sparse datasets by efficiently encoding non-sequential data types. The empirical results when applied to datasets like vector similarity search databases highlight the practicality and efficiency of RPCs in achieving optimal compression.

Implications and Future Directions

The implications of this work are broad, particularly in areas dealing with large-scale, non-sequential data such as database management, network data, and high-dimensional vector representations. By providing a framework to compress data structures without regard to an arbitrary order, RPCs offer a method to reduce the storage and transmission costs significantly.

The paper speculates on future developments, suggesting that the principles developed could be extended to other non-sequential data types, potentially even those that are not naturally represented as CRVs. One possible direction is relaxing the restriction on the equivalence relation to allow for more complex structures or manipulating it to explore new classes of non-sequential objects. Another future avenue could involve integrating machine learning techniques to automatically determine the most efficient equivalence relation for a given dataset.

In summary, this paper presents a comprehensive approach to lossless compression of non-sequential data, offering both theoretical insights and practical algorithms that efficiently encode complex data types as combinatorial random variables. Through Random Permutation Codes, it opens up new possibilities for optimizing data storage and communication, with implications across a variety of domains.