Se Yong Park

The finite-dimensional Witsenhausen counterexample

Recently, we considered a vector version of Witsenhausens counterexample and used a new lower bound to show that in that limit of infinite vector length, certain quantization-based strategies are provably within a constant factor of the optimal cost for all possible problem parameters. In this paper, finite vector lengths are considered with the vector length being viewed as an additional problem parameter. By applying the “sphere-packing” philosophy, a lower bound to the optimal cost for this finite-length problem is derived that uses appropriate shadows of the infinite-length bounds. We also introduce latticebased quantization strategies for any finite length. Using the new finite-length lower bound, we show that the lattice-based strategies achieve within a constant factor of the optimal cost uniformly over all possible problem parameters, including the vector length. For Witsenhausens original problem — which corresponds to the scalar case — lattice-based strategies attain within a factor of 8 of the optimal cost. Based on observations in the scalar case and the infinite-dimensional case, we also conjecture what the optimal strategies could be for any finite vector length.

Linear Function Computation in Networks: Duality and Constant Gap Results

In linear function computation, multiple source nodes communicate across a relay network to a single destination whose goal is to recover linear functions of the original source data. When the relay network is a linear deterministic network, a duality relation is established between function computation and broadcast with common messages. Using this relation, a compact sufficient condition is found describing those cases where the cut-set bound is tight. These insights are used to develop results for the case where the relay network contains Gaussian multiple-access channels. The proposed scheme decouples the physical and network layers. Using lattice codes for both source quantization and computation in the physical layer, the original Gaussian sources are converted into discrete sources and the Gaussian network into a linear deterministic network. Network codes for computing functions of discrete sources across the deterministic network are then found by applying the duality relation. The distortion for computing the sum of an arbitrary number of independent Gaussian sources over the Gaussian network is proven to be within a constant factor of the optimal performance. Furthermore, the constant factor results are extended to include asymmetric functions for the case of two sources.

Approximately Optimal Solutions to the Finite-Dimensional Witsenhausen Counterexample

Recently, a vector version of Witsenhausens counterexample was considered and it was shown that in the asymptotic limit of infinite vector length, certain vector-quantization-based control strategies are provably within a constant factor of the asymptotically optimal cost for all possible problem parameters. While suggestive, a constant factor result for the finite-dimensional problem has remained elusive. In this paper, we provide a resolution to this issue. By applying a large-deviation “sphere-packing” philosophy, we derive a lower bound to the optimal cost for the finite dimensional case that uses appropriate shadows of an existing vector lower bound that is the same for all dimensions. Using this new lower bound, we show that good lattice-quantization-based control strategies achieve within a constant factor of the optimal cost uniformly over all possible problem parameters, including the vector length. For Witsenhausens original problem-which is the scalar case-the gap between regular lattice-quantization-based strategies and the lower bound is provably never more than a factor of 100, and computer calculations strongly suggest that the factor in fact may be no larger than 8. Finally, to obtain a numerical understanding of the possible room for improvement in costs using alternative strategies, we also include numerical comparison with strategies that are conjectured to be optimal. Using this comparison, we posit that there is more room for improvement in our lower bounds than in our upper bounds.

An algebraic mincut-maxflow theorem

Can we design a communication network just like a huge linear time-invariant filter? To answer this question, we generalize the celebrated mincut-maxflow theorem to linear time-invariant networks where edges are labeled with transfer functions instead of integer capacity constraints. We prove that when the transfer functions are linear time-invariant, the fundamental design limit, mincut, is achievable by a linear time-invariant scheme regardless of the topology of the network. Whereas prior works are based on layered networks, our proof has a novel way of converting an arbitrary relay network to an equivalent acyclic single-hop relay network, which we call Network Linearization. This theorem also reveals a strong connection between network coding and linear system theory.Can we design a communication network just like a huge linear time-invariant filter? To answer this question, we generalize the celebrated mincut-maxflow theorem to linear time-invariant networks where edges are labeled with transfer functions instead of integer capacity constraints. We prove that when the transfer functions are linear time-invariant, the fundamental design limit, mincut, is achievable by a linear time-invariant scheme regardless of the topology of the network. Whereas prior works are based on layered networks, our proof has a novel way of converting an arbitrary relay network to an equivalent acyclic single-hop relay network, which we call Network Linearization. This theorem also reveals a strong connection between network coding and linear system theory.

A constant-factor approximately optimal solution to the Witsenhausen counterexample

Despite its simplicity (two controllers and otherwise LQG), Witsenhausens counterexample is one of the long-standing open problems in stochastic distributed control. Recently, it was proved that an asymptotic vector “relaxation” can be solved to within a constant factor of the optimal cost. A parallel result is shown here for the original scalar problem. Between linear strategies and explicit-signalling-based nonlinear strategies, the optimal performance can be obtained to within a constant factor that is uniformly bounded regardless of the problem parameters. The key contribution is a new lower bound that is much tighter thanWitsenhausens bound for some parameter values.

Carry-free models and beyond

The generalized deterministic models recently proposed by Niesen and Maddah-Ali [1] successfully capture real-interference alignment as observed in Gaussian models. Simpler deterministic models, like ADT models [2], cannot demonstrate this phenomenon because they are limited in the set of channel gains they can model. This paper reinterprets the Niesen and Maddah-Ali models through the lens of carry-free operations. We further explore these carry-free models by considering i.i.d. unknown fading networks. In the unknown fading context, a carry-free model can be further simplified to a max-superposition model, where signals are superposed by a nonlinear max operation. Unlike in relay-networks with known fading and linear superposition, we find that decode-and-forward can perform arbitrarily better than compress-and-forward in max-superposition relay networks with unknown fading.

It may be “easier to approximate” decentralized infinite-horizon LQG problems

We consider scalar decentralized average-cost infinite-horizon LQG problems with two controllers. It is shown that when two controllers are asymmetric, the linear controller performance can be an arbitrary factor worse than the optimal performance. To fix this problem, we propose a set of nonlinear controllers parameterized by only a few variables, and prove that the proposed set contains an approximately optimal solution that achieves within a constant ratio of the optimal cost. This insight is conveyed using bit-oriented deterministic models that elucidate the nature of the ongoing implicit communication that must occur.

Network coding meets decentralized control: Capacity-stabilizabililty equivalence

The main difference between centralized and decentralized control is the communication. Controllers in a decentralized system can communicate with each other to achieve their common goal. In this paper, we argue that even linear time-invariant controllers in a decentralized linear system “communicate” via linear network coding to stabilize the plant. To justify this argument, we propose an algorithm to “externalize” the implicit communication between controllers that we believe must be occurring to stabilize the plant. Based on this, we show that the stabilizability condition for decentralized linear systems comes from an underlying communication limit, which can be described by an algebraic mincut-maxflow theorem.

Function computation in networks: Duality and constant gap results

In the linear function computation problem, multiple source nodes communicate across a relay network to a single destination whose goal is to recover a linear function of the original source data. For the case when the relay network is a deterministic network, a duality relation is established between the linear function computation problem and the standard, well-known multicast problem. Using this relation, a compact sufficient condition is found describing those cases where the cut-set bound is tight. Then, these insight are used to develop results for the case where the relay network contains Gaussian superposition channels. Assuming the original source sequences are independent Gaussians, the resulting distortion for the recovery of their sum is found to within a constant gap.

Intermittent Kalman filtering with adversarial erasures: Eigenvalue cycles again

We consider intermittent Kalman filtering with adversarial erasures, and characterize the observability condition. Like intermittent Kalman filtering with random erasures, the concept of eigenvalue cycles turns out to be crucial in the characterization. Moreover, the nonuniform sampling which breaks the eigenvalue cycles can also dramatically increase Kalman filtering robustness against adversarial erasures. Precisely, the system becomes observable as long as the ratio of erasures is strictly less than 1.