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Second-Order Image Characterization with Distortion and Secrecy

Cao Daming

arXiv:2610.09472v1cs.ITmath.IT

TL;DR

Second-order image bounds must preserve joint fluctuations, conditional likelihood losses, and encoder-observable choices. The paper extends image methods to these constraints and reports matching regions in several settings, with explicit bounds and remaining losses in others.

  • Problem

    Second-order image estimates need to retain joint output fluctuations, account for source-cell likelihood changes, and restrict auxiliary choices to encoder-observable information.

    Source: §I · p. 1

  • Method

    The paper develops joint posterior and conditional-composition images, then extends image bounds to weighted reconstruction-policy images and posterior-mass constraints.

    Source: §I · p. 1, §IX · p. 29

  • Results

    Applications yield matching second-order regions for compatible helper intersections, lossless-informed reconstruction, Gaussian common/private descriptions, and prescribed conditional-type extraction.

    Source: §IX · p. 29

  • Takeaways & Limitations

    The rate corrections depend on the active image constraints and on which observations are available to each terminal.

    Source: §I · p. 1, §IX · p. 29

    Remaining losses include arbitrary helper-cell prediction, fractional reconstruction-policy profiles, Gaussian information–distortion bounds, and reliable-key converses with preprocessing and public discussion.

    Source: §IX · p. 29

Abstract

from arXiv · show

Image-size characterization connects coding constraints to the probability carried by source cells and their output images. At second order, this connection must retain the joint fluctuations of all active constraints and distinguish encoder-observable source variation from unobserved channel noise. We extend the single-output posterior characterization to simultaneous posterior and conditional-composition image bounds. The bounds preserve the actual coupling of multiple outputs, quantify conditional likelihood losses for irregular cells, and admit actual-distortion marks. Two further extensions replace disjoint counting by weighted reconstruction-policy images and replace output cardinality by posterior-mass constraints for secrecy. Direct applications yield matching second-order regions for compatible multi-source helper intersections, one lossy reconstruction with several informed lossless reconstructions, Gaussian common/private descriptions, and privacy amplification from a prescribed conditional type. The Gaussian description result also gives joint rate--distortion offsets. For general helper sections, genuinely lossy informed reconstruction, Gaussian Wyner--Ziv coding, and reconciliation followed by extraction, the same image methods give explicit rate bounds with their remaining losses identified. The resulting rate corrections reflect both the image constraints and the observations available to each terminal.

I. Introduction

Second-order image bounds must retain coupled output fluctuations, cell-induced likelihood changes, and source choices limited to encoder-observable information. The paper extends image characterization to joint outputs, distortion, and secrecy, then applies these tools to matching and nonmatching coding regions.

I. Introduction

  • Fixed-error second-order bounds must retain joint output fluctuations, cell-induced conditional likelihood changes, and source choices based only on encoder-observable information.At message-log scale nr + √nL + o(√n), an unspecified o(n) image loss is insufficient.Source: §I · p. 1

A. Contributions

  • The extensions cover simultaneous posterior and conditional-composition images, actual-distortion marks, weighted reconstruction-policy images, and posterior-mass secrecy constraints.The Gaussian covariance is evaluated after the admissible source-side choice is fixed.Source: §I · p. 1
  • Applications include matching second-order characterizations for helper intersections, informed lossless reconstruction, Gaussian Gray–Wyner coding, and conditional-type privacy amplification.General helper sections, both-lossy reconstruction, Gaussian Wyner–Ziv coding, and reconciliation followed by extraction receive bounds with remaining losses identified.Source: A. Contributions · p. 2

B. Related work

  • The paper builds on the single-output posterior characterization of Cao [2] and weighted WAK refinement [3], extending image methods alongside related functional-inequality and side-information approaches.For Gaussian Gray–Wyner coding, the analysis uses a projection inequality and conditional Gram-matrix cover; for Gaussian Wyner–Ziv it fixes the source–auxiliary Gram matrix before side-information fluctuations.Source: B. Related work · p. 2

C. Organization

  • The paper organizes joint-image, distortion, and secrecy extensions in Sections III–VI, matching applications in Section VII, and nonmatching bounds in Section VIII.The stated conventions use finite fixed alphabets unless a Gaussian model is specified, allow channel zeros unless positivity is required, and distinguish posterior measures from auxiliary stochastic matrices.Source: C. Organization · p. 2, A. Operational conventions · p. 4

III. Joint Posterior Images and Compatible Sections

Joint posterior image bounds preserve multiple output and section constraints on the same success event, while using a common source-past decomposition for the joint output law. Their optimizer selection must remain compatible with the helper’s observable source information.

III. Joint Posterior Images and Compatible Sections

  • Separate encoding imposes aggregate and section constraints on one joint success event, with at most the product of the remaining message counts as correct index tuples.The ancillary section spectrum also applies after restricting the source cell to any nonempty subset.Source: §III · p. 5, A. Counting on the original success… · pp. 5–6

A. Counting on the original success event

  • A weighted section spectrum bounds success for arbitrary input laws using one prediction correction, and predictions for finitely many sections share the same exceptional event.The correction is based on the induced output law and an arbitrary complement reference law.Source: A. Counting on the original success… · p. 6

B. A multivariate posterior transfer

  • The simultaneous posterior bound uses a common source-past decomposition and evaluates the joint output Gaussian law only after conditioning on the whole source block.Source-past martingale bounds remove the residual on low-gap histories and localize posterior mass to the complete fixed-rate-pair optimizer family.Source: B. A multivariate posterior transfer · p. 7

C. Preserving section means under an observable helper choice

  • Weighted-spectrum and Slepian–Wolf means agree across every optimizer exactly under the stated complement-law compatibility condition; the condition must hold throughout the complete optimizer family.If contact differences span the full source tangent space, compatibility forces the complement channel to have identical rows; dependence between source and complement remains possible when variations span a proper subspace.Source: C. Preserving section means under an… · pp. 7–8

IV. Conditional-Composition Images with Actual-Distortion Marks

Conditional-composition image bounds handle multiple projected output images while retaining actual reconstruction costs and correcting for conditioning on irregular source cells. Their Gaussian limits preserve the physical coupling among outputs.

A. Projected images and marked success

  • Theorem 6 bounds simultaneous projected-image success together with actual distortion marks, with a conditional-likelihood correction for the source cell.The bound permits dependent channel coordinates, vanishing transitions, and no independence assumption between the auxiliary and complement outputs.Source: A. Projected images and marked success · p. 9
  • For positive retained marginal variances, projected-image success converges to a joint Gaussian probability; singular joint covariance is allowed, and a cell deficit shifts image coordinates only.When lim sup δn/√n ≤ β, the lower and upper limits are ΦΓ(Q)(L) and ΦΓ(Q)(L + βeim).Source: A. Projected images and marked success · p. 9
  • For two output images, the optimal probability that both outputs fall in their respective images has the joint Gaussian limit, with bounded actual-distortion marks included in that limit.The off-diagonal covariance comes from the physical joint channel; applying scalar image bounds separately does not give this joint limit.Source: A. Projected images and marked success · p. 10

B. Conditional likelihood control for irregular cells

Conditional likelihood control measures how irregular source cells affect projected image bounds, while weighted reconstruction policies and successful-subset certificates extend image methods to actual distortion. The resulting constructions include a finite-block rounding guarantee, but a general both-lossy second-order formula still requires evaluating its limiting fractional-image profile.

B. Conditional likelihood control for irregular cells

  • When conditional likelihood loss is o(√n), the Gaussian conclusion persists even without the density assumption, including when the positive part of each row loss divided by √n vanishes in probability.For a complement observing V^n, conditional-type uncertainty can remain even when the cell's full logarithmic density deficit is Θ(n).Source: B. Conditional likelihood control for… · p. 10

V. Distorted Images and Weighted Image Measures

  • Actual block acceptance can be handled with weighted reconstruction-policy images, retaining overlapping acceptable reconstructions rather than requiring disjoint cells.The conditional marks evaluate distortion fluctuations at a fixed source–reconstruction composition; optimizing over policies requires the additional image construction.Source: §V · p. 11, A. Reconstruction policies and a… · p. 11

A. Reconstruction policies and a fractional cell bound

  • The fractional policy relaxation admits rounding: a feasible fractional solution can be converted into a finite code with a controlled success loss.The construction samples cell-policy pairs, and the encoder selects using the source alone, not the realized side information.Source: A. Reconstruction policies and a… · pp. 11–12
  • For general both-lossy Heegard–Berger coding, transferring the fractional-image profile to operational coding still requires evaluating its limiting profile.With δ_n = n^-2, the extra message budget is O(log log n), while transfer requires control of the profile at the shifted tolerance.Source: A. Reconstruction policies and a… · p. 12

B. Weighted acceptance and successful-subset certificates

  • Successful-subset certificates bound success by the probability that the score exceeds its budget threshold, plus an exponential penalty; multiple certificates can be imposed together.For decoder side information, the certificate must hold for every admissible block policy under the actual block-distortion constraints.Source: B. Weighted acceptance and… · p. 12

C. From a distortion cell to a decodable channel image

Image methods preserve overlapping distortion and posterior-mass constraints, then yield applications under specific decoding and extraction structures. The compatible multi-source helper result gives a matching second-order success profile, while several broader coding problems are handled through bounds or reductions with stated scope.

C. From a distortion cell to a decodable channel image

  • The distortion-cell reduction yields a source-cell spectrum for exact informed recovery; genuinely lossy informed reconstructions instead require policy images that retain overlaps.The argument depends on exact recovery at informed decoders, while the weak distortion remains an actual event.Source: C. From a distortion cell to a… · p. 13

VI. Private Images and Posterior-Mass Constraints

  • For extraction, the relevant image consists of key values generated by high-posterior-mass source points, whose actual and ideal uniform-key probabilities can differ substantially.The image may have too few points to carry comparable probability under the ideal uniform key law.Source: §VI · p. 13

A. Two-sided private-image spectra

  • Thresholding points whose posterior mass exceeds e^γ/K yields fewer than Ke^-γ key values, supporting a private-image spectrum for extraction.The cited bound also gives a fixed deterministic hash attaining its upper bound for the given source law.Source: A. Two-sided private-image spectra · p. 13

B. Public messages and private-image composition

  • Extraction after a public message accounts for its effect on posterior mass without assuming equal-probability public bins.The bound applies to every deterministic public message; applying it to reconciliation gives a public-rate/key-rate construction.Source: B. Public messages and private-image… · p. 14

VII. Tight Second-Order Rate Applications

  • Rate applications first identify the code-imposed image constraint, then provide a realization construction, with auxiliary choices limited to encoder-observed information.For the helper theorem, the converse covariance supremum ranges over the complete optimizer fiber.Source: §VII · p. 14, A. Several sources with one… · p. 16

A. Several sources with one rate-limited helper

  • For compatible mixed ancillary and weighted helper intersections, every code's limiting upper success is bounded by F(L), and that profile is attained.The theorem assumes positive retained diagonal covariance entries; the joint covariance may be singular, and zero-rate operational restrictions are included.Source: A. Several sources with one… · pp. 14–15

B. One lossy and several informed lossless reconstructions

The section characterizes a common-message code for one lossy decoder and several informed lossless decoders. Its finite-block success bounds use simultaneous image tests for the receivers’ actual joint observations.

B. One lossy and several informed lossless reconstructions

1) The code-imposed image spectrum:

  • The optimal success probability is bounded by source-type probabilities times a maximum over feasible joint types of simultaneous image tests, with O(log n) threshold slack and polynomially small additive errors.Source: 1) The code-imposed image spectrum · p. 17

2) The attainable means of the image experiment:

The second-order characterization optimizes attainable image means after the encoder observes the source type, while retaining the joint receiver covariance. A heterogeneous-channel example shows that optimizing second-order distortion allocation changes the rate offset.

2) The attainable means of the image experiment:

3) An explicit heterogeneous-channel example:

C. Privacy amplification from a conditional-type image

The paper extends image methods to extraction from a prescribed conditional type and to Gaussian common/private descriptions. It gives a conditional-type extraction result and an exact second-order Gaussian Gray–Wyner region with a supporting converse.

C. Privacy amplification from a conditional-type image

  • For a prescribed conditional-type source, the extraction analysis evaluates secrecy against the observer and public seed and identifies a Gaussian second-order coefficient from conditional information variance.Source: C. Privacy amplification from a… · pp. 20–21

D. Gaussian Gray–Wyner coding from a weighted distortion image

1) Model and the covariance optimization:

2) An exact supporting spectrum:

  • The supporting converse applies to every positive-probability subset of the joint successful source set and does not assume that its conditional source distribution remains Gaussian.Source: 2) An exact supporting spectrum · pp. 22–23

3) Covering on an observed Gram matrix:

The Gaussian construction adapts to the observed Gram matrix using a common-message header and hierarchical covers. Directional evaluation of the optimized boundary then yields the second-order region, including zero-rate boundaries.

3) Covering on an observed Gram matrix:

4) Directional evaluation and proof of the region:

5) Distortion offsets, a curved point, and a zero-common face:

This section gives Gaussian rate–distortion offsets and several nonmatching rate bounds. It identifies the remaining estimates needed to turn those bounds into operational second-order characterizations.

5) Distortion offsets, a curved point, and a zero-common face:

  • Allowed distortion offsets shift each Gaussian rate coefficient Li by ∆i/(2Di), with the same displacement in both achievability and converse.At zero common first-order rate, an O(log n)-nat Gram header and an order-√n description can supply positive private-budget deficits.Source: 5) Distortion offsets, a curved point,… · pp. 24–25

VIII. Nonmatching Rate Bounds

  • The section frames these applications as rate bounds whose remaining estimates are identified, rather than as matching second-order characterizations.The outstanding estimates include conditional prediction loss, fractional reconstruction-policy profiles, and sharper Gaussian information–distortion bounds.Source: §VIII · p. 25

A. Subset-rate bounds for general helper cells

  • General helper-cell bounds retain subset-rate constraints, but matching second-order results require uniform control of conditional prediction loss over code-induced cells.A conditional type permits a sharper conditional-divergence correction; the displayed bound allows a t√n + O(log n) subset-rate loss.Source: A. Subset-rate bounds for general… · pp. 25–26

B. Both-lossy Heegard–Berger coding

  • For both-lossy Heegard–Berger coding, a reconstruction-policy image gives a rate bound preserving overlapping acceptable sets, but an operational second-order coefficient requires a locally uniform, strictly crossing limit.The bound also includes a log log(1/δn) term.Source: B. Both-lossy Heegard–Berger coding · p. 26

C. Gaussian Wyner–Ziv coding with an actual-distortion mark

  • Gaussian Wyner–Ziv coding uses a source-adaptive achievable profile by fixing the encoder-observed source–auxiliary Gram matrix before evaluating side-information fluctuations.The reconstruction constraint is an actual-distortion mark on the joint information–distortion image.Source: C. Gaussian Wyner–Ziv coding with an… · p. 26

1) Source model and the marked identity:

The Gaussian Wyner–Ziv construction separates encoder-observable source variation from unobserved side-information noise and yields a source-adaptive achievable profile. The section also gives a public-rate/key-rate inner bound for reconciliation followed by extraction.

1) Source model and the marked identity:

2) An achievable source-adaptive profile:

  • The Gaussian Gram-conditioned construction provides an achievable profile, while its coefficient remains bracketed unless an optimum exists; closing the interval needs an improved construction or converse.The source-adaptive allocation improves on a fixed allocation within the same Gram construction, but remains separated from the conditional-rate-distortion bound.Source: 2) An achievable source-adaptive profile · pp. 27–28

D. Reconciliation followed by extraction

  • Reconstruction followed by hashing achieves public-rate and key-rate offsets with key disagreement at most ε and total-variation distance from uniformity at most δ.Reliability and secrecy have separate spectral inequalities; tightness requires a converse accounting for preprocessing and the full public transcript.Source: D. Reconciliation followed by extraction · p. 28

IX. Conclusion

  • The conclusion identifies three second-order requirements: preserve active image constraints jointly, control source-cell likelihood corrections, and restrict auxiliary selection to encoder observations.The applications give source-observable Gaussian profiles for compatible helper sections and informed lossless reconstruction, and a reverse-tail secrecy coefficient for conditional-type extraction.Source: §IX · p. 29

Appendix A Posterior Localization and Helper Implementation

The appendix localizes posterior fluctuations to the complete optimizer family and constructs observable helper branches. It also relates source coding with reconstruction constraints to channel coding on finite source types.

A. The simultaneous image bound

  • On low-gap histories, empirical posterior limits have the required barycenter, entropy, and optimizer-family support, enabling a uniform conditional Gaussian comparison for simultaneous image constraints.The comparison accommodates singular joint covariance when retained marginal variances are positive.Source: A. The simultaneous image bound · pp. 30–31

B. Observable helper branches

  • A helper can choose an auxiliary from its observed type and cover compatible source words with a description of nrh + √nLh + O(log n) nats.The local implementation controls the actual conditional likelihood used by the decoder at order √n, beyond matching first-order entropies.Source: B. Observable helper branches · pp. 31–32

Appendix B Reconstruction-Image Decoding and Transitive Covering

  • Cell extraction and transitive covering reduce reconstruction-image decoding to an ordinary channel experiment while preserving original success probability, without requiring equal cell sizes or codeword errors.For memoryless sources, coordinate permutations act transitively on a source type class and preserve additive distortion.Source: Appendix B · p. 32, A. A success-preserving… · pp. 32–33

A. A success-preserving packing–covering reduction

  • The finite-type operational bracket compares source success with channel-code success within each type, and a polynomial source-message allowance suffices at zero first-order source rate.Choosing η = n^-b makes the channel-size thresholds differ by at most a polynomial factor, or O(log n) in logarithmic size.Source: A. A success-preserving… · p. 33

B. Joint spectra for conditional-composition channel codes

Conditional-composition channel codes admit simultaneous finite-type bounds based on the actual joint score law. Uniform score-moment control then supports Gaussian limits and source-marginal transport at support boundaries.

B. Joint spectra for conditional-composition channel codes

  • Simultaneous conditional-type packing bounds joint success for K-word codes using one event in which every receiver’s score exceeds its threshold.The converse and achievability are finite-type bounds; they do not replace joint success with separately constrained marginal successes.Source: B. Joint spectra for… · pp. 34–35

C. Uniform moments and support regularity

  • Averaged score moments are uniformly bounded over conditional types, and the covariance matrix is continuous even when channel probabilities vanish.With positive retained marginal variances, rectangle probabilities converge uniformly over thresholds, including singular covariance limits.Source: C. Uniform moments and support… · p. 35

A. Source-marginal transport

  • Source-marginal transport constructs a nearby feasible law whose score means decrease by at most a constant times the marginal change, uniformly over auxiliary supports and induced zero patterns.The one-sided bound avoids an η log(1/η) loss when cost correction creates new support.Source: A. Source-marginal transport · p. 36

B. Covariance-preserving mean realization

The appendix characterizes feasible first-order mean directions while preserving covariance at the reference law, then establishes the transport and Gaussian-envelope properties needed for a unique threshold.

B. Covariance-preserving mean realization

2) The source slices obey a uniform transport rule:

  • Source slices can be transported to exact nearby marginals with loss bounded by K times their distance, and feasible marginal perturbations yield tangent directions.The common upper mean remains within K∥q−p∥1 of C near p.Source: 2) The source slices obey a uniform… · p. 37

C. The Gaussian envelope

  • The Gaussian-envelope function is continuous and strictly increases wherever positive, giving a unique finite solution to F(L)=1−ε for every 0<ε<1.Its endpoint limits are zero and one.Source: C. The Gaussian envelope · pp. 37–38

A. A Gaussian projection inequality with a quadratic remainder

  • The Gaussian projection inequality retains quadratic source fluctuation and applies to arbitrary finite-relative-entropy laws, including conditional laws.It tensorizes to block laws and does not require the conditioned source to be Gaussian, memoryless, centered, or product-form.Source: A. A Gaussian projection inequality… · pp. 38–39

Appendix E Gaussian Window Estimates and Covering Details

The appendices establish Gaussian-window estimates that remain uniform for small covariance eigenvalues, then use them in a marked-image construction that tracks source and channel fluctuations jointly.

Appendix E Gaussian Window Estimates and Covering Details

  • The uniform Gaussian-window estimate addresses a common layer whose covariance may be of order 1/n, with dimensions fixed as n grows.A compact positive-covariance estimate alone would not justify this regularization step.Source: Appendix E · p. 39

A. Gram windows with polynomially small eigenvalues

  • For bounded regression and covariance eigenvalues in [cn^-a,C], a Gram window has probability at least n^-c2 and likelihood-ratio error O(n^(1+a−b)) when b>a+2.The assumptions also require the target base covariance to differ from BTB/n by at most n^(-b−2).Source: A. Gram windows with polynomially… · p. 39

B. Parameters for the common and private layers

  • The common-layer windows use a=1,b=6 and base-bin accuracy n^-8, while private residual windows use the scalar estimate with a=1,b=4.The private residual design keeps its conditional reproduction variance bounded below by order 1/n.Source: B. Parameters for the common and… · p. 40

Appendix F The Gaussian Marked-Image Construction

  • Auxiliary covering lists, source-power bins, and allocation tags are fixed before channel noise; bin labels are assigned after list generation, and one deterministic code attains the ensemble-average bound.The decoder thresholds information density within the bin.Source: Appendix F · pp. 40–41
  • The marked-image construction yields a joint Gaussian limit uniformly over Gram-constrained source–auxiliary pairs and bounded design windows.Observed source power contributes a Gaussian fluctuation, while the allocation shifts the rate and distortion design means.Source: Appendix F · pp. 40–42
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