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Semidefinite Programming Approach to Gaussian Sequential Rate-Distortion Trade-offs
Takashi Tanaka, Kwang-Ki K. Kim, Pablo A. Parrilo, Sanjoy K. Mitter
TL;DR
For real-time communication, SRD studies the rate-distortion trade-off and provides a tight outer bound on the zero-delay region achievable by uniquely decodable codes. This paper applies sensor-estimator separation to Gaussian SRD and formulates the resulting design as an SDP, obtaining a semidefinite representation of the SRD function.
Problem
SRD provides a way to study fundamental rate-distortion limits for real-time communication, where the zero-delay achievable region is not generally characterized by the standard rate-distortion region.
Source: §I · p. 1Method
The paper uses sensor-estimator separation to express Gaussian SRD as a linear-Gaussian sensor design problem with a Kalman-filter estimator, then formulates that design as an SDP.
Source: §I · p. 2, A. Gaussian SRD problem · p. 3, B. Linear-Gaussian sensor design problem · p. 3Results
The Gaussian SRD problem has a semidefinite representation and can be solved using standard SDP solvers.
Source: §I · p. 2, §VIII · p. 9Takeaways & Limitations
The SDP approach applies to multi-dimensional, time-varying Gauss-Markov sources for which existing I-MMSE formulas do not apply straightforwardly.
Source: §I · p. 2
Abstract
from arXiv · showhide
Sequential rate-distortion (SRD) theory provides a framework for studying the fundamental trade-off between data-rate and data-quality in real-time communication systems. In this paper, we consider the SRD problem for multi-dimensional time-varying Gauss-Markov processes under mean-square distortion criteria. We first revisit the sensor-estimator separation principle, which asserts that considered SRD problem is equivalent to a joint sensor and estimator design problem in which data-rate of the sensor output is minimized while the estimator's performance satisfies the distortion criteria. We then show that the optimal joint design can be performed by semidefinite programming. A semidefinite representation of the corresponding SRD function is obtained. Implications of the obtained result in the context of zero-delay source coding theory and applications to networked control theory are also discussed.
I. INTRODUCTION
Sequential rate-distortion (SRD) characterizes rate-distortion trade-offs for real-time communication and provides a tight outer bound on uniquely decodable zero-delay codes. The paper uses sensor-estimator separation to formulate Gaussian SRD as an SDP, while the scalar Gaussian example reduces reproduction-policy design to choosing an SNR.
I. INTRODUCTION
- The standard rate-distortion region is only a conservative outer bound for zero-delay communication because achieving it generally requires anticipative, non-causal codes.Source: §I · p. 1
- The SRD function minimizes directed information subject to a distortion constraint and provides a tight outer bound on the zero-delay region achievable by uniquely decodable codes.Source: §I · p. 1
- Sensor-estimator separation makes the Gaussian SRD problem an SDP, yielding a semidefinite representation of its function and an efficient numerical solution procedure.Source: §I · p. 2
- The SDP approach covers multi-dimensional, time-varying Gauss-Markov sources for which existing I-MMSE formulas do not apply straightforwardly.Source: §I · p. 2
II. PROBLEM FORMULATION
- For scalar Gaussian rate-distortion, an optimal reproduction policy composes a linear sensor with a least mean square error estimator.Source: §II · p. 2
- The scalar example reduces an infinite-dimensional policy optimization to choosing an SNR, with a closed-form optimal choice for each D > 0.Source: §II · p. 2
A. Gaussian SRD problem
For time-varying Gauss-Markov sources, the section frames SRD as minimizing directed information under per-time mean-square distortion limits. It motivates a linear-Gaussian sensor/Kalman structure and reduces the design to an SDP-solvable problem.
A. Gaussian SRD problem
- The source follows Gauss-Markov dynamics, and its causal reproduction policy minimizes directed information subject to per-time mean-square distortion constraints.The source and reproduction are vector-valued, with Gaussian process noise and causal stochastic kernels.Source: A. Gaussian SRD problem · p. 3
- An optimal SRD policy has a sensor-estimator separation structure: a linear-Gaussian sensor followed by Kalman filtering.The sensor is yt = Ctxt + vt, and the estimator is zt = E(xt|yt).Source: A. Gaussian SRD problem · p. 3
- The infinite-dimensional SRD optimization reduces to finite-dimensional matrix-valued sensor SNR variables, whose optimal values can be obtained by SDP.An analytical expression for the optimal SNRs may not be available.Source: A. Gaussian SRD problem · p. 3
B. Linear-Gaussian sensor design problem
- P-LGS chooses sensor gains Ct and noise covariances Vt to minimize conditional information cost subject to per-time distortion limits.Sensor dimension is also a design choice; perfect observation has zero estimation error but infinite information cost.Source: B. Linear-Gaussian sensor design problem · pp. 3–4
- The paper establishes that optimizing over the linear-Gaussian sensor-design domain preserves an optimal solution to the original SRD problem.Within that domain, the SRD and sensor-design objectives coincide.Source: B. Linear-Gaussian sensor design problem · pp. 3–4
C. Stationary cases
- For stationary infinite-horizon systems, the paper formulates SRD with average distortion and a stationary linear-Gaussian sensor whose dimensions remain design variables.The source has time-invariant dynamics and stationary white Gaussian noise.Source: C. Stationary cases · p. 4
D. Soft- vs. hard-constrained problems
The paper establishes that Gaussian SRD can be solved through an equivalent linear-Gaussian sensor design, then develops an SDP-based synthesis procedure. It also relates hard- and soft-constrained formulations through suitable Lagrange multipliers.
D. Soft- vs. hard-constrained problems
- With suitable Lagrange multipliers, an optimum of the soft-constrained problem is also optimal for the hard-constrained problem.The paper establishes this after transforming both formulations into finite-dimensional convex optimization problems.Source: D. Soft- vs. hard-constrained problems · p. 4
III. SENSOR-ESTIMATOR SEPARATION PRINCIPLE
- The Gaussian SRD problem is equivalent to linear-Gaussian sensor design, whose optimum also solves the original problem.The equivalence holds when the optimal SRD value is finite.Source: §III · p. 4
IV. SDP-BASED SYNTHESIS
- The linear-Gaussian design reduces to an SDP, enabling Gaussian SRD to be solved using standard SDP solvers.The paper treats the covariance as the state and SNR as the control when developing the SDP transformation.Source: §IV · p. 5
A. SRD optimization as max-det problem
The paper converts the linear-Gaussian sensor design problem into a determinant-maximization problem with LMI constraints. It also provides conditions for existence and connects the hard- and soft-constrained formulations.
A. SRD optimization as max-det problem
- The linear-Gaussian sensor design can be solved by determinant maximization with LMI constraints, yielding an SDP-based solution to Gaussian SRD.The transformation replaces a nonlinear equality with a linear inequality and uses covariance matrices as decision variables.Source: A. SRD optimization as max-det problem · p. 5
- When Wt ≻ 0 and Dt > 0 for every t, the max-det problem is strictly feasible and has an optimal solution.The paper constructs an optimal solution to the original SRD problem under these assumptions.Source: A. SRD optimization as max-det problem · p. 6
- The minimum sensor-output dimension for optimality is rt = rank(SNRt).The optimal SNR sequence can be reconstructed from the max-det solution, then used to recover sensor parameters.Source: A. SRD optimization as max-det problem · pp. 5–6
- For Wt ≻ 0 and Dt > 0, suitable Lagrange multipliers make the soft-constrained and hard-constrained problems share an optimum.The paper invokes strict feasibility and strong duality to establish this result.Source: A. SRD optimization as max-det problem · p. 6
B. Max-det problem as SDP
The max-det formulation is solvable via SDP methods, with a specialized complexity improvement suggested but not established. In the single-stage case, the solution reduces to reverse water-filling and the optimal measurement dimension falls as information becomes more costly.
B. Max-det problem as SDP
- The max-det problems are essentially solvable via SDP, a stronger claim than convexity alone because convexity does not guarantee an efficient algorithm.The hard-constrained version can be transformed into a standard SDP, and existing interior-point methods can handle max-det terms.Source: B. Max-det problem as SDP · p. 6
C. Complexity analysis
- A specialized interior-point method is theoretically expected to reduce complexity to O(T log(1/ε)), but analysis and experiments are still needed to verify this conjecture.The general Cholesky step requires O(T^3) operations.Source: C. Complexity analysis · p. 6
D. Single stage problem
- For T = 1, the optimal solution recovers Gaussian reverse water-filling, with p_i = min(1/α, σ_i²) for each mode.The optimal measurement dimension is dim(y) = rank(C⊤V⁻¹C) = card{i : σ_i² > 1/α}, so it decreases as the information price 1/α increases.Source: D. Single stage problem · pp. 6–7
V. STATIONARY PROBLEMS
For detectable stationary Gauss-Markov systems, the infinite-horizon SRD problem admits an equivalent time-invariant sensor-and-Kalman-filter design. The section relates this formulation to rank trends and zero-delay coding bounds.
A. Sequential rate-distortion function
- When (A, Θ) is detectable, the infinite-horizon Gaussian SRD problem is equivalent to a time-invariant linear-Gaussian sensor design with a time-invariant Kalman filter.The optimal sensor has the form y_t = Cx_t + v_t with i.i.d. Gaussian noise.Source: A. Sequential rate-distortion function · p. 7
- For scalar systems, the SRD expression recovers the known closed form, while alternative representations are reported for stationary multidimensional Gauss-Markov processes with Θ = I.The scalar expression is min{0, 1/2 log(a² + w/D)}.Source: A. Sequential rate-distortion function · p. 7
B. Rank monotonicity
- For multidimensional systems, the measurement dimension rank(C⊤V⁻¹C) tends to decrease as D increases, but the paper does not prove this monotonicity.The sensing matrices are recovered from C⊤V⁻¹C = P⁻¹ − (APA⊤ + W)⁻¹.Source: B. Rank monotonicity · p. 7
A. Zero-delay source coding
- The SRD function lower-bounds achievable zero-delay rate, and an entropy-coded dithered quantizer yields a gap bounded by a constant due to quantization and entropy-coding losses.The SRD and optimal zero-delay rates do not generally coincide.Source: A. Zero-delay source coding · pp. 7–8
B. Networked control theory
The paper connects Gaussian SRD to networked control and distinguishes its continuous sensor-design problem from sensor selection and scheduling. Simulations illustrate rate-distortion sensing trade-offs for a double pendulum and time-varying satellite communication constraints.
B. Networked control theory
- Gaussian SRD is connected to LQG control with information-theoretic constraints, including a related sensor-controller design that minimizes directed information to the control input.The cited formulation applies the paper's SDP-based algorithm.Source: B. Networked control theory · p. 8
C. Experimental design/Sensor scheduling
- Unlike sensor selection and scheduling, the linear-Gaussian sensor-design problem continuously chooses sensing matrices to minimize an information-theoretic cost.The measurement dimension can be freely chosen.Source: C. Experimental design/Sensor scheduling · p. 8
A. Optimal sensor design for double pendulum
- For the noisy double pendulum, the designed sensor and Kalman filter trade communication rate against distortion, with stricter distortion producing better tracking.At D = 0.002 the sensor uses 0.3829 bits/sample and tracks well; at D = 0.3 it observes little and tracking is poor.Source: A. Optimal sensor design for double… · pp. 8–9
B. Minimum down-link bandwidth for satellite attitude determination
- For the satellite example, the SDP produces an optimal downlink schedule for a 120-step horizon with a prespecified time-varying distortion profile, including no communication when the distortion constraint is met.The SDP also yields the sensing matrices and noise covariances.Source: B. Minimum down-link bandwidth for… · p. 9
VIII. CONCLUSION
- The conclusion frames Gaussian SRD as an efficiently SDP-solvable determinant-maximization problem with applications in real-time communication, networked control, and sensor scheduling.The approach revisits the sensor-estimator separation principle.Source: §VIII · p. 9
A. Stochastic kernels
The appendix defines stochastic kernels and specifies how zero-delay kernels factor across time. It also states measure-theoretic results used to construct joint distributions and conditional kernels.
A. Stochastic kernels
- A stochastic kernel q(dy|x) assigns a probability measure on Y to each x in X, with measurable dependence on x for each measurable set in Y.The paper uses this notation for a Borel-measurable stochastic kernel.Source: A. Stochastic kernels · p. 9
- A source measure and a stochastic kernel determine a unique joint probability measure, and a joint measure admits a corresponding conditional kernel.These are the two directions stated in Lemma 3.Source: A. Stochastic kernels · p. 9
- A zero-delay kernel factors as a product of stepwise kernels q(dz_t|z^{t−1}, x_t), and the induced joint measure combines this factorization with the source law.The paper interprets the mutual information and expectation in (P-SRD) under that joint measure.Source: A. Stochastic kernels · p. 9
- For probability measures, absolute continuity supplies a Radon–Nikodym derivative; the appendix also states a conditional-measure identity for an absolutely continuous joint law.The identity is established using the density relative to the product of the marginals.Source: A. Stochastic kernels · p. 10
B. Information theoretic quantities
This section introduces relative entropy and related information measures used in the paper, including mutual information and differential entropy.
B. Information theoretic quantities
- The section introduces relative entropy, also called Kullback–Leibler divergence, and states that it is nonnegative.It also introduces conditional relative entropy for stochastic kernels given a probability measure.Source: B. Information theoretic quantities · p. 10
- Mutual information is defined as the relative entropy between a joint measure and the product of its marginals.The section also introduces conditional mutual information for a joint probability measure.Source: B. Information theoretic quantities · p. 10
- Differential entropy is introduced for a Euclidean-valued random variable with a probability measure.The supplied passage specifies the setting and measure used in its definition.Source: B. Information theoretic quantities · p. 10
APPENDIX B PROOF OF LEMMA 1
The proof establishes that the SRD optimization can be represented by linear-Gaussian kernels without increasing cost, and connects that construction to the reverse implication for the relaxed problem.
APPENDIX B PROOF OF LEMMA 1
- A zero-mean jointly Gaussian measure matching the original joint covariance yields a linear-Gaussian kernel sequence whose cost is no greater than the SRD optimum.The proof shifts the policy to zero mean if needed, then uses a Gaussian construction based on least-mean-square estimates and independent Gaussian noise.Source: Appendix B · pp. 10–11
- The Gaussian replacement preserves each time-step marginal of the state-output pair, supporting preservation of the distortion criteria.Proposition 4 states equality of the corresponding marginals; the proof then uses that equality in the distortion comparison.Source: Appendix B · pp. 10–11
- The replacement does not increase mutual information when the original mutual information is finite.Proposition 5 gives the inequality for the jointly Gaussian measure with the same covariance.Source: Appendix B · p. 11
- Conversely, a finite-cost linear-Gaussian solution to the relaxed problem can be represented by kernels whose mutual-information terms enter the SRD objective without increasing cost.The proof uses the Markov structure and an estimator-based representation of the output.Source: Appendix B · pp. 10–12
APPENDIX C PROOF OF LEMMA 2
The proof converts between linear-Gaussian stochastic kernels and admissible sensor equations while preserving the relevant information quantities and distortion constraints.
APPENDIX C PROOF OF LEMMA 2
- Finite cost requires the coefficient of the current state in the conditional estimate to be zero.Otherwise the mutual information between the state and that nonzero linear function is infinite, while the comparison term is bounded.Source: Appendix C · p. 12
- An invertible transformation of the kernel outputs produces sensor equations whose conditional mutual information matches that of the original outputs.The construction uses the transformed output sequence and a nondegenerate Gaussian noise vector independent of the process noise and its past.Source: Appendix C · p. 12
- The transformed output retains the original output information, so the estimator based on the sensor output satisfies the required distortion bound.The proof uses the equality of generated sigma-algebras and the minimum mean-square error property of the estimator.Source: Appendix C · pp. 12–13
- The converse construction uses a Kalman-filter estimate to define a linear-Gaussian kernel with the same joint distribution and distortion as the sensor design.It then compares conditional mutual information using data processing and causal invertibility of the Kalman filter.Source: Appendix C · p. 13