arXiv subjects

Edwin K. P. Chong

Publications and source records attributed to Edwin K. P. Chong.

48 records · Page 3Linked to original sources

arXiv2012

Learning in Hierarchical Social Networks

We study a social network consisting of agents organized as a hierarchical M-ary rooted tree, common in enterprise and military organizational structures. The goal is to aggregate information to solve a binary hypothesis testing problem. Each agent at a leaf of the tree, and only such an agent, makes a direct measurement of the underlying true hypothesis. The leaf agent then makes a decision and sends it to its supervising agent, at the next level of the tree. Each supervising agent aggregates the decisions from the M members of its group, produces a summary message, and sends it to its supervisor at the next level, and so on. Ultimately, the agent at the root of the tree makes an overall decision. We derive upper and lower bounds for the Type I and II error probabilities associated with this decision with respect to the number of leaf agents, which in turn characterize the converge rates of the Type I, Type II, and total error probabilities. We also provide a message-passing scheme involving non-binary message alphabets and characterize the exponent of the error probability with respect to the message alphabet size.

cs.SI↗

arXiv2012

Detection Performance in Balanced Binary Relay Trees with Node and Link Failures

We study the distributed detection problem in the context of a balanced binary relay tree, where the leaves of the tree correspond to $N$ identical and independent sensors generating binary messages. The root of the tree is a fusion center making an overall decision. Every other node is a relay node that aggregates the messages received from its child nodes into a new message and sends it up toward the fusion center. We derive upper and lower bounds for the total error probability $P_N$ as explicit functions of $N$ in the case where nodes and links fail with certain probabilities. These characterize the asymptotic decay rate of the total error probability as $N$ goes to infinity. Naturally, this decay rate is not larger than that in the non-failure case, which is $\sqrt N$. However, we derive an explicit necessary and sufficient condition on the decay rate of the local failure probabilities $p_k$ (combination of node and link failure probabilities at each level) such that the decay rate of the total error probability in the failure case is the same as that of the non-failure case. More precisely, we show that $\log P_N^{-1}=Θ(\sqrt N)$ if and only if $\log p_k^{-1}=Ω(2^{k/2})$.

cs.IT↗

arXiv2012

Error Probability Bounds for M-ary Relay Trees

We study the detection error probabilities associated with an M-ary relay tree, where the leaves of the tree correspond to identical and independent sensors. Only these leaves are sensors. The root of the tree represents a fusion center that makes the overall detection decision. Each of the other nodes in the tree is a relay node that combines M summarized messages from its immediate child nodes to form a single output message using the majority dominance rule. We derive tight upper and lower bounds for the Type I and II error probabilities at the fusion center as explicit functions of the number of sensors in the case of binary message alphabets. These bounds characterize how fast the error probabilities converge to 0 with respect to the number of sensors.

cs.IT↗

arXiv2012

Detection Performance of M-ary Relay Trees with Non-binary Message Alphabets

We study the detection performance of $M$-ary relay trees, where only the leaves of the tree represent sensors making measurements. The root of the tree represents the fusion center which makes an overall detection decision. Each of the other nodes is a relay node which aggregates $M$ messages sent by its child nodes into a new compressed message and sends the message to its parent node. Building on previous work on the detection performance of $M$-ary relay trees with binary messages, in this paper we study the case of non-binary relay message alphabets. We characterize the exponent of the error probability with respect to the message alphabet size $\mathcal D$, showing how the detection performance increases with $\mathcal D$. Our method involves reducing a tree with non-binary relay messages into an equivalent higher-degree tree with only binary messages.

cs.IT↗

arXiv2012

Submodularity and Optimality of Fusion Rules in Balanced Binary Relay Trees

We study the distributed detection problem in a balanced binary relay tree, where the leaves of the tree are sensors generating binary messages. The root of the tree is a fusion center that makes the overall decision. Every other node in the tree is a fusion node that fuses two binary messages from its child nodes into a new binary message and sends it to the parent node at the next level. We assume that the fusion nodes at the same level use the same fusion rule. We call a string of fusion rules used at different levels a fusion strategy. We consider the problem of finding a fusion strategy that maximizes the reduction in the total error probability between the sensors and the fusion center. We formulate this problem as a deterministic dynamic program and express the solution in terms of Bellman's equations. We introduce the notion of stringsubmodularity and show that the reduction in the total error probability is a stringsubmodular function. Consequentially, we show that the greedy strategy, which only maximizes the level-wise reduction in the total error probability, is within a factor of the optimal strategy in terms of reduction in the total error probability.

cs.IT↗

arXiv2012

Greedy Adaptive Compression in Signal-Plus-Noise Models

The purpose of this article is to examine the greedy adaptive measurement policy in the context of a linear Guassian measurement model with an optimization criterion based on information gain. In the special case of sequential scalar measurements, we provide sufficient conditions under which the greedy policy actually is optimal in the sense of maximizing the net information gain. In the general setting, we also discuss cases where the greedy policy is not optimal.

cs.IT↗

arXiv2011

On Bellman's principle with inequality constraints

We consider an example by Haviv (1996) of a constrained Markov decision process that, in some sense, violates Bellman's principle. We resolve this issue by showing how to preserve a form of Bellman's principle that accounts for a change of constraint at states that are reachable from the initial state.

math.OC↗

arXiv2011

Measurement Design for Detecting Sparse Signals

We consider the problem of testing for the presence (or detection) of an unknown sparse signal in additive white noise. Given a fixed measurement budget, much smaller than the dimension of the signal, we consider the general problem of designing compressive measurements to maximize the measurement signal-to-noise ratio (SNR), as increasing SNR improves the detection performance in a large class of detectors. We use a lexicographic optimization approach, where the optimal measurement design for sparsity level $k$ is sought only among the set of measurement matrices that satisfy the optimality conditions for sparsity level k-1. We consider optimizing two different SNR criteria, namely a worst-case SNR measure, over all possible realizations of a k-sparse signal, and an average SNR measure with respect to a uniform distribution on the locations of the up to k nonzero entries in the signal. We establish connections between these two criteria and certain classes of tight frames. We constrain our measurement matrices to the class of tight frames to avoid coloring the noise covariance matrix. For the worst-case problem, we show that the optimal measurement matrix is a Grassmannian line packing for most---and a uniform tight frame for all---sparse signals. For the average SNR problem, we prove that the optimal measurement matrix is a uniform tight frame with minimum sum-coherence for most---and a tight frame for all---sparse signals.

cs.IT↗

arXiv2011

Continuum Limits of Markov Chains with Application to Network Modeling

In this paper we investigate the continuum limits of a class of Markov chains. The investigation of such limits is motivated by the desire to model very large networks. We show that under some conditions, a sequence of Markov chains converges in some sense to the solution of a partial differential equation. Based on such convergence we approximate Markov chains modeling networks with a large number of components by partial differential equations. While traditional Monte Carlo simulation for very large networks is practically infeasible, partial differential equations can be solved with reasonable computational overhead using well-established mathematical tools.

cs.NI↗

arXiv2011

Error Probability Bounds for Binary Relay Trees with Crummy Sensors

We study the detection error probability associated with balanced binary relay trees, in which sensor nodes fail with some probability. We consider N identical and independent crummy sensors, represented by leaf nodes of the tree. The root of the tree represents the fusion center, which makes the final decision between two hypotheses. Every other node is a relay node, which fuses at most two binary messages into one binary message and forwards the new message to its parent node. We derive tight upper and lower bounds for the total error probability at the fusion center as functions of N and characterize how fast the total error probability converges to 0 with respect to N. We show that the convergence of the total error probability is sub-linear, with the same decay exponent as that in a balanced binary relay tree without sensor failures. We also show that the total error probability converges to 0, even if the individual sensors have total error probabilities that converge to 1/2 and the failure probabilities that converge to 1, provided that the convergence rates are sufficiently slow.

cs.IT↗

arXiv2011

Error Probability Bounds for Balanced Binary Relay Trees

We study the detection error probability associated with a balanced binary relay tree, where the leaves of the tree correspond to $N$ identical and independent detectors. The root of the tree represents a fusion center that makes the overall detection decision. Each of the other nodes in the tree are relay nodes that combine two binary messages to form a single output binary message. In this way, the information from the detectors is aggregated into the fusion center via the intermediate relay nodes. In this context, we describe the evolution of Type I and Type II error probabilities of the binary data as it propagates from the leaves towards the root. Tight upper and lower bounds for the total error probability at the fusion center as functions of $N$ are derived. These characterize how fast the total error probability converges to 0 with respect to $N$, even if the individual sensors have error probabilities that converge to 1/2.

cs.IT↗

arXiv2007

A Group Theoretic Model for Information

In this paper we formalize the notions of information elements and information lattices, first proposed by Shannon. Exploiting this formalization, we identify a comprehensive parallelism between information lattices and subgroup lattices. Qualitatively, we demonstrate isomorphisms between information lattices and subgroup lattices. Quantitatively, we establish a decisive approximation relation between the entropy structures of information lattices and the log-index structures of the corresponding subgroup lattices. This approximation extends the approximation for joint entropies carried out previously by Chan and Yeung. As a consequence of our approximation result, we show that any continuous law holds in general for the entropies of information elements if and only if the same law holds in general for the log-indices of subgroups. As an application, by constructing subgroup counterexamples we find surprisingly that common information, unlike joint information, obeys neither the submodularity nor the supermodularity law. We emphasize that the notion of information elements is conceptually significant--formalizing it helps to reveal the deep connection between information theory and group theory. The parallelism established in this paper admits an appealing group-action explanation and provides useful insights into the intrinsic structure among information elements from a group-theoretic perspective.

cs.IT↗