Maximum Data Similarity

• Maximum Data Similarity is a framework that integrates MDS codes and similarity-maximization methods to ensure optimal data redundancy and recoverability.
• It applies rigorous techniques in distributed storage, network coding, and multidimensional scaling to balance redundancy with performance.
• Practical applications include efficient data scheduling, model selection using metrics like MMD, and quantum error correction for robust information systems.

Maximum Data Similarity (MDS) encompasses both a family of powerful concepts in information theory, coding, and data analysis, and a spectrum of rigorous techniques for optimizing data storage, transmission, and representation in high-reliability, high-performance, or high-dimensional settings. The term predominantly refers to maximum distance separable codes, which realize optimal trade-offs between redundancy and recoverability, but also extends to measure-based notions of similarity-maximization in statistical learning and geometric data analysis. MDS-based methods provide foundational solutions for distributed storage, coding for symbol-pair channels, scalable model selection in generative learning, multidimensional embedding, and cryptography.

1. Foundations: Maximum Distance Separable Codes and Maximum Data Similarity

Maximum distance separable (MDS) codes are linear error-correcting codes with parameters $[n, k, d = n - k + 1]$ that meet the Singleton bound with equality. The key property of an $(n, k)$ MDS code is that any $k$ out of $n$ codewords suffice to fully reconstruct the original data. This property is realized, for example, by Reed-Solomon codes used in storage and communication.

In the broader context, “maximum data similarity” is often used as an umbrella term (editor’s term) for settings where each data fragment, stored object, or statistical representation is maximally informative or recoverable with respect to the whole. This redundancy, optimally allocated, underpins robust distributed storage (data fragments are mutually substitutable), cooperative data exchange (any sufficient subset of messages ensures universality), and embedding methodologies where geometric or probabilistic similarity is maximized.

For distributed data storage (Shah et al., 2012), the high “similarity” refers to the property that any $k$ servers’ fragments suffice for data recovery, and every subset of $k$ out of $n$ fragments is equally informative. In model selection for generative models, “maximum data similarity” alludes to choosing models whose distributions are closest to real data under metrics like maximum mean discrepancy (MMD) (Bounliphone et al., 2015). In multidimensional scaling (MDS), the goal is to maintain pairwise similarities (or distances) as faithfully as possible in a lower-dimensional embedding space.

2. MDS Codes in Storage and Communication: Latency, Redundancy, and Scheduling

The efficiency and flexibility of MDS codes stem from their “any-$k$” property: for any subset of $k$ fragments, reconstruction is always possible. In large-scale data centers (Shah et al., 2012), this feature allows for drastically reduced storage overhead compared to naive replication (a $1/k$ per-server overhead, rather than full duplication), while guaranteeing the same reliability.

A central technical development is the analysis of the “MDS queue,” which models requests for data as batches of jobs, each corresponding to reading $k$ fragments from $n$ servers. The challenge is optimizing latency and throughput. Exact analysis is complex due to the combinatorial job allocation, so the paper introduces upper and lower bounding scheduling policies:

• MDS-Reservation($t$) queues (lower bound): Restrict service so that only the first $t$ waiting batches can be scheduled as soon as there are $k$ idle servers, with remaining batches waiting until resources are released. In the $t = 0$ case, all requests wait for full availability, simplifying modeling.
• n($t$) queues (upper bound): Relax the constraint that $k$ jobs per batch go to distinct servers once the backlog exceeds $t$, approaching a standard $M/M/n$ queue for $t = 0$.

Both policies admit a Quasi-Birth-Death Markov process representation, enabling efficient computation of key metrics (mean and percentile latency, waiting probability, system occupancy) using block-structured transition matrices:

$$\begin{bmatrix} B_1 & B_2 & 0 & 0 & \cdots \ B_0 & A_1 & A_2 & 0 & \cdots \ 0 & A_0 & A_1 & A_2 & \cdots \ \vdots & & & \ddots \end{bmatrix}$$

Simulation validates that even with modest $t$ (e.g., $t=3$), these bounds tightly capture true system performance. Explicit stationary distributions and throughput formulas (e.g., $\rho_{\rm MDS}^* = n/k$) directly guide the extraction of optimal code parameters.

For “degraded reads”—requests for partial data, such as in the presence of failure—regenerating and product-matrix codes introduce a parameter $d$ (number of helper fragments). Here, data can be reconstructed by downloading a fraction $1/(d-k+1)$ from $d$ nodes, reducing bandwidth and improving degraded read latency beyond standard erasure codes.

3. MDS Codes, Cooperative Data Exchange, and Network Coding

The concept of maximum data similarity recurs in cooperative data exchange problems (Li et al., 2017), where the objective is for all network nodes to reconstruct a file, starting from partial knowledge. The “(d,K)-Basis” construction achieves minimal broadcasts by ensuring each transmission uses a maximally overlapping subset of exactly $d+1$ packets, engineered so that any node with at least $d$ packets can recover the rest from $K-d$ coded messages. Existence and construction are directly linked to the algebraic structure of MDS codes: Vandermonde matrices over finite fields are used to ensure that every subset of the relevant size has full rank. This structure guarantees each node’s new information is maximally similar (in terms of combinatorial content) to the union of transmitted packets, optimizing universality and efficiency.

Optimization further extends to weighted transmission cost or prioritized knowledge sets, with efficient algorithms selecting $d$ to minimize total complexity. Applications to weighted-cost settings and successive local omniscience leverage the same basis construction, with polynomial-time search strategies ensuring scalability.

4. MDS Symbol-Pair Codes and High-Density Storage Channels

In symbol-pair channels (relevant, for instance, in optical/magnetic storage where data is read via overlapping symbol chunks), MDS symbol-pair codes (Ding et al., 2016) provide maximum error correction for “pair errors.” Here, data similarity refers to the preservation of information even when read units overlap, and maximization involves achieving Singleton-type bounds under new distance metrics.

Constructions rely on parity-check matrices satisfying cyclic consecutive independence, use of projective geometry configurations (e.g., ovoids in $PG(3,q)$), and elliptic curve algebraic geometry codes—reordered to ensure required cyclic independence. The resulting symbol-pair codes bridge classical error correction and geometric combinatorics, and, due to strong pair-distance properties, ensure retrieval reliability even in the presence of extensive local corruption or channel overlaps.

The general relationship, proved formally, is:

$d_H(u, v) + 1 \leq d_p(u, v) \leq 2 d_H(u, v)$

with explicit constructions achieving $d_p = d_H + 2$ for broad parameter ranges.

5. MDS, NMDS, and AMDS Codes: Duality, Near-Optimality, and Combinatorial Structures

Extensions beyond strict MDS parameters include almost MDS (AMDS, $d = n - k$) and near MDS (NMDS, both code and dual AMDS, so $d(\mathcal{C})+d(\mathcal{C}^\perp) = n$) codes (Ding et al., 2019, Wang et al., 2020, Sun et al., 2023). NMDS codes nearly meet the Singleton bound and possess strong duality symmetry; their weight distributions, codeword support structures, and combinatorial alignment with $t$-designs and Steiner systems (e.g., S(3,4,v)) provide mechanisms for both error resilience and structured redundancy essential for cryptography, combinatorial design, and advanced storage. The explicit characterization of their weight distributions and systematic constructions from BCH codes, oval polynomials, or projective geometric configurations reflect the depth of the combinatorial “data similarity” encoded.

In addition, extended MDS cyclic codes (via parity extensions) reveal trade-offs between MDS and NMDS status, with direct implications for weight enumerators vital for performance analysis and design of distributed cloud storage systems.

6. Geometric, Probabilistic, and Bayesian Notions of Maximum Data Similarity

The concept of MDS generalizes to geometric and statistical data science under the terminology “maximum data similarity.” In multidimensional scaling (MDS), the objective is to embed data into low-dimensional spaces while maximally preserving pairwise similarity (Boyarski et al., 2017, Peterfreund et al., 2018, Delicado et al., 2020, Herath et al., 2021, Liu et al., 2022). Least-squares variants, spectral reformulations, blockwise and out-of-sample algorithms, and Bayesian extensions (including in hyperbolic manifolds for hierarchical data) provide scalable, exact, or uncertainty-quantified similarity maximization. Approaches rely on techniques such as majorization (SMACOF), spectral reduction (via Laplace-Beltrami bases), and adaptive out-of-sample mapping (with neural network regressors trained against landmark embeddings).

Statistical learning introduces relative-similarity tests based on maximum mean discrepancy (MMD) (Bounliphone et al., 2015), which rigorously ascertain which generative model produces samples maximally similar to real data. These kernel-based measures, with analytically derived asymptotic variance and covariance, yield efficient, high-power ranking for model selection.

7. Quantum MDS Codes and Cryptographic Constructions

Quantum analogues of MDS codes (Fang et al., 2018) play a parallel role in quantum error correction and cryptography. Employing Hermitian self-orthogonal generalized Reed-Solomon codes, constructions yield quantum MDS codes saturating the quantum Singleton bound ($2d \leq n - k + 2$). The design of semi-involutory MDS matrices in symmetric cryptography (e.g., for use in block cipher diffusion layers) (Chatterjee et al., 18 Jun 2024) addresses the dual priority of maximum diffusion (all nontrivial minors nonzero) and computationally efficient inverses. Analytical results rigorously specify necessary and sufficient conditions for $3 \times 3$ semi-involutory MDS matrices via their diagonal entries and an associated diagonal matrix, and full enumeration over finite fields of characteristic 2 provides explicit design guidance for efficient, secure implementation.

8. Representative Table: Key Flavors of Maximum Data Similarity

Domain	Principle	Main Construction / Metric
Classical Coding	$(n,k)$ MDS code (all $k$-sets reconstruct)	Reed-Solomon, Vandermonde matrix
Data Storage	High-similarity erasure codes, MDS queue	Reservation($t$) scheduling, QBD models
Symbol-Pair Coding	MDS symbol-pair code (max pair-distance)	Parity-check, projective geometry, elliptic
Network Coding	Cooperative exchange via $(d,K)$-Basis	MDS-like support patterns, coverage bounds
Multidim. Scaling	Max similarity low-dim embedding	Stress minimization, spectral subspace
Model Selection	Most similar model under data metric	Max mean discrepancy (MMD) and test
Quantum Coding	Quantum MDS code (max separation)	Hermitian self-orthogonal GRS codes
Cryptography	MDS/semi-involutory matrices in ciphers	Algebraic diag. structure, lightweight inv.

9. Conclusion

Maximum Data Similarity, in its various incarnations, serves as a stringent paradigm for optimal information allocation, recovery, and representation—from the algebraic certainties of code design through queue-theoretic system analysis to metric and probabilistic embedding methodologies. Across storage, communication, generative modeling, combinatorics, and cryptography, MDS-based theories guarantee maximal resilience, efficiency, and fidelity by structuring and quantifying precise forms of redundancy and similarity, shaping both theoretical understanding and practical engineering of modern information systems.