This paper establishes the capacity region for a class of source coding function computation setups, where sources of information are available at the nodes of a tree and where a function of these sources must be computed at its root. The capacity region holds for any function as long as the sources’ joint distribution satisfies a certain Markov criterion. This criterion is met, in particular, when the sources are independent. This result recovers the capacity regions of several function computation setups. These include the point-to-point communication setting with arbitrary sources, the noiseless multiple access network with conditionally independent sources, and the cascade network with Markovian sources.

Distributed Function Computation Over a Rooted Directed Tree

In forthcoming years, the Internet of Things (IoT) will connect billions of smart devices generating and uploading a deluge of data to the cloud. If successfully extracted, the knowledge buried in the data can significantly improve the quality of life and foster economic growth. However, a critical bottleneck for realizing the efficient IoT is the pressure it puts on the existing communication infrastructures, requiring transfer of enormous data volumes. Aiming at addressing this problem, we propose a novel architecture dubbed Condense which integrates the IoT-communication infrastructure into the data analysis. This is achieved via the generic concept of network function computation. Instead of merely transferring data from the IoT sources to the cloud, the communication infrastructure should actively participate in the data analysis by carefully designed en-route processing. We define the Condense architecture, its basic layers, and the interactions among its constituent modules. Furthermore, from the implementation side, we describe how Condense can be integrated into the Third Generation Partnership Project (3GPP) machine type communications (MTCs) architecture, as well as the prospects of making it a practically viable technology in a short time frame, relying on network function virtualization and software-defined networking. Finally, from the theoretical side, we survey the relevant literature on computing atomic functions in both analog and digital domains, as well as on function decomposition over networks, highlighting challenges, insights, and future directions for exploiting these techniques within practical 3GPP MTC architecture.

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CONDENSE: A Reconfigurable Knowledge Acquisition Architecture for Future 5G IoT

In this paper, the problem of function computation with privacy and secrecy constraints is considered. The considered model consists of three legitimate nodes (i.e., two transmitters, Alice and Bob, and a fusion center that acts as the receiver) that observe correlated sources and are connected by noiseless public channels, and an eavesdropper Eve who has full access to the public channels and also has its own source observations. The fusion center would like to compute a function of the distributed sources within a prefixed distortion level under a certain distortion metric. To facilitate the function computation, Alice and Bob will send messages to the fusion center. Different from the existing setups in function computation, we assume that there is a privacy constraint on the sources at Alice and Bob. In particular, Alice and Bob would like to enable the fusion center to compute the function, but at same time, they do not want the fusion center to learn too much information about the source observations. We introduce a quantity to precisely measure the privacy leakage to the fusion center. In addition to this privacy constraint, we also have a secrecy constraint to Eve and use equivocation of sources to measure this quantity. Under this model, we study the tradeoffs among message rates, private information leakage, equivocation, and distortion. We first consider a scenario that has only one transmitter, i.e., the source at Bob is empty, and fully single-letter characterize the corresponding regions. Then, we consider the more general case and provide both outer and inner bounds on the corresponding regions.

On Function Computation With Privacy and Secrecy Constraints

A key aspect of many resource allocation problems is the need for the resource controller to compute a function, such as the max or arg max, of the competing users metrics. Information must be exchanged between the competing users and the resource controller in order for this function to be computed. In many practical resource controllers the competing users' metrics are communicated to the resource controller, which then computes the desired extremization function. However, in this paper it is shown that information rate savings can be obtained by recognizing that controller only needs to determine the result of this extremization function. If the extremization function is to be computed losslessly, the rate savings are shown in most cases to be at most 2 bits independent of the number of competing users. Motivated by the small savings in the lossless case, simple achievable schemes for both the lossy and interactive variants of this problem are considered. It is shown that both of these approaches have the potential to realize large rate savings, especially in the case where the number of competing users is large. For the lossy variant, it is shown that the proposed simple achievable schemes are in fact close to the fundamental limit given by the rate distortion function.

Overhead Performance Tradeoffs - A Resource Allocation Perspective

A receiver wants to compute a function of two correlated sources separately observed by two transmitters. One of the transmitters may send a possibly private message to the other transmitter in a cooperation phase before both transmitters communicate to the receiver. For this network configuration this paper investigates both a function computation setup, wherein the receiver wants to compute a given function of the sources exactly, and a rate distortion setup, wherein the receiver wants to compute a given function within some distortion. 
For the function computation setup, a general inner bound to the rate region is established and shown to be tight in a number of cases: partially invertible functions, full cooperation between transmitters, one-round point-to-point communication, two-round point-to-point communication, and the cascade setup where the transmitters and the receiver are aligned. In particular it is shown that the ratio of the total number of transmitted bits without cooperation and the total number of transmitted bits with cooperation can be arbitrarily large. Furthermore, one bit of cooperation suffices to arbitrarily reduce the amount of information both transmitters need to convey to the receiver. 
For the rate distortion version, an inner bound to the rate region is exhibited which always includes, and sometimes strictly, the convex hull of Kaspi-Berger's related inner bounds. The strict inclusion is shown via two examples.

On Cooperation in Multi-Terminal Computation and Rate Distortion

A receiver wants to compute a function f of two correlated sources X and Y and side information Z. What is the minimum number of bits that needs to be communicated by each transmitter? In this paper, we derive inner and outer bounds to the rate region which coincide in the cases where f is partially invertible and where one of the sources is constant. From the former case we recover the Slepian-Wolf rate region.

Computing a function of correlated Sources: A rate region

A receiver wants to compute a function of two correlated sources separately observed by two transmitters. One of the transmitters is allowed to cooperate with the other transmitter by sending it some data before both transmitters convey information to the receiver. Assuming noiseless communication, what is the minimum number of bits that needs to be communicated by each transmitter to the receiver for a given number of cooperation bits? In this paper, first a general inner bound to the above three dimensional rate region is provided and shown to be tight in a number of interesting settings: the function is partially invertible, full cooperation, one-round point-to-point communication, two-round point-to-point communication, and cascade. Second, the related Kaspi-Berger rate distortion problem is investigated where the receiver now wants to recover the sources within some distortion. By using ideas developed for establishing the above inner bound, a new rate distortion inner bound is proposed. This bound always includes the time sharing of Kaspi-Berger's inner bounds and inclusion is strict in certain cases.

On cooperation in multi-terminal computation and rate distortion

A receiver wants to compute a function f of two correlated sources X and Y and side information Z. What is the minimum number of bits that needs to be communicated by each transmitter? In this paper, we derive inner and outer bounds to the rate region of this problem which coincide in the cases where f is partially invertible and where the sources are independent given the side information. From the former case we recover the Slepian-Wolf rate region and from the latter case we recover Orlitsky and Roche’s single source result.

Milad Sefidgaran

Papers

Distributed Function Computation Over a Rooted Directed Tree

Computing a function of correlated Sources: A rate region

On Cooperation in Multi-Terminal Computation and Rate Distortion

On cooperation in multi-terminal computation and rate distortion

Computing a Function of Correlated Sources