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Relational Compression: A Framework for Relational Fidelity in Constrained Representations

Yaniv Shulman

arXiv:2609.31816v1cs.LGcs.ITmath.IT

TL;DR

The paper asks how compressed representations can preserve relationships among elements rather than only the elements themselves. It formalizes a common relational-compression interface and develops finite-codeword collision; its correspondences and studies illustrate distinct relational and reconstruction-defined requirements within that framework.

  • Problem

    When information lies in relationships among elements, compression must specify which relational distinctions its constrained representation should preserve.

    Source: §1 · p. 1

  • Method

    The paper formalizes relational compression through source relation, description, reconstruction, fidelity, and resource, then develops finite-codeword collision as one realization.

    Source: §1 · p. 2

  • Results

    Same-codeword probability connects local relational requirements with aggregate R´enyi-2 occupancy and spherical geometry, with exact correspondences to centroid reconstruction and normalized graph association and cut.

    Source: §5.3 · p. 17, §9 · p. 43

  • Takeaways & Limitations

    The studies illustrate how reconstruction-defined, graph-defined, and teacher-defined requirements can shape finite representations within a common formulation.

    Source: §9 · p. 43

    At nonzero assignment uncertainty, collision is a graded affinity that generally lacks equivalence-relation transitivity, and a joint decoder need not admit the per-codeword centroid decomposition.

    Source: §4.2 · p. 13, §7.4 · p. 27

Abstract

from arXiv · show

What should a compressed representation preserve when the information of interest lies in relationships among elements rather than in the elements themselves? We formulate relational compression in the classical source-description-reconstruction sense, but with relational structure itself as the fidelity-bearing content. Each instance specifies the source relation, retained description, reconstructed or evaluated relation, fidelity criterion, and constrained resource. We use this interface to situate selected methods from graph summarization, spectral sparsification, similarity-preserving representation, and relational distillation within a common formulation while keeping their different reconstruction and resource assumptions explicit. We develop finite-codeword collision as one concrete realization. Same-codeword probability yields a relational geometry linking pair-specific alignment and separation to aggregate R\'enyi-2 occupancy and the spherical geometry of categorical assignments, with exact objective correspondences to squared-Euclidean centroid reconstruction and normalized graph association and cut. Graph and image studies illustrate complementary routes within the finite-codeword family: graph- and teacher-defined relational requirements act directly on equality or collision, while reconstruction acts through a joint decoder. Together, these results illustrate how distinct relational requirements can be formulated and tested within a common constrained-representation framework.

1. Introduction

The introduction frames compression around deciding which distinctions should survive, with relational structure—not only elements—as the content to preserve. It proposes a common source–description–reconstruction interface and develops finite-codeword collision as one concrete realization.

1. Introduction

  • Relational compression treats relationships among elements as fidelity-bearing content and asks how to represent them economically under a chosen criterion.Examples include graph connections, observation similarities, and teacher-induced affinities.Source: §1 · p. 1
  • Graph summaries, similarity-preserving hashing, and relational distillation [Park et al., 2019; Tung and Mori, 2019] can be examined through this common interface.Their descriptions, fidelity criteria, and constrained resources differ.Source: §1 · pp. 1–2
  • The framework makes the source relation, retained description, reconstructed or evaluated relation, fidelity criterion, and resource explicit while retaining construction-specific assumptions.The same assignments can support different relational decoders.Source: §1 · p. 2
  • Finite-codeword collision links pairwise alignment and separation with aggregate R´enyi-2 occupancy and spherical geometry, with exact correspondences to centroid reconstruction and normalized graph association and cut.Graph and image studies illustrate this realization under different relational and reconstruction-defined requirements.Source: §1 · p. 2

2. A Framework for Relational Compression

This section formalizes relational compression by separating the source relation, retained description, reconstruction, fidelity, and resource. It presents pairwise and selected structural criteria for assessing what relational content a constrained description preserves.

2. A Framework for Relational Compression

  • The framework treats relational compression as a trade-off between description cost and distortion of the relation reconstructed or evaluated from the description.Its core components include the source relation, retained description, reconstruction, fidelity criterion, and resource measure.Source: §2 · p. 3, §2.3 · p. 5

2.2. Descriptions and reconstruction

  • A description and decoder determine the reconstructed relation, so assignments alone need not uniquely determine what relation is recovered.For block summaries, assignments select pairs sharing reproduction values and the table supplies those values.Source: §2 · p. 3, §2.2 · p. 4

2.1. Relational sources

  • A relational source consists of a carrier and a relation among its elements; the compression target is the relation conditional on shared carrier identities.For observations, the relation may encode distances, similarities, or neighborhoods; for graphs, it may encode adjacency or edge weights.Source: §2.1 · pp. 3–4

2.3. Resources and relational fidelity

  • A resource measure records description cost, while a fidelity criterion measures relational distortion; the best attainable distortion is nonincreasing as the budget grows.Zero distortion under a chosen criterion need not imply literal equality of source and reconstruction.Source: §2.3 · p. 5

2.4. Pairwise relational fidelity

  • Pairwise fidelity averages local relation-value distortion over a declared pair distribution, which determines which source pairs receive weight.Uniform pair weights with squared error give average relation-value reconstruction error.Source: §2.4 · p. 5

2.5. Selected structural fidelity constructions

  • Selected structural criteria measure retained content or compare relation-derived operator behavior on probes, allowing fidelity to target properties beyond individual relation values.Content examples include edge weight and path-based values; operator fidelity weights signal directions through a probe distribution or matrix.Source: §2.5 · pp. 6–7

3. Existing Methods Through the Framework

This section maps representative methods to the framework while preserving their distinct fidelity and resource conventions. A block-valued summary illustrates how assignments and a reproduction table reconstruct relational structure.

3.1. Representative mappings

  • Representative methods differ in their relational sources, fidelity criteria, retained descriptions, reconstructions, and resource conventions.The mappings include graph summaries, spectral sparsification, compact codes and coordinates, relational transfer, and objective correspondences.Source: §3.1 · pp. 7–9, Table 2 · p. 8
  • Spectral hashing uses binary codes whose Hamming geometry reflects source similarities, while relational distillation uses normalized teacher distances to supervise student-feature relations.The constrained resource may be code storage, student-feature storage, or a student model, depending on the construction.Source: §3.1 · pp. 8–9
  • Squared-error centroid reconstruction has an exact within-code pairwise form, while normalized cut evaluates graph affinities after accounting for group volumes.The full quantizer retains reproduction vectors as well as assignments; a partition can be evaluated using its equality relation alone.Source: §3.1 · p. 9

3.2. A worked example: block-valued relational summaries

  • For a fixed assignment, the optimal block table averages the source relation values placed together; for binary adjacency, each reproduction is its block’s edge density.The same calculation extends to weighted pair distributions through weighted block means.Source: §3.2 · p. 10
  • In the example, the table decoder gives distortion 2/3, versus 10/3 when the same assignment retains only within-group pairs.The table preserves strong cross-group relations that the equality mask removes.Source: §3.2 · p. 11
  • Grouping and table precision contribute separately to relational distortion, while increasing the allowed number of groups yields nested unquantized approximation families.For a fixed assignment, active table entries use the nearest permitted reproduction value.Source: §3.2 · p. 11

4. Finite-Codeword Collision Geometry

This section develops same-codeword collision as a representation-side relation for finite codes. Probabilistic assignments make collision a soft affinity, while hard assignments, factorized binary codes, and prevalence weighting provide related constructions and interpretations.

4. Finite-Codeword Collision Geometry

  • A finite hard code induces same-codeword equality, while probabilistic assignments use collision probability as a soft representation-side relation.The construction links smooth optimization to hard partitions and codeword-prevalence corrections.Source: §4 · p. 11, §4.1 · p. 12

4.1. Probabilistic finite encoders

  • For two independently encoded inputs, collision is the probability that their codewords match, and it becomes exact same-part membership for deterministic assignments.At nonzero assignment uncertainty, it is a graded affinity and generally lacks equivalence-relation transitivity.Source: §4.1 · p. 12, §4.2 · p. 13

4.2. Hard partitions and their relaxation

  • Collision is a continuous relaxation of hard same-partition membership, but soft and hard relational statistics should be interpreted separately.A soft encoder may distribute mass broadly even when a hardening rule assigns many inputs to one codeword.Source: §4.2 · p. 13

4.3. Factorized binary codes

  • With b conditionally independent bits, collision can be evaluated without enumerating all 2^b complete codewords.The product factors are per-bit agreement probabilities and reduce to complete-code equality as bit probabilities approach zero or one.Source: §4.3 · p. 13

4.4. Reference-weighted codeword affinities

  • Reference-weighted affinity gives greater weight to agreement on codewords with smaller reference mass; using the aggregate codeword distribution corrects for actual prevalence.Inactive codewords are omitted on the declared population, and inputs outside its support require a separate convention.Source: §4.4 · p. 14

5. Codeword Organization and Spherical Geometry

The section uses collision to describe both codeword organization and pairwise agreement, then separates their concentration and angular components through spherical coordinates.

5.1. Population collision and effective occupancy

  • Under iid population sampling, collision equals aggregate squared codeword mass, giving inverse-Simpson effective occupancy; broad occupancy alone need not imply input dependence.The measure depends on the sampling distribution: relation-specific pairs can also reflect dependence between assignments.Source: §5.1 · pp. 15–16

5.2. External references and endogenous mass correction

  • External references specify target marginal organization, while choosing r(z) = qZ(z) corrects local agreement for encoder-induced codeword prevalence; divergence minimization broadens occupancy, but adding +λH2 encourages concentration.Source: §5.2 · p. 16

5.3. Radial normalization and cosine similarity

  • Radial normalization maps categorical probability vectors bijectively to the positive unit sphere, making normalized collision cosine similarity and isolating angular agreement from concentration.Source: §5.3 · pp. 16–17

5.4. Entropy in spherical coordinates

  • In spherical coordinates, angles describe normalized collision, while an individual vector describes categorical concentration; increasing H2 spreads either an input’s assignment or aggregate codeword use, depending on the distribution.Source: §5.4 · p. 18
  • R´enyi-2 has a particularly simple angular description, whereas Shannon entropy depends on the full coordinate profile and generally has different level sets; regularizer choice remains application dependent.Source: §5.4 · p. 18

6. Alignment, Separation, and Relational Requirements

Source-specific pair statistics can align favored pairs, separate disfavored pairs, and optionally control alphabet use. Whether an objective equals relational distortion depends on the reconstruction or evaluation convention.

6.1. From the source relation to a learning objective

  • Pairwise objectives use code-induced statistics such as collision, reference-weighted affinity, normalized collision, or dissimilarity, and may be fidelity measures or training surrogates.For −log k, the monotonic direction reverses; −log collision promotes overlap but has infinite loss at zero overlap unless a numerical surrogate is substituted.Source: §6.1 · pp. 18–19

6.2. Pair-specific separation and aggregate organization

  • Pair-specific terms identify which source relations should survive, while marginal matching organizes population use and cannot distinguish partitions with identical occupancy.The finite categorical case uses a fixed available frame and an explicitly chosen marginal reference, unlike unrestricted continuous hypersphere geometry.Source: §6.2 · p. 19

6.3. Occupancy, fidelity, and resource constraints

  • Occupancy–fidelity relations compare relational distortion under upper or lower bounds on R2, but occupancy alone is not the complete description resource.Other description and decoder components must be held fixed or controlled, or auxiliary payload could carry arbitrary source information despite a small code alphabet.Source: §6.3 · pp. 19–20
  • Upper occupancy bounds can be trivial for direct positive-relation distortion, whereas lower bounds ask how much fidelity survives with a noncollapsed alphabet.At fixed K, occupancy encouragement can accompany increasing empirical distortion as R2 rises; nonconvex optimization and discrete feasible sets may prevent scalar penalties from recovering every frontier point.Source: §6.3 · p. 20

7. Selected Realizations of Relational Fidelity

This section instantiates finite-codeword relational fidelity through source-defined requirements and reconstruction-defined objectives, spanning signed and baseline-relative pairs, graph operators, centroid reconstruction, normalized association, and normalized cut. It also identifies a shared inverse-aggregate-mass affinity whose meaning depends on source-side weighting.

7.1. Selected constructions for relational requirements

  • Collision-based signed requirements can come from intrinsic sign or deviations around a source baseline; that baseline is distinct from occupancy references and collision thresholds, and group inequality need not mean separation.Source: §7.1 · p. 20

7.2. Direct relation preservation and signed distinctions

  • Edge-cut distortion is linear, so soft collision attenuation equals expected hard cut under independent assignments; signed collision fidelity also penalizes collapse but is nonmonotone and relates to correlation clustering.Source: §7.2 · p. 21

7.3. Graph probes and operator fidelity

  • Probe second moments select which signal variations and edges matter; isotropic probes recover normalized edge distortion, while source-normalized all-mode probes average relative losses across positive source Laplacian modes.The all-mode criterion is bounded between zero and one under attenuation, and excludes zero-energy modes.Source: §7.3 · pp. 23–24
  • Source-frozen probes can weight edges by transition-profile differences or lost gradient energy of the local collision-entropy field; on unweighted simple graphs that field is log degree, and constant fields make normalized entropy fidelity undefined.Source: §7.3 · pp. 24–25
  • For connected sources, any nontrivial hard partition removing all cross-class edges has maximal worst-case distortion of one, making this criterion an uninformative ranking among such reconstructions.Source: §7.3 · p. 25

7.4. Centroid reconstruction as an exact objective correspondence

  • Centroid reconstruction error exactly equals inverse-marginally weighted pairwise squared-distance distortion; arbitrary reproductions add centroid deviation, but joint latent decoders need not satisfy this per-codeword identity.The correspondence is for scalar fidelity with optimal per-codeword reproductions; it does not generally extend to decoders that jointly interpret multiple latent variables.Source: §7.4 · pp. 26–27

7.5. Normalized association and normalized cut

  • Degree-weighted inverse-mass affinity recovers a soft normalized-association surrogate and, for hard assignments, classical fixed-K normalized cut; empty classes add zero association and a unit cut penalty.The empty-class correction extends the criterion to hard or limiting assignments with empty classes.Source: §7.5 · pp. 27–28
  • Hard normalized-association and cut objectives have clustering correspondences, and soft surrogates are used by GAP and Gatti et al. (Nazi et al., 2019; Gatti et al., 2022).These objectives are positioned as source-fidelity and codeword-affinity formulations.Source: §7.5 · p. 28
  • The soft normalized-cut ratio uses deterministic soft mass, unlike expected hard normalized cut, whose sampled class volume is random.This ratios-of-expectations versus expected-ratios distinction motivates complementary probabilistic-cut formulations (Ghriss, 2026).Source: §7.5 · p. 28

7.6. Common structure of the exact correspondences

  • The same inverse-aggregate-mass affinity supports distinct fidelities: squared distance penalizes grouping distant observations, while graph weights reward grouping strongly linked vertices.The source-side weighting determines what agreement means without making the underlying source geometries identical.Source: §7.6 · pp. 28–29

8. Experimental Realizations of the Framework

The studies use finite-codeword assignments as a common family to examine relational fidelity, representation, and decoding, including objective correspondences, graph-fidelity choices, inductive normalized-cut learning, and reconstruction-defined image representations. The image studies also examine joint-decoder reconstruction with separate marginal organization and teacher-defined pair relations applied directly to codeword collision without reconstruction or a marginal penalty.

8.1. Study 1: A controlled reconstruction–relational identity

  • Corrected pairwise and centroid objectives agree within 2 × 10^-6 with encoder-gradient cosine 1.000, while uncorrected KDraw has mean absolute error 4.79.With assignments fixed, decoder excess distortion falls from 21.05 to 8.15 × 10^-5, with maximum decomposition residual 1.91 × 10^-6; the identity applies to per-codeword centroid reproduction, not joint neural decoding.Source: §8.1 · p. 30, Figure 1 · p. 31

8.2. Study 2: Exchanging relational fidelity on a fixed representation

  • At organization weight 0.20 on unweighted MalNet, all four fidelities reach hard Keff ≈7.8, with distinct own distortions: DE 0.238, DF 0.153, DC 0.127, and DH2 0.096.The experiment holds the graph-local K = 8 representation and marginal-organization mechanism fixed while exchanging normalized direct-edge, effective-resistance, source-collision, and entropy-field criteria; trajectories are nonconvex-solver results, not certified frontiers.Source: §8.2 · pp. 31–33, Figure 2 · p. 32
  • DF and DC rank partitions similarly on unweighted MalNet and PROTEINS (mean rank correlations 0.975 and 0.981), but weighted MalNet lowers their correlation to 0.724.On PROTEINS, DF versus DC has mean ARI 0.968 at H2 = 1.0 and 0.991 at H2 = 1.25 with near-zero transfer excess; at H2 = 2.0 their ARI is 0.764 despite excesses of 0.0046 and 0.0084.Source: §8.2 · p. 33, Figure 3 · p. 34

8.3. Study 3: An inductive realization of normalized-cut fidelity

  • The shared encoder’s pure normalized-cut test score is 1.160 ± 0.043, close to the spectral reference’s 1.159, but its soft Keff is only 1.42 ± 0.14.At organization weight 0.20, soft Keff rises to 2.45 ± 0.03 while test Ncut remains 1.170 ± 0.030; at 0.50, Keff reaches 3.55 ± 0.10 as Ncut increases to 1.319 ± 0.022.Source: §8.3 · pp. 34–35, Table 5 · p. 36, Table 6 · p. 36
  • At weights 0.03 and 0.20, nearly identical test Ncut values (1.171 ± 0.062 and 1.170 ± 0.030) accompany DF values of 0.0657 ± 0.0179 and 0.1117 ± 0.0017.The weight-0.20 solution has higher DF on 581 of 584 paired test graphs, while the mean Ncut difference is −0.0006, showing that normalized cut and effective-resistance distortion define different fidelities.Source: §8.3 · p. 36

8.4. Study 4: Reconstruction-defined image representations

  • The image study uses reconstruction MSE as source fidelity and hard-token collision entropy and effective code count to describe organization, with a joint spatial decoder rather than a per-codeword centroid identity.The fixed 16-bit bottleneck is varied in marginal organization, and a 32-bit run checks capacity; latent capacities are 0.25 bpp and 0.50 bpp, respectively, before entropy coding. Training adapts a group-mean relaxation to hard-sign spatial latents.Source: §8.4 · pp. 36–38
  • At organization weight 10−6, the best 16-bit reconstruction reached 24.83 dB PSNR and 0.896 MS-SSIM with 13 388 effective codes; weight 10−5 raised occupancy to 29 569 as PSNR fell to 24.56 dB.The single-seed study leaves the few-tenths-of-a-decibel difference unresolved; occupancy can keep increasing after reconstruction quality plateaus. The finite-minibatch smooth collision objective is a surrogate, while reported H2 and effective count use pooled hard tokens.Source: §8.4 · pp. 38–39, Table 7 · p. 39
  • Without organization, reconstruction used only 38 distinct codewords, with effective count about 7.2, hard collision 1.38 × 10−1, and PSNR 19.86 dB, showing that reconstruction alone need not spread occupancy broadly.The 32-bit capacity check reached 25.88 dB PSNR and 0.920 MS-SSIM with effective count about 2.58 × 10^5, but also changed nominal capacity, code space, and uniform-reference target.Source: §8.4 · p. 39

8.5. Study 5: Teacher-defined relations without reconstruction

  • Favored-pair collision was 1.29 × 10−2 versus aggregate collision of 1.04 × 10−3, while disfavored-pair collision was 1.47 × 10−4; enrichment was 12.4× and suppression 85.8%.The hard representation had about 965 collision-effective words across 43 028 active words. Pair-specific collision organized the code without a decoder during representation learning or an explicit marginal entropy penalty.Source: Table 8 · p. 42, §8.5 · p. 42
  • A decoder trained after freezing the teacher-relational encoder achieved test PSNR 19.50 dB and MS-SSIM 0.718, probing compatibility with image reconstruction rather than causal attribution.Pixel fidelity did not shape representation learning; without random-encoder or shuffled-teacher decoder controls, the result is not a comparison with the reconstruction-trained codec.Source: §8.5 · p. 42, Table 8 · p. 42

9. Conclusion

The conclusion presents finite-codeword collision as a realization of relational compression, linking local relational requirements to aggregate occupancy and geometric or objective correspondences. Appendices qualify the scope of these statistics and illustrate alternative relational contents.

9. Conclusion

  • Same-codeword probability links local relational requirements with Rényi-2 occupancy and spherical geometry, with exact correspondences to centroid reconstruction and normalized graph association and cut.The framework makes the source relation, retained description, evaluated relation, fidelity criterion, and resource explicit.Source: §9 · pp. 42–43
  • The paper identifies complete-description budgets, richer relation-level decoders, broader source classes, and operational coding formulations as directions for extending the framework.These are proposed opportunities, not evaluated results.Source: §9 · p. 43
  • Figures 5–8 show representative graph partitions and image or teacher diagnostics; organization broadens code use, but beyond baseline reconstruction slightly worsens along a single-seed trajectory.Figure 5’s partition colors are independently permuted, while Figure 8 separates encoder development from later decoder training.Source: §9 · p. 43, Figure 5 · p. 44, Figure 7 · p. 45

Appendix A. Higher-Order Collision Geometry

  • A triplet-structured objective can compare pairwise normalized collisions without being a third-order statistic; genuine third-order collision requires all three encoded inputs to share a codeword.The source-relation arity, objective arity, and collision-event order are distinct modeling choices.Source: Appendix A · pp. 50–51

Appendix A.1. R´enyi entropy in spherical coordinates

  • The spherical coordinates also express higher-order Rényi quantities, with the order-one limit recovering the corresponding entropy expression.Higher power sums generally depend on more than one ordinary spherical angle.Source: Appendix A.1 · p. 51

Appendix B. Conditional Collision and Dependence

  • Population collision measures code occupancy, not input dependence; endpoint alignment is equality-specific, and the entropy difference has no asserted general data-processing characterization.Broad aggregate occupancy can arise even from an input-independent encoder.Source: Appendix B · p. 52

Appendix C. Additional Monotone Content Examples

  • Relational content can track shortest-path geometry or finite walk structure rather than only removed edges, and monotone contents can be combined with nonnegative coefficients.Shortest-path content needs a specified relationship between weights and lengths when considering weight attenuation.Source: Appendix C · p. 53

Appendix D. Reported Metrics and Numerical Conventions

The appendix defines recurring experimental metrics and aggregation conventions for hard codes, graph partitions, teacher relations, and image reconstruction.

Appendix D.1. Hard finite codes

  • Hard-code metrics use the empirical distribution over evaluated tokens; aggregate collision, collision entropy, and effective codeword count summarize its concentration, while active words count nonzero-mass codes.For unit-range images, the 32-bit run’s nominal raw latent capacity is reported before entropy coding.Source: Appendix D.1 · p. 54

Appendix D.2. Graph partitions

  • Graph metrics use fixed-K hard normalized cut and graph-local, degree-weighted occupancy; occupancy metrics are computed per graph before averaging across five learned seeds.Mean effective count is not generally the exponential of mean collision entropy; the spectral reference is deterministic for the stated solver configuration.Source: Appendix D.2 · pp. 54–55

Appendix D.3. Teacher relations and reconstruction

  • Teacher metrics compare favored and disfavored weighted collision rates with pooled hard-code collision, while image MSE averages squared pixel error and higher PSNR corresponds to lower MSE.Teacher validation and test losses aggregate their alignment and relational-separation numerators using the corresponding relation-weight masses.Source: Appendix D.3 · p. 55

Appendix E. Synthetic Study Details

The synthetic study checks the finite-sample centroid–pairwise identity and decoder decomposition using a fixed dataset and matched encoder initialization.

Appendix E. Synthetic Study Details

  • The study evaluates every pair in one fixed finite dataset and identically initializes two encoders optimized with centroid and inverse-mass-weighted pairwise objectives.A separate decoder calculation holds encoder assignments fixed.Source: Appendix E · p. 56, Table E.10 · p. 56
  • The checks target the centroid–pairwise identity and decoder decomposition; reported values near 10^-6 reflect single-precision arithmetic.The encoder-gradient comparison uses cosine similarity and relative error.Source: Appendix E · p. 56, Table E.9 · p. 56

Appendix F. Transductive Relational-Fidelity Study Details

The appendix records the transductive study’s graph data and optimization procedure for comparing relational-fidelity settings.

Appendix F. Transductive Relational-Fidelity Study Details

  • The two MalNet collections use the same 20 cleaned MalNet-Tiny test topologies; COLLAB and PROTEINS each contribute 20 cleaned largest connected components, with labels removed or ignored.The weighted variant uses deterministic positive lognormal weights with scale 0.75, normalized within each graph.Source: Appendix F · p. 57, Table F.11 · p. 57
  • Optimization uses ten random restarts and one collapse-biased initialization, followed by cross-criterion warm starts and a closure pass when they improve a target.Final selection is by the target soft objective across candidates.Source: Appendix F · p. 57

Appendix F.1. Matched occupancy and cross-distortion transfer

Appendix F.1 describes how hard partitions are matched by collision occupancy before agreement and cross-distortion are compared. Appendix G records MalNet-Tiny cleaning and the spectral reference procedure.

Appendix F.1. Matched occupancy and cross-distortion transfer

  • Matched solutions must lie within 0.10 nats of each target and of one another, with at least five matched graphs required to display a point.Targets span H2(q̄G) values from 0.25 to 2.0 nats; each criterion selects independently from its available sweep.Source: Appendix F.1 · pp. 57–58
  • At 2.0 nats, Table F.12 reports hard-partition agreement and directional transfer differences, while Table F.13 averages excess hard distortion across matched comparisons.Transfer differences use the destination fidelity’s scale; full-table comparisons are made within columns, whose coverage can differ.Source: Table F.12 · p. 58, Appendix F.1 · p. 58

Appendix G. Inductive Normalized-Cut Study Details

  • MalNet-Tiny graphs retain the official split before cleaning and are kept only when their largest connected component has at least 512 nodes.The spectral embedding uses SciPy’s sparse eigsh solver with seeded K-means discretization; the spectral reference is computed separately for each retained validation and test graph.Source: Appendix G · p. 58

Appendix H. Reconstruction-Trained Flowers102 Study Details

Appendix H details Flowers102 reconstruction-study training, evaluation, and collision-based supervision. It also specifies that reported reconstruction metrics use hard signs fed directly to the decoder.

Appendix H. Reconstruction-Trained Flowers102 Study Details

  • The nonstandard split trains on the official Flowers102 test images and evaluates final test performance on the official training images.The split uses 6,149 training, 1,020 validation, and 1,020 final-test images; all five settings use seed 1337, batch size 64, and 150 epochs.Source: Appendix H · pp. 58, 62
  • Collision pairs are independent uniform draws with replacement from minibatch spatial tokens, allowing self-pairs and same-image pairs.These pairs form the minibatch estimate using the threshold collision.Source: Appendix H · p. 62
  • Reported reconstruction metrics use hard signs fed directly to the decoder.The mean-group relaxation centers nonempty positive- and negative-sign groups within each channel over batch and spatial positions, then adds the centered residual to the corresponding hard sign.Source: Appendix H · p. 62

Appendix I. Teacher-Defined Flowers102 Study Details

Appendix I describes teacher-defined representation learning from favored and disfavored pair requirements, followed by secondary synthesis with the encoder frozen. It also documents numerical limits of the smooth collision loss.

Appendix I. Teacher-Defined Flowers102 Study Details

  • Teacher-defined representation learning uses the baseline-relative teacher relation through favored and disfavored pair requirements, then freezes the encoder for secondary synthesis.The teacher receives the same augmented crop as the student after teacher-specific normalization.Source: Appendix I · p. 64
  • The clipped smooth loss can differ from the ideal expression in saturated regions and can have zero gradients through clipping there.For the centered margin to retain a representable region below neutral collision, ε must be less than 2^-b; the reported 16-bit loss is evaluated in float32 and meets that condition.Source: Appendix I · p. 64
  • These numerical operations affect only the smooth training loss; reported hard-collision metrics use hard code equality.The favored-pair enrichment and disfavored-pair suppression use pooled empirical token collision as their baseline.Source: Appendix I · p. 64
  • Figure I.13 pairs Flowers102 test images with reconstructions from a secondary decoder trained on frozen teacher-guided hard codes.Pixel reconstruction did not participate in learning the encoder.Source: Figure I.13 · p. 66

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