Sequential Lossy Compression With Causal Conditional Perception

I. INTRODUCTION

• Low-rate MSE compression can produce blurry or implausible outputs, motivating distributional fidelity that also accounts for temporal structure and causal decoding.The proposed conditional criterion compares source and reconstruction laws given the same past reconstructions.Source: §I · p. 1
• Earlier sequential coding work characterized stagewise rates and minimum sum rates, with the latter connected to the nonanticipative rate-distortion function (NRDF),.The framework in retains the complete rate vector and allows encoding and decoding delays.Source: §I · p. 1
• The conditional self-adaptive criterion extends prior sequential perceptual-compression work, [9] and is motivated by predictive video decoding from previously reconstructed frames.The paper also identifies sequential semantic communication, where interpretation may depend on recovered context, as a relevant setting.Source: §I · p. 1
• The paper formulates a finite-horizon NRDPF with per-frame distortion and stagewise conditional-perception constraints, then derives one-shot sum-rate bounds for first-order Markov sources.The bounds use a strengthened SFRL and common randomness for variable-length prefix-free coding.Source: §I · p. 1
• For time-varying scalar Gauss–Markov sources, the paper proves Gaussian optimality and gives a log-variance characterization with a closed-form solution.The solution includes perfect realism and recovers the Gaussian NRDF when perception is inactive and the Gaussian RDPF for i.i.d. sources.Source: §I · p. 1

II. PROBLEM STATEMENT

• Each of the n + 1 frames is modeled as part of a first-order Markov source, whose joint law factorizes accordingly.The source frames take values in the specified frame spaces.Source: §II · pp. 1–2
• At each stage, the encoder observes source frames through the current time and shared randomness, sends a variable-length prefix-free message, and the decoder reconstructs using messages and that randomness.The conditional codebook is prefix-free given prior messages and common randomness.Source: §II · p. 2
• The fidelity requirements combine per-frame expected distortion with a stagewise perception constraint comparing source and reconstruction distributions under the same reconstruction history.Setting D_t = ∞ removes the distortion constraint at stage t; the perception levels are specified stagewise.Source: §II · p. 2
• The NRDPF minimizes the information-theoretic rate over feasible causal reproduction distributions, generalizing the classical NRDF with stagewise perceptual constraints.Removing or making the perception constraints inactive recovers the classical nonanticipative RDF.Source: §II · p. 2
• For first-order Markov sources, the infimum can be restricted without loss of optimality to Markov reproduction distributions.The resulting expression simplifies the NRDPF while retaining the stagewise fidelity criteria.Source: §II · p. 2

III. ONE-SHOT ACHIEVABILITY

• Theorem 1 establishes lower and upper operational sum-rate bounds in terms of the NRDPF for first-order Markov sources with feasible constraints and unlimited independent common randomness.The theorem concerns variable-length prefix-free coding over the finite horizon.Source: §III · p. 2
• The converse uses prefix-free source-coding inequalities and data processing to relate any feasible causal code's rate to the NRDPF.The induced reproduction law is feasible, and the argument uses the exogenous-source property and directed-information conservation.Source: §III · p. 3
• The achievability proof selects an approximately optimal feasible reconstruction law and takes the approximation parameter to zero to obtain the bound.The stated proof also uses averaging and Jensen’s inequality.Source: §III · p. 3
• For achievability, the strengthened conditional SFRL represents each reconstruction using fresh shared randomness, which need not be communicated, and conditional Shannon coding.Forward induction preserves the target joint law exactly, so distortion and nonlinear conditional-perception constraints remain satisfied.Source: §III · p. 3

IV. CASE STUDY: JOINTLY GAUSSIAN PROCESSES

• Restricting to jointly Gaussian source–reconstruction processes leaves the NRDPF infimum unchanged.The Gaussian-replacement argument preserves the source distribution, causal structure, and MSE; its conditional coupling preserves the perception bound.Source: §IV · pp. 3–4
• A history-dependent translation leaves directed information unchanged and cannot increase either the MSE or conditional perception loss.It allows the common conditional mean of the source and reconstruction to be used in the Gaussian innovations parametrization.Source: §IV · p. 4
• The scalar characterization uses Gaussian innovations, including source prediction variance, reconstruction-innovation variance, and filtering-error variance.The characterization is realized by an orthogonal source–reconstruction construction; when the reconstruction innovation vanishes, the filtering-error variance equals the prediction variance.Source: §IV · p. 4
• In the 1,000-stage example, 526 stages use the MSE-only solution and 474 use the MSE–perception solution.The example applies the closed-form solution recursively across stages.Source: §IV · p. 5
• The optimal solution has a two-branch, stagewise reverse-waterfilling structure: an MSE-only allocation or a joint MSE–perception branch.The MSE constraint sets the classical allocation; the perception water level determines when the joint branch applies.Source: §IV · pp. 4–5
• At P = 0, the paper gives a perfect-realism solution; for i.i.d. sources, this recovers the distribution-preserving RDF.The perfect-realism specialization is the zero-vector perception budget.Source: §IV · p. 5
• With inactive perception, the solution reduces to the classical scalar Gaussian NRDF; for i.i.d. Gaussian sources, it recovers the classical Gaussian RDPF.In the i.i.d. case, the optimal reproduction is memoryless, so conditional perception reduces to unconditional Wasserstein fidelity.Source: §IV · p. 5

• For first-order Markov sources, the paper formulates the NRDPF and establishes one-shot lower and upper bounds on operational variable-length sum-rate using common randomness and a strengthened SFRL.Source: §V · p. 5
• For time-varying scalar Gauss–Markov sources under MSE and conditional squared Wasserstein-2 fidelity, the paper proves Gaussian optimality and derives an explicit characterization with a closed-form solution, including perfect realism.Source: §V · p. 5
• Extensions to vector sources and practical predictive video-coding architectures are left for future work.Source: §V · p. 5