Computation of Channel Capacity Based on Self-Concordant Functions

Abstract

The computation of channel capacity is a classical issue in information theory. We prove that algorithms based on self-concordant functions can be used to deal with such issues, especially when constrains are included. A new algorithm to compute the channel capacity per unit cost is proposed. The same view is suited to the computation of maximum entropy. All the algorithms are of polynomial time.

1. Introduction

The computation of channel capacity is a classical issue in information theory. Much attention has been paid to it [1–3]. Despite some progress [3], the Arimoto-Blahut algorithm [4–7] is still the most used method. This method has been developed into a general algorithm, alternative minimization method [8], and has found applications in various fields [9–11]. Recently, the computation of channel capacity with constraints, such as the computation of capacity-cost function, is becoming increasingly important [5, 12–15]. The Arimoto-Blahut algorithm is very limited in dealing with such issues. Due to lack of appropriate means to eliminate the Lagrange multipliers, the method in [5], which is based on Arimoto-Blahut algorithm, cannot obtain the optimal point.

The computation of channel capacity has always been an optimization problem, but no one optimization algorithm, such as the gradient method and Newton’s method, and so forth, has become the mainstream method. One reason is that so far no one optimization algorithm can be as effective as the Arimoto-Blahut algorithm. However, with advances in optimization theory and with the increasing importance of constrained channel capacity issues, the situation is changing.

In this paper, some new algorithms based on self-concordant function are proposed. Besides the effectiveness—it is of polynomial time—the algorithm can compute constrained channel capacity easily. In Section 2, we introduced the definition of the self-concordant function and the main properties, as well as some algorithms for convex optimization problems. In Section 3, we give some new algorithms for channel capacity with or without constraints and for channel capacity per unit cost. Because it is based on the self-concordant function, these algorithms are of polynomial time. Some conclusions are given in Section 4.

2. Self-Concordant Function and Convex Programming

For convenience, in this section, we make a brief introduction to self-concordant functions, including the definition, main propositions, and algorithms for convex optimization.

2.1. Self-Concordant Function

Self-concordant function theory, published by Nesterov and Nemirovskii in 1994 [16, 17], is of landmark importance for establishing general interior point polynomial-time algorithms for convex programming. The theory reveals the main property of the interior point method for linear programming initiated by Karmarkar in 1984 [18]: self-concordance of the barrier function.

Suppose that 𝑓∶𝐷→𝑅 is three times continuously differentiable and strictly convex, where the domain 𝐷 is an open convex subset in 𝑅^𝑛.

Definition 1 (see [19]). Let 𝑛=1. A convex function 𝑓∶𝑅→𝑅 is self-concordant if ||𝑓^(𝑥)||≤2𝑓^(𝑥)^3/2(1) for all 𝑥∈𝐷.

Definition 2. A function 𝑓∶𝑅^𝑛→𝑅 is self-concordant if it is self-concordant along every line in its domain, that is, if the function 𝑔(𝑡)=𝑓(𝑥+𝑡𝑣) is a self-concordant function of 𝑡 for all 𝑥∈𝐷 and for all 𝑣∈𝑅^𝑛.
Since linear and convex quadratic functions have zero third derivatives, they are trivial self-concordant functions. The most common nontrivial self-concordant function is the logarithmic function −log 𝑥.
We can get more self-concordant functions due to the following simple and useful combination rules [19].

Proposition 3. If 𝑓₁, 𝑓₂ are self-concordant on 𝐷 and 𝑎₁≥1, 𝑎₂≥1, then 𝑎₁𝑓₁+𝑎₂𝑓₂ is self-concordant on D.

Proposition 4. If 𝑓 is self-concordant on 𝐷⊂𝑅^𝑛, and 𝐴∶𝑅^𝑚→𝑅^𝑛 is a linear map, 𝑏∈𝑅^𝑛, then 𝑓(𝐴𝑥+𝑏) is self-concordant on {𝑥∈𝑅^𝑚∣𝐴𝑥+𝑏∈𝐷}.

Proposition 5. If 𝑓_𝑖 is self-concordant on 𝐷_𝑖⊂𝑅^𝑛𝑖,𝑖=1,2,…,𝑚. Let 𝐷=𝐷₁×⋯×𝐷_𝑚. Then, 𝑓(𝑥₁,…,𝑥_𝑚)=𝑓₁(𝑥₁)+⋯+𝑓_𝑚(𝑥_𝑚) is self-concordant on 𝐷.

2.2. Newton’s Method for Self-Concordant Functions

Consider the convex optimization problemmin_𝐷𝑓(𝑥),(2) where 𝐷⊂𝑅^𝑛 is a bounded, closed, convex subset with a nonempty interior. Let 𝑓(𝑥) be a convex function.

Damped Newton Algorithm
Given an initial point 𝑥₀∈int(𝐷), with tolerance 𝜀>0.

Repeat
(1)Compute the Newton step Δ𝑥_𝑛𝑡=−∇²𝑓(𝑥)⁻¹∇𝑓(𝑥),(3) and Newton decrement 𝜆(𝑓,𝑥)=∇𝑓(𝑥)^𝑇∇²𝑓(𝑥)⁻¹∇𝑓(𝑥)^1/2.(4)
(2) Stopping criterion. Quit if 𝜆(𝑓,𝑥)²/2≤𝜀.
(3) If 𝜆(𝑓,𝑥)>1/4, then 𝑥=𝑥+11+𝜆(𝑓,𝑥)Δ𝑥^𝑛𝑡,(5)
else 𝑥=𝑥+Δ𝑥_𝑛𝑡.(6)
If 𝑓(𝑥) is a self-concordant function on 𝐷, then the damped Newton algorithm above possesses the following good properties:(i)all 𝑥∈int(𝐷) if 𝑥₀∈int(𝐷).(ii)if 𝜆(𝑓,𝑥)>0.25, then 𝑓(𝑥_𝑖)−𝑓(𝑥_𝑖+1)≥0.026.(iii)if at some iteration 𝑖 we have 𝜆(𝑓,𝑥_𝑖)≤0.25, then we are in the region of quadratic convergence of the method, that is, for every 𝑗≥𝑖, 𝜆𝑓,𝑥_𝑗+1≤2𝜆²𝑓,𝑥_𝑗≤1𝑓𝑥_𝑗2𝜆𝑓,𝑥^𝑗,−𝑝^∗≤𝜆²𝑓,𝑥_𝑗21−𝜆𝑓,𝑥_𝑗.(7)(iv)the number of Newton steps required to find a point 𝑥∈int(𝐷) with 𝜆(𝑓,𝑥_𝑖)≤𝜀<1 is bounded by𝐶𝑓𝑥⁽⁰⁾−𝑝^∗+log₂log₂1𝜀(8) for some constant 𝐶.
For the convex optimization problem with equality constraints 𝐴𝑥=𝑏, Newton’s method and the bound (8) are still valid as long as the initial point 𝑥₀∈int(𝐷) satisfies 𝐴𝑥₀=𝑏, and the Newton step Δ𝑥_𝑛𝑡 and decrement 𝜆(𝑓,𝑥) are computed as follows: ⎡⎢⎢⎣∇²𝑓(𝑥)𝐴^𝑇𝜆(𝑓𝐴,𝑥)=0Δ⎤⎥⎥⎦𝑥⎡⎢⎢⎣Δ_𝑇𝑛𝑡𝑤𝑥^𝑛𝑡⎤⎥⎥⎦=⎡⎢⎢⎣−∇𝑓0(𝑥)⎤⎥⎥⎦,∇²𝑓(𝑥)Δ𝑥_𝑛𝑡^1/2.(9)
For general convex optimization problems are misn.t𝑓𝑓_0𝑖(𝑥),𝐴𝑥(𝑥=)≤𝑏,0,𝑖=1,…,𝑚,(10) where 𝑓₀,…,𝑓_𝑚∶𝑅^𝑛→𝑅 are convex and twice continuously differentiable, and 𝐴 is a 𝑝×𝑛 matrix with rank (𝐴)=𝑝<𝑛. By means of the logarithmic barrier function, (10) can be formulated approximately as an equality constrained problem as follows: misn.t𝐴𝑡𝑓𝑥₀(=𝑥)𝑏+,𝜙(𝑥),(11) where 𝜙(𝑥)=−^𝑚_𝑖=1log−𝑓_𝑖(𝑥)(12) is called the logarithmic barrier function for (10). Let 𝑥^∗(𝑡) be the optimal point of (11) and let 𝑝^∗ be the optimal value of (10), then 𝑓₀(𝑥^∗(𝑡))−𝑝^∗≤𝑚/𝑡. Therefore, we can simply take 𝑡=𝑚/𝜀 and solve the equality constrained problem (11) using Newton’s method to get an 𝜀-solution of (10), and if the objective function of (11) is self-concordant, then the number of Newton’s iterate is bounded by (8). However, this method usually requires a large 𝑡, which may bring about a large 𝑓(𝑥⁽⁰⁾)−𝑝^∗ and a large number of Newton’s iterate, thus it is rarely used. A commonly used method is the path-following method as follows.

Path-Following Method
Given strictly feasible 𝑥,𝑡=𝑡₀>0,𝜇>1, with tolerance 𝜀>0.

Repeat
(1)Centering step.Compute 𝑥^∗(𝑡) by minimizing 𝑡𝑓₀(𝑡)+𝜑, subject to 𝐴𝑥=𝑏, starting at 𝑥.(2)Update. 𝑥=𝑥^∗(𝑡).(3)Stopping criterion. Quit if 𝑚/𝑡<𝜀.(4)Increase 𝑡. 𝑡=𝜇𝑡.

If for any 𝑡>0,𝑓(𝑥)=𝑡𝑓₀(𝑥)+𝜑(𝑥) is self-concordant, then the total number of Newton steps in the path-following method, not counting the initial centering step, is𝑁=log𝑚/𝑡⁰𝜀log𝜇×𝑚(𝜇−1−log𝜇)+log²log₂1𝜀.(13)

The term in brackets is the iterate number of 𝑡 from 𝑡₀ to 𝑚/𝜀, and the term in parentheses is the iterate number of Newton’s method per centering step.

3. Algorithms Based on Self-Concordant Functions for Channel Capacity

In this section, it is proved that the channel capacity, with or without constrains, can be computed by the path-following method in polynomial time. Meanwhile, we prove that channel capacity per unit cost is a single-peak function of the expected cost.

3.1. Channel Capacity without Constrains

Let 𝑋={𝑥₁,…,𝑥_𝑚} and 𝑌={𝑦₁,…,𝑦_𝑛} be the input and output alphabets, respectively. Let 𝑝(𝑥)=(𝑝(𝑥₁),…,𝑝(𝑥_𝑚)) and 𝑞(𝑦)=(𝑞(𝑦₁),…,𝑞(𝑦_𝑛)) be the distributions of 𝑋 and 𝑌, respectively. Let 𝑇=(𝑝(𝑦_𝑗∣𝑥_𝑖))_𝑚×𝑛=(𝑝_𝑖𝑗)_𝑚×𝑛be a transition matrix from 𝑋 to 𝑌. Hence,𝑞𝑦_𝑗=𝑝𝑥₁𝑝_1𝑗+⋯+𝑝𝑥_𝑚𝑝_𝑚𝑗,𝑗=1,…,𝑛.(14)

If 𝐼(𝑋;𝑌) is the mutual information between 𝑋 and 𝑌, then𝐼(𝑋;𝑌)=_𝑖,𝑗𝑝𝑥_𝑖𝑝𝑦_𝑗∣𝑥_𝑖log𝑝𝑦^𝑗∣𝑥_𝑖𝑞𝑦_𝑗(15)=_𝑖,𝑗𝑝𝑥_𝑖𝑝𝑦_𝑗∣𝑥_𝑖log𝑝𝑥^𝑖∣𝑦_𝑗𝑝𝑥_𝑖.(16)

Channel capacity is defined as follows:𝐶=max_𝑝(𝑥)𝐼(𝑋;𝑌).(17) Equation (16) is a function of 𝑝(𝑥) and 𝑝(𝑥|𝑦) and we denote it by 𝐼(𝑝(𝑥);𝑝(𝑥|𝑦)).

The Arimoto-Blahut algorithm utilized (16) as follows. In (16), both 𝑝(𝑥) and 𝑝(𝑥|𝑦) are unknown, but given 𝑝(𝑥), (16) can get its maximum at𝑝𝑥_𝑖∣𝑦_𝑗=𝑝𝑥^𝑖𝑝𝑦_𝑗∣𝑥_𝑖𝑖=1,∑…,^𝑚𝑘=1𝑚𝑝.𝑗𝑥=_𝑘1𝑝,…𝑦_𝑗,𝑛∣,𝑥_𝑘,(18) and given 𝑝(𝑥|𝑦), (16) can get its maximum at𝑝𝑥_𝑖=∏^𝑗𝑝𝑥_𝑖∣𝑦_𝑗^{𝑝(𝑦𝑗∣𝑥𝑖)}∑^𝑚𝑖=1∏_𝑗𝑝𝑥_𝑖∣𝑦_𝑗^{𝑝(𝑦𝑗∣𝑥𝑖)},𝑖=1,…,𝑚.(19) If 𝑝⁰(𝑥)=(1/𝑚,…,1/𝑚), then the channel capacity can be approximated by computing 𝑝⁰(𝑥|𝑦),𝑝¹(𝑥),𝑝¹(𝑥|𝑦),… through (18) and (19), alternatively. Let 𝐶(𝑁+1,𝑁) be the mutual information 𝐼(𝑝^𝑁+1(𝑥);𝑝^𝑁(𝑥|𝑦)), then,𝐶−𝐶(𝑁+1,𝑁)≤log𝑚𝑁.(20)

In order to get an algorithm based on the self-concordant function, we utilize (15). Let𝑐_𝑖=−^𝑛_𝑗=1𝑝𝑦_𝑗∣𝑥_𝑖log𝑝𝑦_𝑗∣𝑥_𝑖.(21) By (15), the channel capacity can be expressed as follows:min^𝑚_𝑖=1𝑐_𝑖𝑝_𝑖+^𝑛_𝑗=1s.t𝑞_𝑗𝑞_𝑗log𝑞_𝑗,=^𝑚_𝑖=1𝑝𝑝_𝑖𝑝𝑦_𝑗∣𝑥_𝑖,𝑗=1,…,𝑛,₁𝑝+⋯+𝑝_𝑚=1,₁>0,…,𝑝_𝑚>0.(22) It is a standard convex optimization problem. Unfortunately, the objective function of (22) is not self-concordant, even when the logarithmic barrier −log 𝑝_𝑖 is added.

It seems that we should add 𝑞_𝑗>0 as constraints for (22) since there are log𝑞_𝑗 in the objective function of (22). But the 𝑞_𝑗 are not independent variables. It can be computed by (14) and maintain positive as long as all the 𝑝_𝑖>0,𝑖=1,2,…,𝑚, hold. Nevertheless, we still add the logarithmic barrier −log 𝑞_𝑗 in the objective of (22) to get a self-concordant objective function.

For 𝑡>0, consider the convex optimization problem as follows:min𝑡^𝑚_𝑖=1𝑐_𝑖𝑝_𝑖+𝑡^𝑛_𝑗=1𝑞_𝑗log𝑞_𝑗−^𝑚_𝑖=1log𝑝_𝑖−^𝑛_𝑗=1s.t𝑞_𝑗log𝑞_𝑗,=^𝑚_𝑖=1𝑝𝑝_𝑖𝑝𝑦_𝑗∣𝑥_𝑖,𝑗=1,…,𝑛,₁+⋯+𝑝_𝑚=1.(23) In order to show the self-concordance of the objective function of (23), we need Proposition 6.

Proposition 6. For any 𝑡>0,𝑓(𝑥)=𝑡𝑥log𝑥−log𝑥 is a self-concordant function on 𝑥>0.

Proof. We have 𝑓^′(𝑥)=𝑡log𝑥+𝑡−1𝑓_𝑥,(𝑥)=𝑡𝑥+𝑥1²,||𝑓^(𝑥)||=𝑡𝑥²+2𝑥³.(24) Thus ||𝑓^(𝑥)||𝑓^(𝑥)^3/2=(𝑡𝑥+2)(𝑡𝑥+1)^3/2≤2.(25)

By Propositions 3–6, the objective function of (23) is self-concordant, so we can solve it by the path following method, and the number of Newton iterations is bounded by (13).

Equation (13) shows that solving (23) by the path following method is a polynomial time algorithm. In addition, a main advantage is that the algorithm can deal with constrains very easily.

3.2. Channel Capacity with Constrains and Channel Capacity Per Unit Cost

Consider the channel capacity with constraints as follows:masx.t𝐼_𝑚(𝑋;𝑌),_𝑗=1𝑝𝑎_𝑖𝑗𝑝_𝑗≤𝐸_𝑖,𝑖=1,…,𝑟,₁𝑝+⋯+𝑝_𝑚=1,₁≥0,…,𝑝_𝑚≥0.(26) For 𝑟=2, [5] gave some algorithms based on the Arimoto-Blahut method. Because one cannot eliminate the Lagrange multiplier, the algorithms in [5] cannot get the optimal solution for (26).

By adding logarithmic barriers for the inequality constraints in the objective of (26) as follows, we can get a polynomial time algorithm for (26):min𝑡^𝑚_𝑖=1𝑐_𝑖𝑝_𝑖+𝑡^𝑛_𝑚𝑗=1_𝑖=1𝑝_𝑖𝑝𝑦_𝑗∣𝑥_𝑖log^𝑚_𝑖=1−_𝑚𝑝_𝑖𝑝𝑦_𝑗∣𝑥_𝑖_𝑖=1log𝑝_𝑖−^𝑛_𝑗=1log^𝑚_𝑖=1−_𝑟𝑝_𝑖𝑝𝑦_𝑗∣𝑥_𝑖_𝑘=1log𝐸_𝑘−^𝑚_𝑖=1𝑝_𝑖𝑎_𝑟𝑖s.t𝑝₁,+⋯+𝑝_𝑚=1.(27) The objective function of (27) is self-concordant, so we can solve (27) by the path-following method.

Channel capacity per unit cost is one of the typical problems of channel capacity with constraints [15, 20–22]. By [15], channel capacity per unit cost can be computed by𝐶=sup_𝛽>0𝐶(𝛽)𝛽,(28) where 𝐶(𝛽) is the capacity-cost function, which is the solution of the following constrained problem [12]:masx.t𝐼_𝑚(𝑋;𝑌),_𝑗=1𝑝𝑎_𝑗𝑝_𝑗≤𝛽,₁𝑝+⋯+𝑝_𝑚=1,₁≥0,…,𝑝_𝑚≥0.(29) Equation (29) is a special case of (26) (𝑟=1). Here we discuss the channel capacity per unit cost in a more general manner.

Theorem 7. Let 𝐷⊆𝑅^𝑛 be a convex set. Let 𝐹(𝑥) be a concave function on 𝐷 and 𝐺(𝑥) a convex function on 𝐷. Let 𝑎=inf{𝐺(𝑥)|𝑥∈𝐷},𝑏=sup{𝐺(𝑥)|𝑥∈𝐷}, and suppose 𝑎≥0. Let 𝐶(𝛽)=sup{𝐹(𝑥)|𝐺(𝑥)≤𝛽}. Then,(1)𝐶(𝛽)is a concave increasing function,(2)let 𝜌(𝛽)=𝐶(𝛽)𝛽,(30) then 𝜌(𝛽) is either a unimodal function or a monotone function. sup_𝐷𝐹(𝑥)𝐺(𝑥)=sup_{𝑎<𝛽<𝑏}𝐶(𝛽)𝛽.(31)

Proof. (1) Let 𝛽₁>𝛽₂, then {𝑥|𝐺(𝑥)≤𝛽₂}⊆{𝑥|𝐺(𝑥)≤𝛽₁}. Therefore, 𝐶(𝛽₂)≤𝐶(𝛽₁), For any 𝜀>0, there are 𝑥₁ such that 𝐺(𝑥₁)≤𝛽₁ and 𝐶(𝛽₁)=𝐹(𝑥₁)+𝜀, and there are 𝑥₂ such that 𝐺(𝑥₂)≤𝛽₂ and 𝐶(𝛽₂)=𝐹(𝑥₂)+𝜀. Since 𝐺(𝑥) is convex and 𝐹(𝑥) is concave, we get 𝐺𝛼𝑥₁+(1−𝛼)𝑥₂≤𝛼𝐺𝑥₁+(1−𝛼)𝐺𝑥₂≤𝛼𝛽₁𝛼𝐶𝛽₁+(1−𝛼)𝛽₂,+(1−𝛼)𝐶𝛽₂≤=𝛼𝐹𝐹𝛼𝑥𝑥₁₁+(1−𝛼)𝐹𝑥₂+𝜀≤𝐶𝛼𝛽₁+(1−𝛼)𝑥₂+𝜀+(1−𝛼)𝛽₂+𝜀.(32) Since 𝜀 is arbitrary, 𝐶(𝛽) is concave.
(2) Without loss of generality, we can assume 𝐶(𝛽) is differentiable. If not, we can approach it by a differentiable function to any precision [23]. Suppose there is a 𝛽₀∈(𝑎,𝑏), such that 𝜌^′(𝛽)|||_𝛽=𝛽0=0,thatis𝐶^′𝛽₀𝛽₀−𝐶𝛽₀=0.(33)
Let 𝛽₁>𝛽₀; since 𝐶(𝛽) is concave, we have 𝐶^(𝛽₁)≤𝐶^(𝛽₀). Let 𝜌^(𝛽)||_𝛽=𝛽0=0,thatis𝐶^′𝛽₀𝛽₀−𝐶𝛽₀=0.(34) It is the equation of a straight line that has slope 𝐶^(𝛽₁) and passes through the (𝛽₀,𝐶(𝛽₀)). Since 𝐶(𝛽) is increasing and concave, we get 𝑦₁(𝛽₁)≤𝐶(𝛽₁) and 𝐶^𝛽₁−𝐶𝛽¹𝛽₁≤𝐶^𝛽₁−𝑦¹𝛽₁𝛽₁=𝐶^𝛽₁𝛽⁰𝛽₁−𝐶𝛽⁰𝛽₁≤𝐶^𝛽₀𝛽⁰𝛽₁−𝐶𝛽⁰𝛽₁=0,(35) that is, 𝜌^(𝛽₁)≤0.
Let 𝛽₂<𝛽₀, since 𝐶(𝛽) is concave, 𝐶^(𝛽₂)≥𝐶^(𝛽₀). Let 𝑦₀(𝛽)=𝐶^𝛽₀𝛽−𝛽₀+𝐶𝛽₀.(36) It is the equation of a straight line that has slope 𝐶^(𝛽₀) and passes through (𝛽₀,𝐶(𝛽₀)). Since 𝐶(𝛽) is increasing and concave, we get 𝑦₀(𝛽₂)≥𝐶(𝛽₂) and 𝐶^𝛽₂−𝐶𝛽²𝛽₂≥𝐶^𝛽₂−𝑦⁰𝛽₂𝛽₂≥𝐶^𝛽₀−𝑦⁰𝛽₂𝛽₂=𝐶^𝛽₀𝛽⁰𝛽₂−𝐶𝛽⁰𝛽₂=0,(37) that is, 𝜌^(𝛽₂)≥0.
Therefore, 𝛽₀ is a maximum point of function 𝜌(𝛽)=𝐶(𝛽)/𝛽.
(3) If sup_𝐷𝐹(𝑥)𝐺(𝑥)=+∞,(38) then for any 𝑁>0, there are 𝑥₀, such that 𝐹𝑥₀𝐺𝑥₀>𝑁.(39) Let 𝐺(𝑥₀)=𝛽₀, then 𝐶(𝛽₀)=sup{𝐹(𝑥)|𝐺(𝑥)≤𝛽₀}≥𝐹(𝑥₀), so sup_{𝑎<𝛽<𝑏}𝐶(𝛽)𝛽≥𝐶𝛽⁰𝛽₀≥𝐹𝑥⁰𝐺𝑥₀>𝑁.(40) Since 𝑁 is arbitrary, we get sup_{𝑎<𝛽<𝑏}𝐶(𝛽)𝛽=+∞.(41) In turn, if sup_{𝑎<𝛽<𝑏}𝐶(𝛽)𝛽=+∞,(42) then for any 𝑁>0, there are 𝛽₀, 𝑎<𝛽₀<𝑏, such that 𝐶𝛽₀𝛽₀>𝑁.(43) Since sup_𝐷𝐹(𝑥)𝐺(𝑥)≥sup_{𝐺(𝑥)≤𝛽0}𝐹(𝑥)𝐺(𝑥)≥sup_{𝐺(𝑥)≤𝛽0}𝐹(𝑥)𝛽₀=𝐶𝛽⁰𝛽₀>𝑁(44) so sup_𝐷𝐹(𝑥)𝐺(𝑥)=+∞.(45) Therefore, if sup_𝐷𝐹(𝑥)𝐺(𝑥)<+∞(46) is valid, then so is sup_{𝑎<𝛽<𝑏}𝐶(𝛽)𝛽<+∞(47) and vice versa.
Suppose (46) is valid, for any 𝜀>0, there are 𝑥₀∈𝐷, such that 𝐹𝑥₀𝐺𝑥₀+𝜀=sup^𝐷𝐹(𝑥)𝐺(𝑥).(48) Let 𝐺(𝑥₀)=𝛽₀, then sup_𝐷𝐹(𝑥)𝐺(𝑥)=𝐹𝑥⁰𝐺𝑥₀+𝜀≤𝐶𝛽⁰𝛽₀+𝜀≤sup_{𝑎<𝛽<𝑏}𝐶(𝛽)𝛽+𝜀.(49) Since 𝜀 is arbitrary, we get sup_𝐷𝐹(𝑥)𝐺(𝑥)≤sup_{𝑎<𝛽<𝑏}𝐶(𝛽)𝛽.(50)
On the other hand, since (47) is valid too, for any 𝜀>0, there are 𝑎<𝛽₀<𝑏, such that 𝐶𝛽₀𝛽₀+𝜀=sup_{𝑎<𝛽<𝑏}𝐶(𝛽)𝛽.(51) Let 𝑥₀∈𝐷, such that 𝐺(𝑥₀)≤𝛽₀ and 𝐹(𝑥₀)+𝜀𝛽₀=sup{𝐹(𝑥)|𝐺(𝑥)≤𝛽₀}=𝐶(𝛽₀), then sup_𝐷𝐹(𝑥)𝐺(𝑥)≥sup_{𝐺(𝑥)≤𝛽0}𝐹(𝑥)𝐺(𝑥)≥𝐹𝑥⁰𝛽₀=sup_{𝑎<𝛽<𝑏}𝐶(𝛽)𝛽−2𝜀.(52) Since 𝜀 is arbitrary, we get sup_𝐷𝐹(𝑥)𝐺(𝑥)≥sup_{𝑎<𝛽<𝑏}𝐶(𝛽)𝛽.(53)

If 𝐹(𝑥)=𝐼(𝑋,𝑌) is the average mutual information defined by (15) and 𝐺(𝑥)=𝐸[𝑏(𝑋)], then (3) of Theorem 7 is Theorem  2 in [15].

In addition to average mutual information, entropy is another frequently used function in applications [24]. It is obvious that Theorem 7 is valid when 𝐹(𝑥) is an entropy function (as a function of the distribution of input symbols). It is useful to point out that Theorem 7 is still valid when 𝐺(𝑥) is a positive definite quadratic function [25].

Making use of (2) of Theorem 7, we can write an algorithm to compute the sup(𝐹(𝑥)/𝐺(𝑥))as follows. The algorithm is based on the 0.618 method used to locate the optimal point of a single peak function.

Algorithm 8 (0.618 method). Let 𝑎=inf{𝐺(𝑥)|𝑥∈𝐷},𝑏=sup{𝐺(𝑥)|𝑥∈𝐷}, and assume 𝑎≥0.
(1) Given search interval [𝑎,𝑏], and tolerance 𝜀>0. compute 𝜆 and 𝜇 as follows:
𝜆=𝑎+0.382(𝑏−𝑎),𝜇=𝑎+0.618(𝑏−𝑎); evaluate 𝐶(𝜆)/𝜆 and 𝐶(𝜇)/𝜇. Let 𝑘=1.
(2) If 𝐶(𝜆)/𝜆>𝐶(𝜇)/𝜇, then go to 3, else go to 4.
(3) If 𝑏−𝜆≤𝜀, then output 𝜇 and 𝐶(𝜇)/𝜇, else, let 𝑎=𝜆,𝜆=𝜇,𝐶(𝜆)/𝜆=𝐶(𝜇)/𝜇,𝜇=𝑎+0.618(𝑏−𝑎); evaluate 𝐶(𝜇)/𝜇, go to 5.
(4) If 𝜇−𝑎≤𝜀, then output 𝜆 and 𝐶(𝜆)/𝜆, else, let 𝑏=𝜇,𝜇=𝜆,𝐶(𝜇)/𝜇=𝐶(𝜆)/𝜆,𝜆=𝑎+0.382(𝑏−𝑎); evaluate 𝐶(𝜆)/𝜆, go to 5.
(5)  𝑘=𝑘+1, go to 2.
Even if 𝑎=0, that is when there are zero cost input symbols, the algorithm is still valid.

3.3. Maximum Entropy

A similar but simpler problem is the computation of maximum entropy. The typical form is as follows:min^𝑚_𝑖=1𝑝_𝑖log𝑝^𝑖𝜋_𝑖s.t_𝑚,_𝑗=1𝑝𝑎_𝑖𝑗𝑝_𝑗=𝐸_𝑖,𝑖=1,…,𝑟,₁𝑝+⋯+𝑝_𝑚=1,₁≥0,…,𝑝_𝑚≥0,(54) where 𝜋_𝑖 are known, and 𝐸_𝑖 can be computed by sample data, 𝑖=1,…,𝑟. The GIS algorithm [26–28] is one of the most common algorithms for (54), which is only suited to linear constraints. In spite of that there is a good discussion on the computation in [27]; however, we cannot ensure that GIS is a polynomial time algorithm.

In [25] a maximum entropy problem with nonlinear constraints is considered. It is as follows:min^𝑚_𝑖=1𝑝_𝑖log𝑝^𝑖𝜋_𝑖s.t_𝑚,_𝑖=1𝑝𝑝_𝑖𝑎_𝑖−_𝑖,𝑗𝜎_𝑖𝑗𝑝_𝑖𝑝_𝑗≥𝑏,₁𝑝+⋯+𝑝_𝑚=1,₁≥0,…,𝑝_𝑚≥0,(55) where (𝜎_𝑖𝑗)_𝑚×𝑚 is a positive definite matrix.

We can solve (54) and (55) in polynomial time. In fact, for (54), we add logarithmic barriers −log𝑝_𝑖 in its objective function, for (55), we add logarithmic barrier −log𝑝_𝑖 and−log^𝑚_𝑖=1𝑝_𝑖𝑎_𝑖−_𝑖,𝑗𝜎_𝑖𝑗𝑝_𝑖𝑝_𝑗−𝑏(56) in its objective function, and the problems can be transformed into (3). It is good to know that the function in (56) is self-concordant. Therefore, the path following method is valid.

4. An Example

Let transmission matrix 𝑃 be as follows: 𝑃=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝00000.00000....095111000000..010000....12112100000000..000......0513125110000.0000000...1491000000..00000....011221100000..000000...011411000000..00000....01121110000000..0000.....011121110000..000000.00.511100000..0000000...14212⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.(57)

Let 𝜀=0.00001. Suppose that the initial distribution is discrete uniform, with penalty factor 𝜇=2 and initial penalty parameter 𝑡₀=10. Iterating 70 times by the path-following method, we get channel capacity 𝐶=1.370656 and the optimal distribution:(0.0893518,0.2414400,0.1743530,0.1548483,0.0000003,0.0811754,0.1946234,0.0000004,0.0642042,0.0000080).

With the Arimoto-Blahut algorithm, iterate 180 times, we get the same results.

Let 𝑏=(1,2,5,7,6,8,3,9,3,4)^𝑇 be cost vector of the input symbols. Let 𝐺(𝑝)=𝑏^𝑇⋅𝑝 be the expect cost of the input symbols. The curve of 𝐶(𝛽)/𝛽 is shown in Figure 1; By Algorithm 8, the channel capacity per unit cost is 0.5801, and it is attained at 𝛽=1.8325. The search interval is [1, 5.8]. The number of iterations of the path-following is 17. The number of iterations of the 0.618 method is 19. The number of iterations of Newton’s algorithm is 5.

Figure 1 

Curve of 𝐶(𝛽)/𝛽, where 𝐺(𝑥) is the expected cost of the input symbol.

Let 𝐺(𝑥)=𝑏^′𝑥+𝑥^𝐷𝑥 be a positive definite quadratic function, where𝐷=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝4333233334..........26315210277325997121999037730030959771773433233323..........64662731173586579663938462113106546525613342231224..........36889996712848695569983509097995627499403324223223..........16812886635680794170045072508154243046632222322123..........52926203409567374898360766958596732448453332252333..........27982492009799712158729265418675404813003313226333..........13980908247954422635710594971872944129723322133433..........01663289451651816222319051780315968390413222233353..........21764024162667953285037888105476464074634343333336..........7713004565139088525901915683457103502133⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.(58)The curve of 𝐶(𝛽)/𝛽 is shown in Figure 2. The maximum is 0.2156, and it is attained at 𝛽=5.9093. The number of iterations of the path-following is 17. The number of iterations of the 0.618 method is 19. The number of iterations of Newton’s algorithm is 4.

Figure 2 

Curve of 𝐶(𝛽)/𝛽, where 𝐺(𝑥) is a positive definite quadratic function.

5. Conclusion

By means of self-concordant function theory, the computation of channel capacity, especially, when there are constraints and when constraints are nonlinear, becomes very simple. When the numerator is a general concave function and the denominator is a general convex function, the formula about channel capacity per unit cost is still valid. Furthermore, the function 𝐶(𝛽)/𝛽 is single peak, hence we can get some new algorithms for channel capacity per unit cost.