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Discrete Convexity and Log-Concave Distributions in Higher Dimensions

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Discrete Convexity and Log-Concave Distributions in Higher Dimensions Discrete Convexity and Log-Concave Distributions In Higher Dimensions 1 Basic Definitions and notations n 1. R : The usual vector space of real n− tuples n 2. Convex set : A subset C of R is said to be convex if λx + (1 − λ)y 2 C for any x; y 2 C and 0 < λ < 1: n 3. Convex hull : Let S ⊆ R . The convex hull denoted by conv(S) is the intersection of all convex sets containing S: (or the set of all convex combinations of points of S) n 4. Convex function : A function f from C ⊆ R to (−∞; 1]; where C is a n convex set (for ex: C = R ). Then, f is convex on C iff f((1 − λ)x + λy) ≤ (1 − λ)f(x) + λf(y); 0 < λ < 1: 5. log-concave A function f is said to be log-concave if log f is concave (or − log f is convex) n 6. Epigraph of a function : Let f : S ⊆ R ! R [ {±∞}: The set f(x; µ) j x 2 S; µ 2 R; µ ≥ f(x)g is called the epigraph of f (denoted by epi(f)). 7. The effective domain (denoted by dom(f)) of a convex function f on S n is the projection on R of the epi(f): dom(f) = fx j 9µ, (x; µ) 2 epi(f)g = fx j f(x) < 1g: 8. Lower semi-continuity : An extended real-valued function f given n on a set S ⊂ R is said to be lower semi-continuous at a point x 2 S if f(x) ≤ lim inf f(xi) for every sequence x1; x2; :::; 2 S s.t xi ! x and limit i!1 of f(x1); f(x2); ::: exists in [−∞; 1]: 1 2 Convex Analysis Preliminaries In this section, we recall some basic concepts and results of Convex Analysis. 1.( Alternative definitions for convexity) n (a) f is convex on S ⊂ R (given S is convex) iff epi(f) is convex as a n+1 subset of R . (b) Convexity of f is equivalent to that of the restriction of f to dom(f): n (c) A non-empty subset D ⊂ R is convex iff its indicator function is con- vex. Note that in the context of Convex Analysis, the indicator function n of a non-empty subset S ⊂ R can be defined as, ( 0; x 2 S S(x) = +1 otherwise n 2. (Rockafellar, Corollary 10.1.1)A convex function finite on R is neces- sarily continuous. 3. A convex function is proper if epi(f) = ; and contains no vertical lines. In other words, a proper convex function is obtained by taking a finite convex n function on convex set C and then extending it to all of R by setting f(x) = 1 for x2 = C: n 4. A convex function f : R ! [−∞; 1] is said to be closed if its epigraph is closed. 5.( Topological properties of convex functions)Let f be an arbitrary n function from R to [−∞; 1]: Then, TFAE. n (a) f is lower semi-continuous throughout R : (b) fx j f(x) ≤ αg is closed for every α 2 R: n+1 (c) The epigraph of f is a closed subset in R : 2 n 6. Let f 2 C (C); where C ⊂ R is an open convex set. Then, f is convex on C iff its Hessian matrix is positive semi-definite for all x 2 C: 3 Discrete convexity in higher-dimensions In higher-dimensions, the definition of discrete convexity (equivalently, concavity) d is not clear. In Z ; d ≥ 2; there are multiple definitions of discrete convexity. The following generalization of the concept of log-concavity for discrete mul- tivariate distributions was proposed by Bapat '88. 2 d Definition 1. (generalized log-concave) Let S ⊂ N ; d ≥ 2 and let f : S ! d (0; 1): f is said to be generalized log-concave on S if for any k 2 N such that k + ei + ej 2 S; i; j = 1; 2; 3:::; d; then the positive symmetric d × d matrix ((f(k + ei + ej))) has exactly one(simple) positive eigenvalue. The following theorem follows from the above definition and another interme- diary lemma (the interested reader may refer to 'Bapat, Discrete Multivariate Distribution and Generalized Log-Concavity '88). d Theorem 2. Let S ⊂ N ; d ≥ 2 and let f : S ! (0; 1): Suppose there exist log-concave functions fi : N ! (0; 1) for all i = 1; 2; :::; d such that f(x) = Qd i=1 fi(xi); where x = (x1; x2; :::; xd) 2 S; then f is generalized log-concave. Remarks 1. 1. Using Theorem 2, it can be verified that distrbutions such as Hypergeo- metric, negative hypergeometric, multinomial are g.l.c. d 2. The definition of g.l.c need not be restricted to N and can easily be extended d to Z : Murota and Shioura , '01 provided a detailed survey about convex func- tions and sets in higher-dimensions. Among them, we are particularly interested in three definitions which can be defined as follows. d Definition 3. (Separable-convex) Consider a function h : Z ! R [ f1g and d define the domain, dom(h) = fz 2 Z j h(z) < 1g: h is said to be separable- Pd d convex if h(z) = i=1 hi(zi)(z 2 Z ) for a finite family of discrete convex functions hi : Z ! R [ f1g; i 2 f1; 2; 3; :::; dg: That is, (∆hi)(z) ≥ 0 for all z 2 Z and for all i. In here, ∆ is the discrete Laplacian operator. d Definition 4. (Discretely-convex) Define the set N0(x) = fz 2 Z j bxc ≤ z ≤ dxeg: h is said to be discretely-convex if for any x; y 2 dom(h) and α 2 [0; 1]; minfh(z) j z 2 N0(αx + (1 − α)y)g ≤ αh(x) + (1 − α)h(y) d Similarly, a set S ⊆ Z is said to be discretely-convex if, for any x; y 2 S and any α 2 [0; 1];N0(αx + (1 − α)y)g \ S 6= ;. Note 1. Alternatively, Murota and Shioura defined S to be discretely convex if δS is discretely convex. Definition 5. ( Convex-extendible) h is said to be convex-extendible if h(z) = d h(z) for all z 2 Z ; where h is the convex closure of h which is defined as, d h(x) = sup fα + β|x j α + β|z ≤ h(z) for all z 2 Z g α2R; β2Rd 3 d d Similarly, a set S ⊆ Z is said to be convex-extendible if S \ Z = S; where S is the convex closure of S. Remarks 2. 1. For a discrete convex-extendible function h, its convex extension is a R d R closed, continuous convex function h defined on R s.t h (z) = h(z) for d all z 2 Z . 2. Convex closure h(x) is convex and it is the greatest convex extension of h. 3. Separable-convex ⊂ (discretely-convex \ convex-extendible) (By Murota and Shioura '01). 4. The following result gives a criterion to check a given discrete function is convex-extendible. d Lemma 6. (Murota and Shioura) Let h : Z ! R[f1g be some function.Then, h(z) = h(z) iff there exists a closed convex extension of h. d Example 1. Consider h(z) = z|Az ; where z 2 Z and A is a symmetric d × d R positive semi-definite matrix. h (x) = x|Ax; x 2 R is a convex extension of h (note that convexity of h follows from Rockafellar, theorem 4.5 on page 27). Since h is continuous, h is closed. Thus, it is convex-extendible. Example 2. Consider the set, 3 S = fz 2 Z j z1 + z2 + z3 = 2; zi ≥ 0; i = 1; 2; 3g [ f1; 2; 0g [ f0; 1; 2g [ f2; 0; 1g Define h(z) = 0 on S and 1 on Z n S. 1 1 Note that, S and h are discretely-convex; however, (1; 2; 0) + (0; 1:2) + 3 3 1 (2; 0; 1) = (1; 1; 1) 2 S \ d but (1; 1; 1) 2= S.Thus, h is not convex-extendible 3 Z Example 3. Let S = fx = (0; 0); y = (2; 1)g and let the function h be defined as before. d The convex closure of S is the segment between (0; 0) and (2; 1). Thus, S\Z = S. We conclude that h and S are convex-extendible. On the other hand, h is not discretely-convex, since N0(0:5x + 0:5y) = N0((1; 0:5)) = f(1; 0); (1; 1)g so N0(αx + (1 − α)y)g \ S = ;. 4 4 Discrete log-concave distributions in higher dimen- sions Throughout this, we focus on PMFs defined on Z: First, we introduce the notion of log-concave probability mass function in the one-dimensional setting. This is due to Balabdaoui et ea, '13. Definition 7. (Balabdaoui et ea, '13) Let p 2 Z ! [0; 1] denote a PMF. p is said to be log-concave if for any z 2 Z; (∆h)(z) = h(z − 1) − 2h(z) + h(z + 1) ≥ 0; where h(z) = − log p(z): Remarks 3. ∆ denotes the discrete Laplacian operator, which can also be ex- pressed as (∆h)(z) = h(z − 1) − h(z) − (h(z − h(z + 1)). This is the second difference of h; thus it matches well that of the continuous setting. d d Definition 8. (R. Bapat) A PMF p on N with the support S = fz 2 N j p(z) > 0g is said to be generalized log-concave if, d Y p(z) = pi(zi) ; z 2 S; i=1 where each pi satisfies (∆ log pi)(zi) ≤ 0: Note that each pi is a univariate discrete log-concave function (though, not necessarily a PMF; so this definition is slightly different than separable log- concavity).
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