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Transforms for LLM Quantization: The Great Inversion and Format Co-Design

Ehsan Jokar

arXiv:2608.25188v1cs.LGcs.IT

TL;DR

The paper addresses a gap in understanding which transform direction suits different quantization regimes. It formalizes the contrast between concentration and flattening, showing that their optimality depends on allocation and format, with no guarantee transferring across regimes.

  • Problem

    Existing transform-based LLM quantization lacks a dedicated account of why classical concentration and shared-scale flattening prescribe opposite directions.

    Source: §1 · p. 4, §2.5 · p. 12

  • Method

    The paper formalizes the contrast using majorization-based surrogate objectives and synthesizes transform methods across quantization regimes and formats.

    Source: §1 · p. 4, §13 · p. 86

  • Results

    The proofs establish that concentration and flattening are optimal for their respective objectives, but no optimality guarantee transfers to a generic spectrum.

    Source: §1 · p. 4, §4.7 · p. 32

  • Takeaways & Limitations

    Transform choice should follow the downstream allocation regime and number format rather than import a single classical prescription.

    Source: §1 · p. 4, §8.2 · p. 58, §13 · p. 86

    No single transform is established as jointly optimal across deployed number formats and rounding stages.

    Source: §4.7 · p. 32

Abstract

from arXiv · show

Most competitive 4-bit LLM research pipelines now open the same way: apply a linear, function-preserving transform (rotation, scaling, permutation, non-orthogonal affine) so the outlier mass sits more favorably against the group scales, and only then round. Yet we are aware of no survey dedicated to this transform stage, and its literature is quietly re-deriving an older theory. We identify and formalize the principle that organizes it, the Great Inversion: allocation-flexible coding rewards energy concentration, whereas the grouped shared-scale quantization a deployed matrix instruction performs rewards within-group flattening. Classical transform coding (1963: decorrelate, allocate bits, quantize) spends different bits per coordinate at a fixed total rate; for a Gaussian source at high rate the Karhunen-Loeve transform's concentration minimizes distortion. A deployed operand tile instead carries one absolute-maximum scale per group and equal bits everywhere, with no allocation; on a uniform grid that objective rewards flattening, approached by Hadamard incoherence. We prove that opposition under within-group majorization: the prescriptions point in opposite directions, each backed by a proof against its own objective, and for a generic spectrum no optimality guarantee transfers. A second axis is the number format: the non-uniform FP4 grid makes flattening buy less, MXFP4's power-of-two block scale still rewards a rotation confined to that block, and NVFP4's mantissa-carrying scale largely removes that pull, so the target pole depends jointly on allocation regime and format. We survey 200 works to a June 2026 cutoff; classify 43 transform methods by structure, data-awareness, searched-versus-constructed, and runtime cost; record, where reported, how they compose with GPTQ rounding; distill a first-choice guide by deployment regime; and close with the open problems it exposes.

1 Introduction

Low-bit LLM deployment is constrained by shared-scale outliers, making function-preserving transforms a central pre-rounding intervention that preserves dense low-bit kernels. This survey organizes the field around the Great Inversion: allocation-flexible coding rewards energy concentration, whereas shared-scale quantization rewards within-group flattening, with the preferred direction further depending on number format and deployment regime.

Motivation and transform stage

  • Shared group scales let outliers stretch the grid, so transform-then-round redistributes energy before quantization while preserving the layer function and dense low-bit execution.Randomized Hadamard rotations can reduce crest factor from O(d) to O(√log d).Source: §1 · p. 2

Organizing principle

  • The Great Inversion separates two objectives: variable-rate coding concentrates energy, while equal-bit shared-scale quantization on a uniform grid favors within-group flattening.The contrast is formalized through within-group majorization, and for a generic spectrum neither regime’s optimality guarantee transfers to the other.Source: §1 · pp. 3–4

Format and deployment regime

  • The preferred transform depends jointly on allocation freedom and number format: Hadamard-like flattening suits shared-scale integer grids, while floating-point formats weaken or can reverse that prescription.The survey notes that NVFP4 activation releases use group size 16 without a named rotation, while other shipped paths differ by block-scale behavior.Source: §1 · pp. 2, 4

Survey scope and taxonomy

  • The survey covers 200 works and classifies 43 transform methods by structure, data awareness, search strategy, runtime cost, and reported composition with rounding.Its scope includes diagonal scalings, permutations, fixed and learned rotations, non-orthogonal affine maps, and sequence-axis or frame transforms, while treating GPTQ-related rounding only where it interacts with transforms.Source: §1 · pp. 4–5

Empirical and systems evidence

  • The survey combines optimality theory with published head-to-head evidence on transform necessity, transform–rounder overlap, integer-to-FP4 changes, KV-cache effects, and deployment speed and memory.It distinguishes published measurements from the survey’s own mechanism illustrations and checks the theory against shipped 4-bit releases, where weight and activation paths remain split rather than universal.Source: §1 · p. 5

Open problems

  • The inversion motivates four open problems: joint transform-and-rounding optimality, guarantees for the deployed extreme-value objective, allocation-flexible inference coding, and transform–format co-design.These problems follow from the gap between surrogate optimality results and the formats, kernels, and deployment choices used in practice.Source: §1 · pp. 4–5

2 Foundations and Background … 2.4 Quantization noise: Bennett’s model and SQNR

This foundation section frames quantization transforms as a tradeoff between classical energy concentration and shared-scale flattening, and develops the layer, error proxy, scalar quantizer, and Bennett-model machinery needed to analyze it. It shows how crest factor affects high-resolution distortion and SQNR, motivating transforms that reduce crest factor while noting the model’s caveats.

2 Foundations and Background

  • Classical transform coding concentrates energy, whereas shared-scale low-bit kernels favor flattening within groups—the survey’s central “Great Inversion.”This opposition organizes the subsequent geometric and quantization analysis.Source: §2 · p. 5, §2.2 · p. 8

2.1 The decoder-only layer, and where the bits go

  • Decoder-only layers contain seven quantized projection matrices, while attention’s QK⊤ and PV products have no static operand to absorb a transform.The seven projections are natural transform sites because their weight-side transforms can fold offline; the two attention matmuls require separate instruments.Source: §2.1 · pp. 6–7
  • Quantization targets weights, activations, and KV-cache independently, with W4A16, W4A4, and W4A4KV4 corresponding to progressively broader low-bit coverage.Weight-only quantization suits memory-bound decoding, activation quantization supports tensor-core throughput, and cache quantization addresses growing KV memory.Source: §2.1 · pp. 6–7

2.2 Notation, the linear layer, and the layer-output proxy

  • The layer-output proxy weights quantization error by activation and weight second moments, reducing to the exact weight-side objective used by OBC/GPTQ for weight-only quantization.For bilateral quantization, the clean additive proxy relies on independent unbiased errors and drops the higher-order second-order term, which is only approximately justified at aggressive W4A4 widths.Source: §2.2 · p. 8
  • Function-preserving transforms redistribute error without changing Y = XW⊤, but their activation-side computation is costly unless the transform can be absorbed into static weights.This asymmetric cost structure makes cheap or free absorption central to transform placement; RMSNorm’s learned diagonal gain is one zero-cost example.Source: §2.2 · p. 8

2.3 The scalar quantizer zoo

  • Grouped AbsMax quantization makes the crest factor—the peak-to-typical ratio—control shared-scale cost, so one outlier coarsens the grid for every ordinary value in its group.The crest factor ranges from 1 for a flat group to √G for energy concentrated in one of G coordinates; the sixteen-value example has CF = 3.7, while real outlier channels can be an order of magnitude or more above typical channels.Source: §2.3 · p. 10

2.4 Quantization noise: Bennett’s model and SQNR

  • Bennett’s high-resolution model supplies the missing link from crest-factor geometry to quantization error, converting the grid’s ∆2/12 noise estimate into decibel and bit penalties.The model treats values within a sufficiently fine cell as nearly uniform because the input density varies little across the cell.Source: §2.4 · p. 10
  • Bennett’s integral gives D = 1/12 ∫Δ(x)^2p(x) dx, reducing for uniform quantization to additive noise with D = Δ²/12 and approximately uniform, signal-decorrelated error.The approximation assumes high resolution and negligible overload; it is exact only as Δ→0 and merely good at four bits.Source: §2.4 · p. 11
  • The resulting SQNR law is SQNR_dB = 6.02b + 4.77 − 20 log10 CF, so doubling crest factor costs 6 dB, equivalent to one high-resolution bit.At deployed widths, the exact factor (2^b−1 − 1)^2 makes the fourth bit worth 7.4 dB and the fifth 6.6 dB; at b = 4, 6 dB buys about 0.8 bit.Source: §2.4 · p. 11
  • An 18 dB tax from CF = 8 leaves flat 8-bit groups ample SQNR but consumes most of a flat 4-bit group’s roughly 28 dB budget, creating the 4-bit wall.The passage reports about 53 dB for flat 8-bit groups and just under 28 dB for flat 4-bit groups.Source: §2.4 · p. 11
  • In the 16-value example, a per-group Hadamard reduces CF from 3.7 to 1.9 and raises measured SQNR from 17.2 to 21.6 dB.The figure and text note that the deterministic Hadamard flattens this group, while the worst-case guarantee requires random signs.Source: §2.4 · p. 11, Figure 2 · p. 12
  • The crest-factor law applies specifically to uniform grids pinned to the group peak; quantizers with flexible level placement or codeword lengths follow a different law.The section therefore treats crest-factor reduction as a transform’s value under the shared-scale uniform-grid regime, not as a universal quantization principle.Source: §2.4 · pp. 11–12

2.5 Rate regimes, rate–distortion, and majorization · 2.6 Incoherence and the flattening law

This section formalizes how allocation regimes shape the opposing objectives of energy concentration and within-group flattening under majorization, and examines incoherence as a route to flattening in shared-scale quantization. It covers Hadamard and randomized rotations, their guarantees and limitations, and their interaction with error-feedback rounding.

2.5 Rate regimes, rate–distortion, and majorization

  • Classical rate–distortion theory permits unequal coordinate allocation, so reverse water-filling spends more bits on high-variance coordinates and the KLT maximizes concentration through Schur–Horn majorization.Every deployed LLM GEMM kernel instead uses shared-scale, fixed-allocation quantization; entropy coding is optional to the allocation-flexible regime.Source: §2.5 · pp. 12–13
  • Shared-scale quantization rewards groups whose crest factor approaches 1, but achieving that flatness depends on the transform family and on how energy is placed across groups.The per-group maximum is bounded below by the group mean, with equality only for internally uniform groups; across-group placement is a separate lever.Source: §2.5 · p. 14
  • Proposition 2.11 proves that variable-rate distortion is minimized by concentration, while shared-scale uniform distortion is minimized by within-group flattening.The coding objective depends on the geometric mean of transformed variances, whereas the shared-scale objective sums per-group maxima; neither global optimality transfers because grouping matters.Source: §2.5 · p. 14

2.6 Incoherence and the flattening law

  • Incoherence addresses spread but not alignment: under the stated noise assumptions, alignment contributes a separate SNR factor unchanged by orthogonal transforms and requiring a non-orthogonal, data-aware map.The alignment-optimal transform balances weight and activation second moments, motivating whitening methods and leaving realized deployed error unresolved.Source: §2.6 · p. 15
  • A randomized Haar rotation or randomized Hadamard makes every transformed coordinate sub-Gaussian, reducing a worst-case coherent vector from CF = d to CF = O(√log d) in expectation.The expected reduction is by at least a factor of √(d/(2 ln 2d)); deterministic Hadamards and signed permutations lack this arbitrary-vector guarantee.Source: §2.6 · pp. 15–16
  • For shared-scale groups, the same rotation bounds expected group maxima, but per-group crest-factor guarantees additionally require realized group RMS balance, which can fail through Hadamard cancellation.A Sylvester Hadamard example with two active coordinates yields group RMS 0 or √2τ with equal probability, making the crest factor undefined half the time.Source: §2.6 · p. 16
  • Hadamard is the default fixed-rate transform because its incoherence benefit justifies its multiplier-free O(d log d) cost.The quantitative case weighs the bits associated with incoherence against the cost of applying the transform.Source: §2.6 · p. 16
  • The relevant invariant is crest factor, an ℓ∞/ℓ2 extreme-value measure, rather than kurtosis, because AbsMax scales are controlled by extremes.A sub-Gaussian marginal bound suffices for an AbsMax scale but is weaker than what a codebook requires.Source: §2.6 · pp. 16–17
  • A single randomized Hadamard can leave adversarial coordinates non-Gaussian, while two improve marginal Gaussianity and three may be needed for covariance decay in vector quantization.The lattice-codebook deployment described here uses one randomized Hadamard per matrix axis, so it does not rely on the two-fold or three-fold guarantees.Source: §2.6 · pp. 16–17
  • 3.7 to 1.9: a per-group Hadamard nearly halves the crest factor and yields +4.4 dB on the running group (Figure 2).The reduction approaches the ≈6 dB value priced by Corollary 2.7, making the flat pole essentially free to reach in this example.Source: §2.6 · p. 17
  • Error-feedback rounding may already capture much of the rotation’s intended benefit, leaving open whether transforms and rounders compose or substitute for each other.The flattening analysis assumes an innocent rounder, but coupled quantization error can also be reduced by the rounding procedure.Source: §2.6 · p. 17

2.7 Why rounding is not independent: an error-feedback primer · 2.8 Where a transform can live: absorption, invariance, and cost

Rounding is coupled through the layer objective, making error feedback a second lever that can overlap with transform gains but cannot remove the shared-scale dynamic-range wall. Transform deployment cost depends on where the map lives: orthogonal residual-stream rotations can be absorbed at zero runtime under RMSNorm invariance, while structured and dense maps range from cheap online computation to a full matmul.

2.7 Why rounding is not independent: an error-feedback primer

  • Error-feedback rounding exploits off-diagonal curvature in HX to compensate neighboring coordinates, rather than rounding each weight independently.Its optimal single-coordinate update follows a constrained least-squares solution under the layer proxy, assuming HX is invertible with practical damping.Source: §2.7 · p. 17
  • Greedy application yields Optimal Brain Compression, while GPTQ scales it with fixed column order and lazy blocked feedback, inheriting Babai’s no-clipping error bound in reverse order.This establishes error feedback as a substantive rounding mechanism rather than an afterthought to transformation.Source: §2.7 · p. 17
  • Error feedback can capture part of a transform’s benefit, but fixed-grid compensation cannot remove the shared AbsMax dynamic-range wall or its crest-factor noise floor.With frozen scales, excursions beyond the range clip; GPTQ instead refits each group’s scale on compensated weights using the same AbsMax rule.Source: §2.7 · pp. 17–18

2.8 Where a transform can live: absorption, invariance, and cost

  • Transform inference cost depends on placement: weight-side transforms are absorbed offline, while input-axis transforms range from zero-cost folding through O(d log d) structured maps to O(d^2) dense matmuls per token.A mid-block transform is absorbable only when both sides reach static weights; otherwise its online cost depends on the transform structure.Source: §2.8 · p. 18
  • Under gain-free RMSNorm, a global orthogonal residual-stream rotation preserves the function and folds into surrounding weights at zero inference cost.The result relies on rms(xQ) = rms(x) and coordinated replacements for weights and stream biases; mean-subtracting norms would break this invariance.Source: §2.8 · p. 18
  • Fixed-rate formats share per-group scales, but scale representation changes how flattening is priced: INT4 permits arbitrary scales, MXFP4 uses power-of-two E8M0 scales, and NVFP4 uses finer E4M3 scales.The element grid and scale type jointly determine whether flattening helps or hurts, so the uniform-grid crest-factor result does not transfer unchanged across formats.Source: §2.8 · p. 19
  • For uniform-grid AbsMax quantization, crest factor costs 6 dB per doubling, while concentration and flattening are opposing within-group extremes; random rotations approach flattening, and error feedback contests part of that gain.Residual-stream orthogonal transforms obtain this potential benefit for free, whereas mid-block orthogonal maps remain online unless both adjacent sides are static weights.Source: §2.8 · p. 19

3 The Classical Inheritance: Transform Coding, 1963–2021

Classical transform coding rigorously favors energy concentration through the KLT and water-filling when rates and entropy coding remain allocation-flexible. This tradition extends into neural-weight compression but stops where fixed-rate shared-scale inference begins.

3.1 The high-rate law and the variable-rate distortion

  • High-rate variable-rate coding separates fixed-length density shaping from entropy-constrained uniform quantization, introducing the survey’s concentrate-versus-flatten split.Entropy coding removes the peak-dependent constraint of shared-scale grids while retaining the 2^-2R distortion slope.Source: §3.1 · p. 20

3.2 Entropy coding and the space-filling gap

  • Entropy-coded uniform scalar quantization approaches the Shannon bound within 0.255 bits/sample, with the remaining loss caused by interval geometry rather than source statistics.The gap is the normalized-second-moment difference between one-dimensional intervals and high-dimensional spherical cells.Source: §3.2 · p. 20

3.3 The KLT, coding gain, and water-filling

  • For Gaussian vectors with positive-definite covariance, the KLT maximizes coding gain, while reverse water-filling allocates rates toward higher-variance coordinates.Together, these prescriptions concentrate energy and then spend bits where that energy resides; both require per-coordinate rate allocation.Source: §3.3 · pp. 21–22

3.4 Beyond the linear-Gaussian ideal

  • The classical guarantees are bounded by source dependence, high-rate assumptions, and Gaussian second-order structure, while prior shared-scale rotations reshape peaks without answering the transform-design question.Hung & Meng (1998) use Walsh–Hadamard rotation to improve overload characteristics, and Suresh et al. (2017) reduce shared-scale error from Θ(d) to O(log d).Source: §3.4 · p. 22

3.5 Neural network weights as a rate–distortion source

  • Neural-weight compression transferred classical coding through entropy-coded codebooks and curvature-weighted distortion, making output or loss sensitivity more relevant than raw weight MSE.Deep Compression (Han et al., 2016) exemplifies entropy-coded weight sharing, while Choi et al. (2017) and SqueezeLLM (Kim et al., 2024) use Hessian or Fisher-based weighting.Source: §3.5 · p. 23

3.6 Industrial codification and the LLM-scale descendant

  • DeepCABAC (Wiedemann et al., 2019) and MPEG Neural Network Compression provide production entropy-coded, rate–distortion-optimized weight coders without introducing their own transform.The standard’s open implementation is NNCodec, while Radio (Young, 2025) operationalizes reverse water-filling across LLM weight groups.Source: §3.6 · p. 23

3 The Classical Inheritance: Transform Coding, 1963–2021

  • Classical coding therefore prescribes KLT concentration, water-filling allocation, and entropy-coded scalar quantization within 0.255 bits of the Shannon bound, all assuming allocation-flexible rates.The fixed-rate shared-scale inference datapath removes the per-coordinate allocation that makes the KLT optimal, motivating the inversion developed next.Source: §3 · p. 19, §3.6 · pp. 23–24

4 The Great Inversion and the Optimality Theory

This section formalizes the Great Inversion and examines how transform optimality depends on the objective, contrasting energy concentration in variable-rate coding with flattening in deployed shared-scale quantization. It establishes that guarantees for KLT- and WUSH-like regimes do not transfer for generic spectra, while transform attainability, realized error, non-uniform formats, and model-level justification remain separate qualifications.

4.1 Why the deployed kernel is fixed-rate

  • Deployed low-bit GEMM kernels require equal-width operands and at most one shared scale per block, excluding per-coordinate bit allocation and reversing the classical variable-rate prescription.This fixed-rate constraint reflects today’s tensor-core datapaths rather than a theorem about computation, and applies to INT4-per-group, MXFP4, and NVFP4 operand tiles.Source: §4.1 · p. 24

4.2 The uniform-grid fixed-rate optimum is flatness

  • For a uniform element grid, fixed-rate distortion is minimized exactly when every group is internally uniform, with crest factor 1 as the absolute floor.The shared-scale cost lies between E/G and E: the lower bound occurs for internally uniform groups, while concentration toward a single coordinate reaches the upper bound.Source: §4.2 · p. 25

4.3 The Great Inversion

  • Theorem 4.3 proves that concentration strictly lowers the variable-rate coding surrogate, while within-group flattening minimizes the uniform-grid shared-scale AbsMax surrogate.The result establishes opposing directions under within-group majorization, not globally optimal KLT and Hadamard transforms; reachable optima depend on grouping and admissible transforms.Source: §4.3 · pp. 26–27
  • The inversion covers the uniform-grid AbsMax surrogate; non-uniform formats, lattice codes, and KV-cache transforms are extensions requiring separate models or measurements rather than corollaries.MXFP4’s block scale retains an AbsMax-like penalty, whereas NVFP4’s mantissa-carrying scale weakens it; the section does not establish a common deployed-format statistic.Source: §4.2 · p. 25, §4.3 · p. 27

4.4 A cautionary consequence: when the L2 proxy lies

  • The smooth L2 proxy ranks transforms like AbsMax only when crest factors lie in a sufficiently tight band; concentrating transforms can change crest factors and invert the ranking.A sufficient condition is c̄^2/c^2 < S_A/S_B, but orthogonal rotations preserve aggregate Frobenius energy, so the condition is unavailable when S_A = S_B.Source: §4.4 · p. 29
  • For deployed AbsMax quantization, crest-factor spread—not kurtosis or summed energy—can make an L2-optimal transform surrogate-pessimal.Concentration minimizes the coding proxy while spreading within-group crest factors charged by the deployed surrogate.Source: §4.4 · p. 29

4.5 From layer error to model error

  • Perplexity increases decompose into weighted per-layer relative squared weight errors only to leading order under block-diagonal and isotropic-Hessian approximations.The theorem does not directly justify the data-aware output proxy tr(∆W H_X ∆W^T), except when H_X is isotropic; aggressive quantization can break the ranking.Source: §4.5 · p. 30

4.6 The optimality landscape, read through the inversion

  • WUSH is optimal for the stochastic FP-AbsMax model and near-optimal for INT-AbsMax, combining data-dependent whitening with a post-whitening Hadamard across quantization groups.The KLT instead optimizes allocation-flexible coding; format and noise model determine which side applies, while the whitening-plus-Hadamard combination—not raw Hadamard flattening—reaches WUSH’s modeled floor.Source: §4.6 · p. 31

4.7 Synthesis: no optimum transfers

  • No optimality guarantee transfers between allocation-flexible and fixed-rate regimes for a generic spectrum; simultaneous optimality occurs only in degenerate within-group spectral plateaus.The corollary concerns surrogate guarantees, not every deployed instance, and no surveyed result establishes one transform as jointly optimal across formats and rounding stages.Source: §4.7 · p. 32

5 What Is Proven, and What Is Not

The optimality ledger proves concentration and flattening under different objectives and assumptions, so guarantees do not transfer across the Great Inversion. Regime-neutral rounding and bridging results clarify the boundary, while deployed-format optimality remains unproved.

What is proven

  • The ledger proves concentration for allocation-flexible coding and flattening for shared-scale grids, but neither guarantee transfers across the boundary.Each result is valid only for its stated objective and assumptions; the KLT and WUSH optimize different functionals.Source: Table 1 · p. 33, §5 · pp. 33, 35

Format-dependent boundary

  • Format changes whether flattening helps: ideal FP-grid rotation can be harmful, whereas block-confined rotation helps deployed MXFP4’s power-of-two scale, with no proof for either contrast.The ideal-grid effect depends on joint concentration ∆FP and is measured only on the tested matrices; the MXFP4 comparison lacks a cross-family surrogate evaluation.Source: §5 · pp. 34–35

What is proven

  • WUSH is model-optimal for FP-AbsMax and near-optimal for INT-AbsMax only under a stochastic-noise model, combining Hadamard flattening with essential non-orthogonal whitening.Dropping either component returns modeled error to the orthogonal-transform value, while the alignment-optimal map depends on weight and activation second moments.Source: §5 · pp. 34–35

Regime-neutral machinery

  • Regime-neutral results establish GPTQ’s LDLQ interpretation and Babai nearest-plane equivalence under explicit integer, rank, domain, and grid assumptions.LDLQ is optimal among H-dependent, W-independent linear-feedback rounders for integer rounding, while inherited Babai bounds require an unbounded grid.Source: §5 · p. 34

Regime bridge

  • Water-filling provably beats GPTQ’s equal-rate allocation by 1/2 log of the pivots’ AM/GM ratio, while per-layer relative weight error locally links to perplexity.A random rotation shrinks the allocation gap to ≈0.1 bit on Llama-3-8B covariances.Source: §5 · p. 35

What is not proven

  • No theorem establishes one transform as optimal for deployed LLM inference across all formats and rounding stages, leaving the measured MXFP4 contrast unexplained.The missing tests include evaluating the AbsMax surrogate on the published deployed maps and reducing existing mechanistic accounts to a comparable statistic.Source: §5 · p. 35

6 A Taxonomy of Transforms

This taxonomy organizes transform methods by algebraic structure, data dependence, search versus construction, deployment cost, granularity, and rounding composability. It spans permutations, diagonal scalings, rotations, non-orthogonal maps, sequence-axis and overcomplete transforms, highlighting their complementary trade-offs and the growing use of deployable hybrid stages.

6 A Taxonomy of Transforms

  • The taxonomy spans diagonal, permutation, orthogonal, affine, sequence-axis, and frame transforms, whose degrees of freedom generally increase toward affine maps but do not form a strict total order.Fixed and learned orthogonal transforms share a feasible set; permutations are discrete and therefore incomparable to diagonal maps.Source: §6 · p. 35, Figure 4 · p. 36
  • The survey classifies 43 methods by structure, data awareness, search versus construction, deployment cost, granularity, and composability with error-feedback rounding.Table 2 tabulates the first four axes, while granularity and rounding composability are recorded in method descriptions and Section 7.Source: §6 · p. 37, Table 2 · p. 37
  • Under the floating-point model, the optimum is Hadamard applied to the whitened second moment, not raw-operand flattening; diagonal scaling, rotation, and whitening reshape distributions differently.Figure 5 illustrates migration between operands, crest-factor collapse, and covariance isotropization on a real layer.Source: §6 · p. 37, Figure 5 · p. 41

6.1 Diagonal scalings: migrating the difficulty

  • Diagonal scaling preserves the layer function and folds offline by dividing activations by s_j while multiplying the matching weight column, thereby migrating quantization difficulty between operands.SmoothQuant balances activation and weight ranges, while AWQ protects activation-salient channels through a searched exponent; AWQ’s oracle improves OPT-6.7B INT3 perplexity from 23.5→11.4 versus 23.5→22.4 for weight-magnitude selection.Source: §6.1 · pp. 40–42
  • Diagonal maps cannot destroy dynamic range or spread a single-channel outlier across shared-scale coordinates, so they only equalize group burden and are inadequate alone for aggressive W4A4.They are competitive through W8A8/W6A6 and weight-only W4A16, while rotations and non-orthogonal maps address W4A4 more directly.Source: §6.1 · p. 42
  • The FFN down-projection exposes this ceiling: its activation outliers can be roughly a thousand times the median, so smoothing merely creates matching weight outliers and may backfire (Lin et al., 2024a).A rotation-permutation-rotation repair exemplifies why diagonal scaling is often used as a zero-cost first stage rather than a complete solution.Source: §6.1 · p. 42

6.2 Permutations: relabeling the coordinates

  • Permutations preserve the global magnitudes but control which coordinates share group scales; RPTQ reorders activation clusters, while Atom isolates 128 high-energy channels with residual runtime below 0.5%.GPTQ’s activation ordering is effectively free with one scale per row, whereas group-wise scales make ordering affect inference; PermLLM learns permutations for N:M pruning rather than quantization.Source: §6.2 · p. 43

6.3 Fixed orthogonal transforms: incoherence for free

  • Fixed orthogonal transforms use random rotations or Hadamards to spread outliers and achieve near-optimal flattening without calibration, with computational invariance folding much of QuaRot’s global rotation offline.QuaRot reaches W4A4KV4 on Llama-2-70B at under half a perplexity point, while online Hadamards, frames, Kashin, and runtime max-smoothing remain deployment costs; the remaining gap is data-optimality.Source: §6.3 · pp. 44–45

6.4 Learned orthogonal transforms: adapting the rotation

  • Learned orthogonal methods recover only the residual between random-flat and data-optimal rotations, but HARP improves Llama-2-7B 2-bit WikiText perplexity from 8.95 to 7.85 at context 2048.HARP’s proxy aligns curvature with quantizer blocks rather than maximizing incoherence, while ResQ instead exploits mixed precision by keeping roughly one-eighth of high-variance coordinates in INT8 and the rest in INT4.Source: §6.4 · pp. 45–46

6.5 Non-orthogonal affine transforms: spending the last degrees of freedom

  • Non-orthogonal transforms add whitening to rotations: WUSH attains the shared-scale optimum under the floating-point AbsMax model, while its online activation transform stays within about 1.3% of an optimized Hadamard pipeline.WUSH’s closed-form, function-preserving construction is near-optimal for integer grids under stated distribution conditions; BASE-Q reports per-channel means explain up to 85% of rounding error and clipping discards about 18% of activation energy.Source: §6.5 · pp. 46–48

6.6 Sequence-axis and overcomplete transforms

  • Sequence-axis transforms complement feature-axis methods: PrefixQuant removes token-wise outliers, reducing maximum-to-median ratios from several hundred to a few and enabling static per-tensor scales atop QuaRot.The method prepends a fixed high-precision prefix derived from frequently outlier-bearing token types, while the taxonomy also places redundant frame methods alongside feature-axis transforms.Source: §6.6 · p. 48
  • Overcomplete frames use redundancy either to average quantization noise or to spread coefficients, but incur storage overhead and remain online rather than folding into weight matrices.FrameQuant and Kashin are non-square, direct low-bit heirs of redundant-representation and tight-frame constructions.Source: §6.6 · p. 48
  • Sequence-axis and frame methods complement rather than displace channel-axis transforms, with frontier pipelines stacking channel rotations, token prefixes, and residual-balancing scales.These stages target different axes or failure modes and can be combined in a single deployment pipeline.Source: §6.6 · p. 48
  • The taxonomy trends from early diagonal scalings toward non-orthogonal affine maps, constrained throughout by whether transforms fold into weights or run in cheap online kernels.The most competitive recent methods are hybrid, combining two or three levers rather than relying on one transform family.Source: §6.6 · pp. 48–49
  • Hybrid levers target distinct failures: scales rebalance operands, permutations balance groups, rotations flatten groups, whitening balances proxy sides, and prefixes handle unseen token outliers.Their interaction with rounding algorithms and number formats is deferred to the following section.Source: §6.6 · p. 49

7 Composition: Rounding and Codebooks

Transforms interact with rounding and codebooks through composition, substitution, co-optimization, or enabling rather than operating independently. At high rate rotation can substitute for allocation, while Gaussianization enables fixed vector and lattice codebooks; data-aware codebooks instead fit raw weights.

7.1 The error-feedback rounding line

  • GPTQ provides scalable error-feedback rounding without changing basis, while Qronos additionally corrects activation mismatch and runs on transformed weights.GPTQ uses shared column ordering, Cholesky stabilization, and blockwise lazy updates; Qronos alternates activation-error correction with GPTQ-style diffusion and recovers GPTQ when eX = X.Source: §7.1 · pp. 49, 51
  • QuIP’s LDLQ is optimal only for a restricted quadratic proxy and integer grid, with bounded finite grids requiring a repaired algorithm.Its guarantee does not directly establish end-to-end model-loss optimality and fails under clamping to bounded levels.Source: §7.1 · p. 51

7.2 Substitute or compose? The transform–rounding relationship

  • At high rate, rotation and per-coordinate allocation are near-substitutes: rotated GPTQ comes within about 0.1 bit of WaterSIC, itself within approximately 0.25 bit of the rate–distortion bound.WaterSIC achieves the basis-free geometric-mean optimum provably, whereas rotation empirically closes most of GPTQ’s arithmetic-to-geometric-mean gap; their equivalence is not proven in the worst case and weakens at 2–4 bits.Source: §7.2 · pp. 51–52
  • At 2 bits, QuIP’s pre-processing—not its LDLQ rounder—accounts for the main gain, reducing Llama-2-70B perplexity from 123.9 to 6.33.The package combines Hessian-derived rescaling, random orthogonal conjugation, permutation, and an altered clipping range; all substeps were reported necessary, with permutation alone worth 74 perplexity on OPT-125m.Source: §7.2 · p. 52

7.3 Codebooks enabled by incoherence

  • Gaussianization enables fixed codebooks: QTIP’s 2-bit Gaussian-source distortion falls from 0.118 for Lloyd–Max scalar quantization to 0.069 at dimension 256, approaching 0.063.QuIP# reaches 0.089 with its eight-dimensional E8P codebook, while TurboQuant shows rotated scalar quantizers can approach the optimum within a factor of about 2.7 without lattice codebooks.Source: §7.3 · pp. 53–54
  • PCDVQ separately codes Gaussianized directions and magnitudes, reaching 5.81 versus QuIP#’s 6.19 WikiText-2 perplexity at 2 bits while QuIP# retains better C4 perplexity.On Llama-2-7B, PCDVQ’s zero-shot average is 58.60 versus QuIP#’s 58.23, while C4 is 8.16 versus 8.37; evaluation contexts differ across the cited Table 12 comparison.Source: §7.3 · p. 54
  • Data-aware codebooks such as GPTVQ, VPTQ, AQLM, and SqueezeLLM substitute Hessian-weighted fitting to raw weights for a Gaussianizing transform.They trade calibration cost for codebooks adapted to non-Gaussian weights, whereas fixed rotated codebooks rely on approximately Gaussian blocks; fast lookup kernels remain necessary for deployment.Source: §7.3 · pp. 54–55

8 Number Format and Transform–Format Co-Design

Number format changes the transform objective: integer grids favor within-group flattening, ideal floating-point grids can reverse that preference, while quantized shared scales restore format- and block-dependent benefits. Co-design methods therefore jointly adapt element grids, scales, transform support, and even training architectures.

8.1 Low-precision floating-point and microscaling formats

  • E8M0 power-of-two scaling makes MXFP4 vulnerable to binade waste and clipping, whereas NVFP4’s mantissa-carrying scale shrinks that error to a few percent.MXFP4 maps block maxima into [4, 8) against an E2M1 ceiling of 6; NVFP4 uses E4M3 scales over smaller blocks.Source: §8.1 · p. 55
  • 0.85%, 0.88%, and 1.19% relative loss gaps beat MXFP4’s 1.44–1.79% in a HiFloat4 pretraining study at 4.5 versus 4.25 bits per value.HiFloat4 moves dynamic range into a richer scale hierarchy and uses a uniformly spaced E1M2 element grid.Source: §8.1 · p. 57

8.2 The integer–floating-point flip

  • For ideal floating-point grids, joint operand energy concentration can make rotation harmful: a Hadamard lowers INT8 error by about 1.6 effective bits but raises FP8 error by about 0.2.The reversal arises from destroying anti-alignment between weight and activation energy, not merely from either operand’s marginal concentration.Source: §8.2 · pp. 57–58
  • MXFP4’s coarse block scale restores rotation’s value locally: block-confined Hadamards help, whereas global QuaRot falls 6.4 points below per-block RTN and 7.7 below GPTQ.NVFP4’s mantissa-carrying scale makes inference-time flattening close to neutral; WUSH nevertheless leads by 0.70 MXFP4 points and 0.26 NVFP4 points in the reported suite.Source: §8.2 · pp. 58–59

8.3 Co-designing the transform with the format

  • Format-aware methods size rotations to scale blocks, adapt scales or grids, and selectively place transforms; NVIDIA’s recipe reaches MMLU 76.6 versus 77.4 for FP8 at 12B and 10T tokens.Quartet uses 32-element block rotations, MR-GPTQ tunes k ∈ {16, 32, 64, 128}, and NVIDIA rotates only 16-element weight-gradient inputs.Source: §8.3 · pp. 59–60
  • MR-GPTQ brings MXFP4 within 1–2% of NVFP4; at 70B, MXFP4 reaches 98.4% of FP16 versus NVFP4’s 98.3% under matched MR-GPTQ.Its modified E8M0 scale recovers 6.5 RTN points on Llama-3-8B and 1.3 points atop MR-GPTQ’s rotation.Source: §8.3 · p. 60
  • Normalized architectures can eliminate the inference Hadamard: NVFP4 training remains stable without random rotation or dynamic per-tensor scaling, while dot-product SNR rises by roughly seven decibels.Normalization improves signal accumulation rather than individual-value quantization, whose element-level and per-term SNR remains essentially unchanged.Source: §8.3 · pp. 60–61

9 Beyond the Weight Matrix

Beyond weight matrices, transform choice must follow each tensor’s outlier structure, axis, format, and deployment constraints rather than applying flattening universally. The survey extends rotation, scaling, permutation, additive shifts, reparameterization, low-rank correction, and selection across caches, attention, state-space, MoE, diffusion, and training tensors.

9 Beyond the Weight Matrix

  • The toolkit extends unevenly beyond weights: each tensor’s outlier structure determines which lever applies, on which axis, or whether any transform helps.Table 6 classifies rotation, scaling, additive, reparameterization, routing or sampling, additive-lowrank, and no-transform approaches across these tensors.Source: §9 · p. 61, Table 6 · p. 62

9.1 The key–value cache

  • KV-cache methods either choose axes and granularity, rotate before RoPE, or reparameterize cache vectors; OSCAR instead aligns rotations to downstream attention geometry, reaching 69% versus 10% for a same-budget data-agnostic Hadamard.KVQuant and KIVI use channel-wise keys and token-wise values, RotateKV uses adapted Hadamard transforms, and polar methods make distributions known or exploit RoPE structure.Source: §9.1 · pp. 61, 63

9.2 The attention matmuls and state-space models

  • Attention and state-space tensors require structure-specific fixes: SageAttention subtracts softmax-invariant bias, later variants rescale or route precision, while Quamba and MambaQuant rotate or scale according to tensor-specific outliers.MambaQuant’s KLT-preconditioned Hadamard equalizes correlated channel variances and yields under one point of W8A8 accuracy loss on Vim and Mamba.Source: §9.2 · pp. 63–64

9.3 Mixture-of-experts

  • MoE leaves the transform itself unchanged but makes per-expert calibration essential, because rarely routed experts lack sufficient statistics; MoEQuant addresses this with data-free, expert-balanced self-sampling.Expert-wise bit allocation adds an orthogonal allocation axis that can compose with the surveyed transforms.Source: §9.3 · pp. 64–65

9.4 Diffusion transformers and the time axis

  • Diffusion quantization centers on time variation: methods use timestep-aware or static smoothing, adaptive rotations, and low-rank correction, with SVDQuant reaching near-BF16 FID on FLUX.1-dev at 3× the speed of W4A16.Runtime adaptive normalization prevents free residual-stream rotation folding, while SVDQuant stores dominant rank-32 components in a 16-bit branch and quantizes the residual to W4A4.Source: §9.4 · p. 65

9.5 Training-time gradients, and where flattening fails

  • Training exposes the limit of feature-side flattening: forward activations benefit from Hadamard spreading, but row-sparse gradients require leverage-score selection or row-axis rotation rather than channel-axis flattening.Full INT4 training samples important gradient rows, while Ashkboos et al. (2025) use left-hand Hadamard rotation for INT8 fine-tuning and omit error rotation at FP6.Source: §9.5 · pp. 65–66

10 Systems and the Variable-Length Lane

Section 10 shows that transform-based quantization is now shaped by deployment hardware: fused kernels make online transforms viable, while scale granularity and fixed-rate tensor-core inputs determine which accuracy and coding strategies survive. Variable-length coding remains a storage or feeding mechanism rather than an operation inside the matrix instruction, and shipped 4-bit systems mix transform-free trained weight paths with transforms where folding is impossible.

10.1 Kernels that make transforms cheap

  • Fused kernels make online transforms deployable, with Sun et al. (2025) measuring 0.07× slowdown for fused affine maps versus 0.26× for QuaRot’s separate Hadamards.HadaCore also accelerates online rotations by 1.1–1.4× on A100 and 1.0–1.3× on H100 at comparable quantization error, while TensorRT Model Optimizer and vLLM ship transform stages as composable tooling.Source: §10.1 · pp. 66–68

10.2 The dequantization bottleneck and the GPU datapath

  • Per-group W4A4 scales impose a mid-reduction dequantization tax, reaching 20–90% of runtime, whereas W4A8 avoids it and QServe reports 2.5–2.9× higher A100 throughput than the W4A4 systems it compares against.Native FP4 and microscaling move the block-scale multiply into tensor cores, explaining the shift toward FP4 formats, although installed Ampere and Hopper systems still face software dequantization costs.Source: §10.2 · pp. 69–70

10.3 Group size and the hardware scale multiplier

  • Hardware microscaling fixes the free scale granularity at 32 elements for MX formats and 16 for NVFP4, making block size a discrete accuracy–metadata tradeoff rather than a continuous software knob.MXFP4 costs 4.25 effective bits and NVFP4 4.5, while NVFP4’s mantissa-carrying scale reduces scale-rounding error; the transform, format, and block size therefore require joint co-design.Source: §10.3 · pp. 70–71

10.4 The surviving variable-length lane

  • Variable-length coding survives as lossless storage and bandwidth compression, but tensor-core MMA still receives dense equal-width tiles, so entropy coding must decode before or as data enters the matmul.ZipNN compresses already-quantized checkpoints only to about 85–91% of their size, while BF16 checkpoints shrink by roughly a third; ZipServ instead uses fixed-length decoding in the mainloop, and Huff-LLM pipelines constant-rate hardware decoding.Source: §10.4 · pp. 71–72

10.5 What the shipped 4-bit models actually do

  • Shipped 4-bit systems use controlled training and selective tensor coverage for transform-free weight paths, while AWQ, SpinQuant, and selected activation paths retain transforms where folding is unavailable.DeepSeek rotates the FP4 query–key activation path but not its ordinary attention path; Meta’s paired release uses SpinQuant with GPTQ versus QLoRA without a transform, and NVIDIA’s NVFP4 recipe applies a random Hadamard only to weight-gradient GEMM inputs.Source: §10.5 · pp. 72–73

11 The Empirical Landscape

The published record supports regime-specific transform benefits rather than a universal leaderboard: integer activation quantization favors richer transforms, while format, compression target, KV-cache axes, and deployment cost materially change the outcome. Comparisons are protocol-bound, and reported numbers establish consistent shapes rather than protocol-neutral rankings.

11 The Empirical Landscape

  • Each comparison fixes its protocol, so W4A4 perplexities and FP4 accuracies are not commensurable, and authors’ self-tuned margins should not be read as universal.Table 10 combines two sources, while Tables 11–14 transcribe each proposing paper’s best-performing row; comparisons are valid within blocks, not across them.Source: §11 · pp. 73–74

11.1 Activation quantization: climbing the transform ladder

  • 83.12 versus 5.47 FP16 perplexity shows diagonal SmoothQuant collapsing, whereas QuaRot, SpinQuant, FlatQuant, and related transforms reach 5.78–6.19 on W4A4.The ladder across five models preserves the ordering, with FlatQuant reaching 78.6% versus QuaRot’s 70.4% on Llama-3-70B; stronger transforms also leave less for GPTQ to improve.Source: §11.1 · pp. 74–76, Table 10 · p. 75, Table 11 · p. 75

11.2 Weight-only quantization: the low-bit regime

  • At 2 bits, QuIP# holds at 6.66 perplexity within 1.2 of FP16, while scalar AWQ and OmniQuant diverge to 24.0 and 37.4 at 3 and 2 bits.Its incoherence rotation and scalar LDLQ baseline provide most of the gain, while the lattice codebook and inter-layer fine-tuning jointly support extreme compression; metadata is excluded from nominal bit-width.Source: §11.2 · pp. 76–77, Table 12 · p. 77

11.3 The format flip

  • On MXFP4, global QuaRot falls to 62.9% versus 69.3% plain RTN, while block-confined Hadamards improve to 70.5% RTN and 73.7% GPTQ.On NVFP4, QuaRot recovers to 74.1% versus 74.7% RTN, and WUSH reports the best averages: 74.35% on MXFP4 and 76.1% on NVFP4.Source: §11.3 · pp. 77–79, Table 13 · p. 78

11.4 Beyond the weight matrix: the KV cache

  • At 2-bit KV quantization, naive per-token integer scaling reaches 11,779 perplexity, whereas KVQuant reaches 6.01, within about a third of FP16.Pre-RoPE per-channel keys, a non-uniform codebook, attention-sink handling, and 1% full-precision retention drive the result; removing retained outliers raises perplexity to 7.23.Source: §11.4 · p. 79, Table 14 · p. 79

11.5 Is the transform worth it? Speed and memory

  • 0.07× end-to-end slowdown for five fused affine transforms contrasts with 0.26× for QuaRot’s three online Hadamards, while both retain prefill speed-ups near or above 2×.Reported system gains include QuaRot’s 3.33× prefill speed-up at 70B and FlatQuant’s 1.76× decoding speed-up, but Table 15 metrics and baselines are not directly comparable.Source: §11.5 · p. 80, Table 15 · p. 81

12 Evaluation Pitfalls and Practical Guidance

Reliable transform comparisons must match the deployed quantizer, rounder, format, metadata, calibration protocol, runtime, and evaluation horizon rather than optimizing mismatched surrogates. The resulting deployment guide treats transform choice as scenario- and format-dependent, with recommendations synthesized across heterogeneous evidence rather than a single normalized benchmark.

12.1 Evaluation pitfalls

  • Validate transforms on deployed perplexity or downstream accuracy at the actual bit-width, group size, and scale rule, because RMS, pre-quantization, and layer-error surrogates can rank transforms incorrectly.AbsMax error depends on group maxima and crest factors, while exact layer-output error need not predict end-to-end loss at aggressive bit-widths.Source: §12.1 · p. 81
  • Hold granularity, rounder, and format fixed: per-token, per-group, and per-tensor scales differ; GPTQ can erase apparent transform gains; and INT4 results do not transfer automatically to FP4 or MXFP4.Effective bits must include shared scales, zero-points, high-precision outliers, codebooks, and stored transform parameters.Source: §12.1 · pp. 81–82
  • Avoid calibration leakage, report non-absorbable runtime cost, test cross-corpus robustness, and evaluate cache quantization at long generation lengths rather than short-context perplexity alone.“Zero-cost” applies only when transforms are absorbed into adjacent weights; online fused kernels still incur comparison-relevant cost.Source: §12.1 · p. 82

12.2 A which-transform-when guide

  • Table 16 recommends flattening rotations for integer shared-scale grids, block-confined rotations for coarse MXFP4 scales, little or no rotation for mantissa-carrying NVFP4 scales, and no rotation for bounded tensors.These are starting hypotheses synthesized across heterogeneous models, tasks, group sizes, calibration sets, and kernels, not a normalized head-to-head verdict.Source: §12.2 · p. 82, Table 16 · p. 83
  • For Mamba, use decorrelation before Hadamard when channels are correlated; for training gradients, exploit row sparsity with leverage-score selection or row-axis mixing instead of feature-axis flattening.MambaQuant equalizes the rotated diagonal to trΣ/d, while the NVFP4 pretraining recipe rotates only weight-gradient GEMM inputs and is not invertible across other paths.Source: §12.2 · p. 84

13 Open Problems and Conclusion

The survey’s inversion clarifies why transform choices depend on allocation, scale, and format, while exposing unresolved joint optimization, deployed-objective guarantees, hardware regimes, and transform-format co-design. Its conclusions are bounded by surrogate proofs, transcribed evidence, a June 2026 cutoff, and unmeasured safety effects.

13 Open Problems and Conclusion

  • The central open problem is provably optimizing the transform and rounder or codebook jointly for deployed error, rather than optimizing each stage separately.The lattice view interprets transforms as basis changes and rounding as nearest-plane decoding, but scale-aware or LLL-style basis-reduction theory remains open.Source: §13 · p. 84
  • No available optimality theorem targets realized shared-scale AbsMax error on heavy-tailed data; existing results optimize surrogates, while WUSH guarantees are restricted by distribution and model assumptions.Randomized Hadamard and Kashin methods bound crest factors, but the missing result is a data-dependent transform minimizing realized deployed error.Source: §13 · pp. 84–85
  • Whether future inference hardware restores allocation-flexible coding is unsettled: current decoders and block-scaled tensor cores recover only parts of concentration or fine-grained scaling.If allocation-flexible representations become as cheap as shared-scale ones, concentration-optimal transforms could regain relevance; otherwise flattening may remain dominant.Source: §13 · p. 85
  • Jointly designing transforms, element grids, shared scales, and block sizes remains open, with the optimal transform depending on hardware format and granularity.Existing work explores two-level and sub-block scaling, but the profitable scale granularity and corresponding transform must be weighed against hardware area.Source: §13 · p. 85
  • The survey reports no original benchmark leaderboard: results are transcribed from cited sources, comparisons inherit their protocols, and the corpus and rankings may date beyond the June 2026 cutoff.Its original measurements are illustrative, and the 43 classified transform methods do not constitute an audited table of all 200 cited works.Source: §13 · pp. 85–86

Broader Impact Statement

  • Quantization can alter safety-relevant behavior, yet no cited work measures how transform choice affects safety, red-teaming, or alignment properties after low-bit deployment.The literature primarily optimizes perplexity and task accuracy or proxy reconstruction error, which do not automatically transfer full-precision safety evaluations.Source: Broader Impact Statement · pp. 86–87

A Notation

This section fixes the survey’s notation in Table 17, including context-dependent symbols, matrix proxies, quantization errors, transform-coding quantities, and incoherence parameters.

Notation

  • Table 17 collects the symbols used throughout the survey and fixes their meanings from Section 2.The table is the survey’s central notation reference.Source: A Notation · p. 99, Table 17 · p. 100

Context-dependent symbols

  • Q denotes either an orthogonal transform, the quantizer, or a query matrix, while P denotes attention probabilities, a FlatQuant factor, or a microscaling value depending on context.Context distinguishes these overloaded symbols across transform, attention, FlatQuant, and microscaling sections.Source: A Notation · p. 100

Core quantities

  • The notation defines activation and weight Gram matrices, quantization errors and step-size noise, classical transform-coding objects, and incoherence quantities including µ, H, G, and α.HX and HW represent activation- and weight-side Gram matrices; ∆W and ∆X are quantization errors, while UKLT, CF, GTC(U), R, µ, H, G, and α support later analyses.Source: A Notation · p. 100

B Glossary of concepts and formats

This glossary distinguishes fixed-rate shared-scale quantization from allocation-flexible coding, formalizes the Great Inversion between concentration and flattening, and defines transform composition, folding, precision lanes, and formats.

Rate and coding

  • Fixed-rate quantization uses equal bits and one shared per-group scale, whereas variable-rate coding allocates different bits under a fixed total rate.The distinction is per-coordinate allocation, not code length; transform coding follows decorrelate–allocate–quantize.Source: B Glossary of concepts and formats · p. 101

The inversion

  • The Great Inversion opposes concentration for variable-rate coding with flattening for fixed-rate uniform-grid quantization, with optimality transferring only for group-constant spectra.Flattening minimizes the AbsMax crest factor; Hadamard is near-optimal, while WUSH combines whitening with Hadamard under its model.Source: B Glossary of concepts and formats · p. 101

Transforms and composition

  • Function-preserving transforms leave layer outputs unchanged, and orthogonal residual-stream rotations can fold offline at zero runtime after RMSNorm gain absorption.Absorption folds T^-1 into consuming weights while pushing T^⊤ through the preceding RMSNorm or linear layer, so the transformed activation is not materialized.Source: B Glossary of concepts and formats · p. 101

Precision lanes and formats

  • Precision lanes specify weight, activation, and KV-cache bit widths, while INT4 is uniform and FP4-family formats use non-uniform E2M1 elements with distinct shared-scale schemes.MXFP4 uses E8M0 power-of-two scales over microscaling blocks, whereas NVFP4 uses E4M3 scales over 16-element blocks plus a per-tensor scale.Source: B Glossary of concepts and formats · p. 101

C Complete index of cited works

This complete index accounts for all 200 cited works, classifying each by role and linking it to the survey sections where its contributions are treated. It spans transform and non-transform methods, numerical formats, rounding and codebooks, architectures and applications, systems and deployment, and the foundational theory framing their analysis.

Complete index

  • Table 18 classifies all 200 cited works once by role, contribution, and treating section, while per-domain tables provide additional method-level views.The index is intended as the single auditability-oriented location where every cited work is accounted for.Source: Table 18 · p. 102, C Complete index of cited works · p. 102

Classical foundations

  • The classical foundation spans KLT transform coding, high-resolution quantization, Lloyd–Max codebooks, rate-distortion theory, and lattice or trellis methods, with explicit non-Gaussian limitations.The KLT can be strictly suboptimal for non-Gaussian sources (Federici et al., 2026; Effros et al., 2004), while nonlinear transform codes can surpass it (Ballé et al., 2021).Source: C Complete index of cited works · p. 103

Weight/activation transforms

  • Section 6 indexes weight and activation transforms ranging from affine, diagonal, permutation, and Hadamard constructions to learned rotations, whitening, overcomplete frames, and function-preserving scaling.The methods vary in data awareness, parameterization, runtime placement, and computational cost; examples include QuaRot (Ashkboos et al., 2024c), SpinQuant (Liu et al., 2025c), and FlatQuant (Sun et al., 2025).Source: C Complete index of cited works · pp. 104–105

Rounding and codebooks

Number format and co-design

  • Section 8 indexes format and transform co-design, including adaptive and asymmetric FP4 variants, NVFP4 block rescaling, and Hadamard rotation confined to microscaling blocks.Block Rotation (Shao et al., 2025b) reports that a block-confined Hadamard captures most available MXFP4 benefit, while AMXFP4 (Lee et al., 2025a) reports gains over MXFP4 and rotation-based INT4 without a transform.Source: C Complete index of cited works · p. 106

Microscaling and FP4

  • Microscaling and FP4 works show that format and scale design materially shape transform benefits, spanning MXFP4 block scales, NVFP4 mantissa-carrying scales, and Hadamard-based methods.Reported examples include MR-GPTQ recovering up to 98–99% of FP16 on large models, while NVFP4 normalization can stabilize 4-bit pretraining without Hadamard or dynamic per-tensor scaling.Source: C Complete index of cited works · pp. 106–107

Beyond the weight matrix

  • Beyond weight matrices, the index covers rotations, smoothing, permutations, and low-rank repairs for diffusion transformers, KV caches, attention, and state-space models, as well as methods that deliberately use no transform.Examples include ConvRot and DiRotQ for diffusion, RotateKV and KVLinC for KV caches, MambaQuant for channel equalization, and SVDQuant’s rank-32 FP16 outlier branch.Source: C Complete index of cited works · pp. 107–108

Systems, kernels, and coders

  • Systems and coder works connect transform choices to actual matrix-core behavior, dequantization, compression, and hardware-native formats across AMD and NVIDIA platforms.HadaCore reports a 1.1–1.4x faster tensor-core Walsh–Hadamard kernel on A100, while Hopper’s INT4 mma compiles to CUDA-core IMAD and Blackwell provides native FP4/microscaling.Source: C Complete index of cited works · pp. 108–109

Deployed models

  • Deployed-model entries demonstrate heterogeneous practice: shipped systems variously use MXFP4, FP4, AWQ, SpinQuant, online Hadamard transforms, or no transform on selected parameter paths.The index includes gpt-oss MXFP4 routed experts, DeepSeek-V4-Pro FP4 experts with online Hadamard activation preprocessing, and Llama-3.3-70B-Instruct-FP4 variants with and without SpinQuant rotations.Source: C Complete index of cited works · p. 109

Background, surveys, and cross-cutting theory

  • Foundational references supply the survey’s context: GQA and RoPE determine KV-cache structure, PBA and majorization provide the concentration and flattening poles, and prior surveys mostly treat transforms as one step rather than the organizing axis.Marshall and Olkin provide the separable-sum Schur-concavity and maximum Schur-convexity components, while their pairing along the within-group order is identified as the survey’s contribution.Source: C Complete index of cited works · pp. 109–110

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