MathDB

Lyapunov positivity conjecture for coded Kalman filtering

Let B=diag⁡(b1,…,bk)B=\operatorname{diag}(b_1,\ldots,b_k) with bi<1b_i<1 for all ii, let H=diag⁡(h1,…,hn)H=\operatorname{diag}(h_1,\ldots,h_n), and let Γ∈Rn×k\Gamma\in\mathbb{R}^{n\times k} and J∈Rk×kJ\in\mathbb{R}^{k\times k}. For fixed BB and HH, let JJ be the unique solution of

J=BJB−ΓT(I+HΠH)−1Γ+BΓTΓB.J=BJB-\Gamma^T(I+H\Pi H)^{-1}\Gamma+B\Gamma^T\Gamma B.

Here Π\Pi is diagonal. Lyapunov positivity conjecture. For every feasible problem instance, the infimum

inf⁡Π⪰0, J≻0, Γ:the Lyapunov equation holds, Π is diagonalTr⁡(Π)\inf_{\substack{\Pi\succeq 0,\ J\succ 0,\ \Gamma:\\ \text{the Lyapunov equation holds},\ \Pi\text{ is diagonal}}}\operatorname{Tr}(\Pi)

has an optimal solution Γ∗\Gamma^* with exactly one nonzero entry in each column. This conjecture would establish the necessity of the partitioning property for vector sources and MIMO channels, a question that is unknown in the paper; it is a matrix-algebraic conjecture linking positivity in the Lyapunov optimization to a sparse encoder structure.

Sources & referencesView supporting material

Primary source

Barron Han, Oron Sabag, Victoria Kostina and Babak Hassibi, “Coded Kalman Filtering over MIMO Gaussian Channels with Feedback”, arXiv:2406.17196 (2025).

kalman-filteringlyapunov-equationsmimo-channels

Progress summary

Refreshed
Open

The conjecture remains open in the general multi-dimensional case, with only numerical evidence and proofs in scalar special cases.

The conjecture asserts that an optimal encoder can always be chosen with exactly one nonzero entry in each column, which would establish the necessity of partitioning for vector sources and MIMO channels. It appears as Conjecture 1 in the 2024 paper on coded Kalman filtering over MIMO Gaussian channels with feedback.

Known results

  • The conjecture is proved when either the source or the channel is scalar (2024).
  • Under the conjecture, partitioning is necessary and sufficient for finite asymptotic error with linear zero-delay JSCC (2024).
  • Numerical tests support the conjecture for all simulated systems with n,k∈{2,3}n,k\in\{2,3\} (2024).
  • Necessity for general vector sources and MIMO channels remains conditional on the conjecture (2024).

Current status (as of August 2026): The conjecture is open for general vector sources and MIMO channels; scalar cases are proved, and numerical evidence is the only reported support beyond them.

Sources

Solutions 0

No solutions have been posted yet.