LLogarithmic Voronoi cells
Yulia Alexandr and Alexander Heaton University of California, Berkeley Max Planck Institute for Mathematics in the Sciences, Leipzig Technische Universität Berlin
Abstract
We study Voronoi cells in the statistical setting by considering preimages of the maxi-mum likelihood estimator that tessellate an open probability simplex. In general, log-arithmic Voronoi cells are convex sets. However, for certain algebraic models, namelyfinite models, models with ML degree 1, linear models, and log-linear (or toric) models,we show that logarithmic Voronoi cells are polytopes. As a corollary, the algebraicmoment map has polytopes for both its fibres and its image, when restricted to thesimplex. We also compute non-polytopal logarithmic Voronoi cells using numerical al-gebraic geometry. Finally, we determine logarithmic Voronoi polytopes for the finitemodel consisting of all empirical distributions of a fixed sample size. These polytopesare dual to the logarithmic root polytopes of Lie type A, and we characterize their faces.
For any subset X ⊂ R n , the Voronoi cell of a point p ∈ X consists of all points of R n which are closer to p than to any other point of X in the Euclidean metric. In this articlewe discuss the analogous logarithmic Voronoi cells which find application in statistics. Adiscrete statistical model is a subset of the probability simplex M ⊂ ∆ n − , since probabilitiesare positive and sum to . The maximum likelihood estimator Φ (MLE) sends an empiricaldistribution u ∈ ∆ n − of observed data to the point in the model which best explains thedata. This means p = Φ( u ) maximizes the log-likelihood function (cid:96) u ( p ) := (cid:80) ni =1 u i log( p i ) restricted to M . Note that (cid:96) u is strictly concave on ∆ n − and takes its maximum value at u .Usually, u / ∈ M , and we must find the point Φ( u ) ∈ M which is closest in the log-likelihoodsense. For p ∈ M we define the logarithmic Voronoi cell log Vor M ( p ) = { u ∈ ∆ n − : Φ( u ) = p } . a r X i v : . [ m a t h . S T ] J un nformation Geometry [5] considers MLE in the context of the Kullbach-Leibler divergenceof probability distributions, sending data to the nearest point with respect to a Riemannianmetric on ∆ n − . Algebraic Statistics [13] considers the case where M can be described aseither the image or kernel of algebraic maps. Recent work in Metric Algebraic Geometry[9, 11, 12, 21] concerns the properties of real algebraic varieties that depend on a distancemetric. Logarithmic Voronoi cells are natural objects of interest in all three subjects.As an example, consider flipping a biased coin three times. There are four possible outcomes,3 heads (hhh), 2 heads (hht,hth,thh), 1 head (htt,tht,tth), and 0 heads (ttt). Parametrically,the twisted cubic is given by t (cid:55)→ p ( t ) = (cid:0) t , t (1 − t ) , t (1 − t ) , (1 − t ) (cid:1) ∈ M . For this model’s many lives, see [29]. We compute logarithmic Voronoi cells log
Vor M ( p ( t )) with parameter values t ∈ (cid:26) , , . . . , (cid:27) which live inside the simplex ∆ ⊂ R , and whose orthogonal projections into -space areshown in Figure 1. In this case, the logarithmic Voronoi cells are polytopes, and we getboth triangles and quadrilaterals, depending on the point p ( t ) ∈ M . The fact that thesepolytopes are equal to the logarithmic Voronoi cells will follow from Theorem 10 below.Figure 1: Logarithmic Voronoi cellsAfter giving the basic definitions in Section 2, Section 3 describes the relationship betweenlogarithmic Voronoi cells and logarithmic polytopes in the context of algebraic statistics.In particular, we show that ML degree implies that the logarithmic Voronoi cells arepolytopes, and give counterexamples to the converse statement. We also consider both linearmodels and log-linear (toric) models, showing that both families of statistical models havethe property that logarithmic Voronoi cells are polytopes. These include the twisted cubicof Figure 1, decomposable graphical models [28], Bayesian networks [17], staged tree models[10, 34], multinomial distributions, phylogenetic models, hidden Markov models, and manyothers arising in applications [32]. Corollary 11 states that both the image and fibres of thealgebraic moment map are polytopes. In Section 4 we show how to compute a (not necessarily2olytopal) logarithmic Voronoi cell using numerical algebraic geometry. By calculating Φ( u ) for , points u with respect to a model of ML degree 39, we demonstrate that logarithmicVoronoi cells can be reliably computed using numerical methods. Finally, in Section 5 wediscuss the historical motivation of Georgy Voronoi and adapt it to the statistical settingby analyzing a model with finitely many points, namely all possible empirical distributionson n states with d trials. We call the polytopes that arise logarithmic root polytopes of type A n − , show they are dual to the logarithmic Voronoi cells in Theorem 20, and characterizetheir faces in Theorem 18. We work with the open probability simplex ∆ n − ⊂ R n defined by ∆ n − := (cid:40) u ∈ R n : n (cid:88) i =1 u i = 1 , u i > for all i ∈ [ n ] (cid:41) . A statistical model M is a subset of the probability simplex. When M is defined as theintersection of ∆ n − with an algebraic variety or the image of rational map, we say that M is an algebraic statistical model [32, 37]. For any point u ∈ ∆ n − , the log-likelihood function (cid:96) u : R n> → R is defined by (cid:96) u ( p ) = (cid:80) ni =1 u i log( p i ) . For any model M ⊂ ∆ n − , we define therelation Φ ⊂ ∆ n − × M by ( u, p ) ∈ Φ ⇐⇒ p ∈ argmax q ∈M { (cid:96) u ( q ) : q ∈ M} . If ( u, p ) ∈ Φ then we also write Φ( u ) = p . We write ∆ M n − for the set of u ∈ ∆ n − such that Φ( u ) exists. Describing the set ∆ M n − and how it extends to the boundary of ∆ n − is an activearea of research, especially with respect to zeros in the data [16, 20]. MLE existence is alsoconnected to polystable and stable orbits in invariant theory [3]. For the important family oflog-linear (toric) models, [15] shows that positive data u ∈ ∆ n − guarantees existence, andin general the MLE exists exactly when the observed margins belong to the relative interiorof a certain polytope. See also [37, Theorem 8.2.1].Finally, we note that for models with more complicated geometry, Φ( u ) cannot always becomputed by finding critical points of (cid:96) u restricted to manifold points of M . The presentarticle takes the first step of computing logarithmic Voronoi cells for models where criticalpoints of (cid:96) u succeed in finding the MLE, as well as some interesting finite models. We statenecessary assumptions where required. More complicated examples outside the scope of thepresent article include models of nonnegative rank r matrices, which were studied in [27].Whenever p ∈ M ⊂ R n admits a tangent space at the point p , we denote by N p M itsorthogonal complement with respect to the Euclidean inner product on R n . We are alsointerested in the log-normal space at the point p ∈ M , defined by log N p M := { u ∈ R n : ∇ (cid:96) u ( p ) ∈ N p M} . ∇ (cid:96) u ( p ) is the vector whose entries are given by the partial derivatives of (cid:96) u with respectto each of the variables p , . . . , p n . For an algebraic statistical model M , the ML degree isthe number of complex critical points of (cid:96) u on M for generic data u ∈ ∆ n − [37, p. 140]. Lemma 1.
The log-normal space log N p M is a linear subspace of R n . Proof.
The normal space N p M is a linear subspace. Arrange a basis as the rows of a matrix.Adjoin another row with entries u i /p i , the partial derivatives of (cid:96) u ( p ) with respect to each p i . The maximal minors of the resulting matrix are linear equations in the variables u i andtherefore cut out a linear space of such u ∈ R n . This space is the log-normal space at p .By Lemma 1, the intersection of the log-normal space at a point p ∈ M with the closedprobability simplex ∆ n − is a polytope log Poly M ( p ) , which we call its log-normal polytope .In what follows, when we say that a logarithmic Voronoi cell equals its log-normal polytope,we mean that they are equal as sets, excepting the points in the boundary of the simplex. Proposition 2.
Let M be any finite statistical model. Then the logarithmic Voronoi cells log Vor M ( p ) are polytopes for each p ∈ M . Proof.
Fix p ∈ M . The set of all points u ∈ ∆ n − such that (cid:96) u ( p ) ≥ (cid:96) u ( q ) for all q ∈ M is the logarithmic Voronoi cell of p . Consider some q (cid:54) = p but q ∈ M . Then (cid:96) u ( p ) ≥ (cid:96) u ( q ) becomes the condition that n (cid:88) i =1 u i log (cid:18) p i q i (cid:19) ≥ . But this is linear in u and so defines a closed halfspace. Since there are finitely many pointsin M , we see that the logarithmic Voronoi cell is an intersection of finitely many closedhalfspaces (including those defining ∆ n − ). Therefore it is a polytope.For infinite models, the logarithmic Voronoi cells are, in general, not polytopes. However, ifthe model is smooth at p , the logarithmic Voronoi cell will be contained in the log-normalpolytope. Figure 2 shows a logarithmic Voronoi cell for p ∈ M ⊂ ∆ ⊂ R which isnot a polytope, but is contained in a polytope. In this case it is the hexagon given by log Poly ( p ) = log N p M ∩ ∆ . Since the log-normal space is -dimensional, by choosing anorthonormal basis agreeing with this subspace we can visualize the logarithmic Voronoi cell,despite it living in R . We discuss this example in detail in Section 4. For more on finitemodels, see Section 5. Lemma 3.
Let Φ( u ) = p for some p ∈ M ⊂ ∆ n − such that U ∩ M is a manifold for some p -neighborhood U in R n . Then u lies in the logarithmic normal space log N p M and log Vor M ( p ) ⊂ log Poly M ( p ) . Proof.
Note that (cid:96) u ( x ) := (cid:80) u i log( x i ) is a smooth function on any neighborhood of p ∈ M contained in ∆ n − . Consider the gradient ∇ (cid:96) u ( p ) . R n = T p M ⊕ N p M and if ∇ (cid:96) u ( p ) had anynonzero tangential component then there would exist some q ∈ M such that (cid:96) u ( q ) > (cid:96) u ( p ) ,contradicting the fact that Φ( u ) = p . Proposition 4.
Logarithmic Voronoi cells are convex sets.
Proof.
As in the proof of Proposition 2, the logarithmic Voronoi cell of p is defined by theinequalities (cid:80) i ∈ [ n ] u i log( p i /q i ) ≥ for every q ∈ M , each linear in u . Hence, the logarithmicVoronoi cell of p is an intersection of (possibly infinitely many) closed half-spaces, and theresult follows.The following theorem concerns algebraic models with ML degree . These were character-ized in [23] and studied further in [14]. They include, for example, Bayesian networks anddecomposable graphical models. Theorem 5.
Let M be any algebraic model with ML degree which is smooth on ∆ n − .Then the logarithmic Voronoi cell at every p ∈ M equals its log-normal polytope on ∆ M n − . Proof.
We will show that log
Vor M ( p ) = log N p M ∩ ∆ M n − . Let u ∈ ∆ n − be an element of log Vor M ( p ) . Then Φ( u ) = p and since M is smooth, u ∈ log N p M ∩ ∆ M n − by Lemma 3.For the reverse direction, let u ∈ log N p M ∩ ∆ M n − . Recall that Φ( u ) is the argmax of (cid:96) u ( q ) over all points q ∈ M . Since Φ( u ) exists and M is smooth, this argmax must be among thecritical points of (cid:96) u restricted to M , which include p . But since the ML degree is , thereis only one complex critical point, and hence Φ( u ) = p . Therefore u is in the logarithmicVoronoi cell of p , and the result follows. 5 xample 6. Consider M = V ( f ) for f : C → C given by the polynomial system f ( x ) = (cid:20) x x − x x x + x + x + x − (cid:21) : C → C A parametrization of this model is given by ( p , p ) (cid:55)→ ( p p , p (1 − p ) , (1 − p ) p , (1 − p )(1 − p )) . This is the independence model on two binary random variables, and also the Segre embed-ding of P × P . The points of this -dimensional model live in the 3-dimensional hyperplane (cid:80) x i = 1 inside R , so we can choose a basis agreeing with this hyperplane to plot them.For each x ∈ M , we construct an ( m + 1) × n matrix A ( x ) by augmenting the row ∇ (cid:96) u tothe Jacobian matrix df : A ( x ) = x − x − x x u /x u /x u /x u /x . Since our model has codimension two, the × minors of A ( x ) give linear equations describingthe log-normal space. u − u − u x x + u x x + u x x − u x x u − u − u x x + u x x − u x x + u x x u − u + u x x − u x x − u x x + u x x u − u + u x x − u x x − u x x + u x x . Restricting this space to its intersection with the simplex u + u + u + u − to computethe log-normal polytope, we find that the polytopes are line segments. We plot them forvarious points on the model in Figure 3. Since M has ML degree 1, Theorem 5 tells us thatlog-Voronoi cells equal log-normal polytopes, so they are also line segments.Figure 3: One-dimensional log-normal polytopes at various pointsThe following Theorem 7 shows that the ML degree 1 condition in Theorem 5 is sufficient,but not necessary for the equality of logarithmic Voronoi cells and interiors of respective6og-normal polytopes. First consider the independence model of two identically distributedbinary random variables. The natural parametrization in a statistical context leads to theHardy-Weinberg curve defined by x − x x , which has ML degree [24]. A similar-lookingmodel, which has been called the cousin of the Hardy-Weinberg curve [22], is defined by thepolynomial f = x − x x . In this case n = 3 and M ⊂ ∆ ⊂ R . It turns out that the MLdegree of this model is [22, p. 394]. Theorem 7.
The algebraic model defined by the polynomial f = x − x x has ML degree , yet the logarithmic Voronoi cells are equal to their log-normal polytopes. Proof.
Calculate the Jacobian matrix of Lemma 1 by taking the gradients of f = x − x x and g = x + x + x − , augmenting this matrix with an additional row of the u i /x i .Consider the equation of the plane given by the determinant of this matrix. Note that M is a curve in ∆ , so the log-normal space at each point is defined by the vanishing of thedeterminant at that point. This plane has normal vector given by x x − x x + x x x + x x − x x − x x x − x x x x + x x x + 2 x x − x x where ( x , x , x ) is any point in the common zero set of f and g . Consider the cross-productof this vector with the all ones vector, which will give us the direction vector of the log-normalpolytope at ( x , x , x ) . Computing and simplifying each coordinate in the quotient ring Q [ x , x , x ] / (cid:0) x + x + x − , − x + x x (cid:1) Q [ x , x , x ] , we find that this cross product is given by − ( x + x − x x + x − x − ( x + x − x = x x − . This means that regardless of the point on the curve, the log-normal polytopes will be linesegments whose direction vector is ( − , , − . We claim that for any distinct p, q ∈ M thecorresponding line segments are disjoint. Consider the tangent space at some point x in theintersection of ∆ and the common zero set of f and g . Applying Gaussian elimination tothe × Jacobian matrix, it can be shown that if x + x (cid:54) = 0 then all tangent vectors aremultiples of (cid:18) x − x x + x − , x − x x + x , (cid:19) , (1)while if x + x = 0 then all tangent vectors are multiples of ( − , , . In neither case isit possible that a tangent vector is parallel to (1 , − , . For ( − , , this is obvious, butfor (1), a contradiction can be derived by showing that if the vector is parallel to (1 , − , the first and the last coordinates in (1) are equal, forcing x + 4 x + x = 0 . But on ∆ all coordinates are positive. Thus no line parallel to (1 , − , meets the model in twodistinct points. We conclude the log-normal polytopes are disjoint, and the result followsfrom Theorem 8. 7 heorem 8. Let M be any model smooth on ∆ n − . If all log-normal polytopes for each point p ∈ M are disjoint, then the logarithmic Voronoi cells equal log-normal polytopes on ∆ M n − . Proof.
We will show that log
Vor M ( p ) = log N p M ∩ ∆ M n − . The ⊂ direction follows fromLemma 3. For the reverse direction, let u ∈ log N p M ∩ ∆ M n − . Recall that Φ( u ) is theargmax of (cid:96) u ( q ) over all points q ∈ M . Since Φ( u ) exists and M is smooth, this argmaxmust be among the critical points of (cid:96) u restricted to M , which include p . If Φ( u ) were notequal to p then u would be in the intersection of ∆ n − with the log-normal space to the point Φ( u ) ∈ M . But the log-normal polytopes were assumed to be disjoint by the hypothesis.Therefore Φ( u ) = p , which means that u ∈ log Vor M ( p ) , and the result follows.Let f ( θ ) , · · · , f r ( θ ) be nonzero linear polynomials in θ such that (cid:80) ri =1 f i ( θ ) = 1 . Let Θ be the set such that f i ( θ ) > for all θ ∈ Θ and suppose that dim Θ = d . The model M = f (Θ) ⊆ ∆ r − is called a discrete linear model [37, p.152]. Linear models appear in [32,Section 1.2]. An example is DiaNA’s model in Example 1.1 of [32]. Theorem 9.
Let M be a linear model. Then the logarithmic Voronoi cells are equal totheir log-normal polytopes. Proof.
We will show that log
Vor M ( p ) = log N p M ∩ ∆ n − . The ⊂ direction follows fromLemma 3 since an affine linear subspace intersected with ∆ n − is smooth. For the reversedirection, let u ∈ log N p M ∩ ∆ n − . We must show Φ( u ) = p . Since (cid:96) u is strictly concave on ∆ n − , it is strictly concave when restricted to any convex subset, such as the affine-linearsubspace M . Therefore there is only one critical point. Since M is smooth, u must be inthe log-normal space of Φ( u ) , and so Φ( u ) must be p .Next we consider log-linear, or toric, models. These include many important families ofstatistical models, such as undirected graphical models [18], independence models [37], andothers as mentioned in the introduction. For an m × n integer matrix A with ∈ rowspan ( A ) ,the corresponding log-linear model M A is defined to be the set of all points p ∈ ∆ n − suchthat log( p ) ∈ rowspan ( A ) [37, p. 122]. Theorem 10.
Let A ∈ Z m × n be an integer matrix such that ∈ rowspan A . Let M be theassociated log-linear (toric) model. Then for any point p ∈ M , the log-Voronoi cell of p isequal to the log-normal polytope at p . Proof.
We will show that log
Vor M ( p ) = log N p M ∩ ∆ n − . The forward direction followsfrom Lemma 3, since these models are smooth off the coordinate hyperplanes (see [37, p.150]and [2]). For the reverse direction, let u ∈ log N p M . Although the log-likelihood functioncan have many complex critical points, it is strictly concave on log-linear models M forpositive u , in particular for u ∈ ∆ n − . This means that there is exactly one critical point inthe positive orthant, and it is the unique solution p ∈ M to the linear system Ap = Au . [13,Prop. 2.1.5]. This is known as Birch’s Theorem . It follows that Φ( u ) = p , as desired.8s a corollary, the polytopes shown in Figure 1 and Figure 3 are logarithmic Voronoi cells.Following [30], define the map sending a point in projective space to a convex combinationof the columns a i of A , so that the image is a polytope, namely φ A : P n − C → R m z (cid:55)→ (cid:80) ni =1 | z i | n (cid:88) i =1 | z i | a i . This restricts to what [30, p.120] calls the algebraic moment map φ A | M A = µ A : M A → R m ,where M A is the projective toric variety associated to A . The maximum likelihood estimator,then, is the map µ − A ◦ φ A restricted to ∆ n − , identified as a subset of P n − C by extendingscalars and using the quotient map defining projective space. The fact [30, Corollary 8.24]that there is a unique preimage, allowing the definition of µ − A , played a crucial role inTheorem 10. Thus we have the following Corollary 11.
For toric models, the logarithmic Voronoi cells are the preimages φ − A ( µ A ( p )) intersected with ∆ n − . Thus, φ A | ∆ n − is a map whose image is a polytope and whose fibresare also polytopes.For the Segre of Example 6, the image is a square and the fibres are line segments, depictedin Figure 4, which adjoins our Figure 3 with [30, Figure 2, p.121]. For more on the algebraicmoment map, see [36].Figure 4: The fibres and image of the moment map for the Segre of Example 6 Some open questions.
When M does not equal its log-normal polytope, an interestingopen question is how to describe the boundary of the logarithmic Voronoi cells. For Eu-clidean Voronoi cells of algebraic varieties, this was studied in [9]. In particular, are theboundaries algebraic or transcendental? Initial investigations suggest they are transcenden-tal. In addition, when models include singular points, what can we say about the Voronoicells of the singular locus? This is relevant for the important families of mixture modelsand secant varieties as in Example 16, discussed in Section 4. Also, for matrices and tensorsof fixed nonnegative rank the geometry is more complicated, and it would be interesting tostudy logarithmic Voronoi cells in this context, possibly in relation to the basins of attrac-tion of the EM algorithm [27]. Finally, we have focused on the discrete case, but continuous9istributions could also be investigated. One promising case is linear Gaussian covariancemodels [4], since their maximum likelihood estimation is an algebraic optimization problemover a spectrahedral cone. An implicit algebraic statistical model
M ⊂ ∆ n − is equal to the intersection of ∆ n − withthe zero set of some polynomial map f : R n → R m , which means that each of the m component functions f , . . . , f m are polynomials in n variables with real coefficients. Definition 12.
Let f be the × m row vector whose entries are the polynomials f , . . . , f m in the variables x , . . . , x n . We assume that the first polynomial defines the simplex, i.e. f = (cid:80) ni =1 x i − . Let the algebraic set defined by f , . . . , f m have codimension c . Let df denote the m × n Jacobian matrix whose rows are the gradients of f , . . . , f m . Let A be a c × m matrix whose entries are chosen randomly from independent normal distributions. Let B be a similarly chosen random ( m − c ) × ( n + c ) matrix. Let [ λ − be the row vector oflength c + 1 whose first c entries are variables λ , . . . , λ c and whose last entry is − and let I n + c be the identity matrix of size n + c . We are interested in the following vector equationwhose components give n + c polynomial equations in n + c unknowns: (cid:20) [ λ − (cid:20) A · df ∇ (cid:96) u (cid:21) f (cid:21)(cid:124) (cid:123)(cid:122) (cid:125) × ( n + m ) (cid:20) I n + c B (cid:21) = [0 · · · (cid:124) (cid:123)(cid:122) (cid:125) × ( n + c ) . (2) Theorem 13.
Let M be the intersection of ∆ n − and an irreducible algebraic model givenby the polynomial map f : R n → R m . Let u ∈ ∆ n − be fixed and generic. With probability , all points p ∈ M such that u ∈ log N p M are among the finitely many isolated solutionsto the square system of equations given in (2). Proof.
We first refer to [24, Theorem 1.6], which defines the projection map pr and provesthat it is generically finite-to-one. As a consequence, if u ∈ ∆ n − is generic, then withprobability there will be finitely many critical points of (cid:96) u restricted to M . If the algebraicset defined by f has codimension c then the dimension of the rowspace of df will be equalto c and the rows will span N x M for any generic x ∈ M [33, p.93]. With probability ,multiplying by the random matrix A will result in a c × n matrix of full row rank, whoserows also span N x M . Appending the row ∇ (cid:96) u and multiplying the resulting matrix bythe row vector [ λ − produces n polynomials which evaluate to zero whenever ∇ (cid:96) u is inthe normal space N x M . Appending the polynomials f , . . . , f m gives a × ( n + m ) rowvector of polynomials evaluating to zero whenever x ∈ M and ∇ (cid:96) u lies in the normal space N x M . However, this system of equations is overdetermined. Applying Bertini’s theorem[6, Theorem 9.3] or [35, Theorem A.8.7] we can take random linear combinations of these10olynomials using I n + c and B , and with probability , the isolated solutions of the resultingsquare system of polynomials will contain all isolated solutions of the original system ofequations. The result follows. Remark 14.
Numerical algebraic geometry [6, 35] can be used to efficiently find all isolatedsolutions of a square system of polynomial equations (square means equal number of equa-tions and variables). The system of equations given in Theorem 13 formulates our problemspecifically to take advantage of these tools.
Remark 15.
If we are interested in computing the logarithmic Voronoi cell of a specificpoint p ∈ M , then we can generate a generic point u ∈ log N p M by taking a random linearcombination of the gradients of f , . . . , f m . Using this point u we can formulate our systemof equations (2), one of whose solutions we already know, namely p . Using monodromy, wecan quickly find many other solutions p (cid:48) by perturbing our parametrized system of equationsthrough a loop in parameter space. For more details, see [1]. This is especially useful in thecase where the ML degree is known a priori, since we can stop our monodromy search afterfinding ML degree many solutions. This process yields an optimal start system for homotopycontinuation, allowing us to almost immediately compute solutions for other data pointssince we need only follow the ML degree-many solution paths via homotopy continuation.In the next example, we utilize the formulation in Theorem 13 to numerically compute alogarithmic Voronoi cell in a larger example of statistical interest, a mixture of two binomialdistributions, also known as a secant variety. Example 16.
Bob has three biased coins, one in each pocket, and one in his hand. He flipsthe coin in his hand, and depending on the outcome, chooses either the coin in his left orright pocket, which he then flips 5 times, recording the total number of heads in the last 5flips. To estimate the biases of Bob’s coins, Alice treats this situation as a -dimensionalstatistical model M ⊂ ∆ ⊂ R . Using implicitization [30, Section 4.2], Alice derives thefollowing algebraic equations describing M : f ( x ) = x x x − x x − x x + 4 x x x − x x x x − x x x − x x + 4 x x x + 2 x x − x x x x x − x x − x x x + 4 x x x + 2 x x − x x x x x − x x − x x + 4 x x x − x x + x + x + x + x + x − . For a concrete example, consider the point which arises by setting the biases of the coins to b = , b = , b = . Explicitly this point p ∈ M is p = (cid:16) , , , , , (cid:17) . The log-normal space log N p M is -dimensional, becoming a -dimensional polytope whenintersected with ∆ ⊂ R . This intersection is the log-normal polytope, in this case, a11exagon. In fact, this hexagon is the (2-dimensional) convex hull of the following six vertices: (cid:18) , , , , , (cid:19)(cid:18) , , , , , (cid:19)(cid:18) , , , , , (cid:19)(cid:18) , , , , , (cid:19)(cid:18) , , , , , (cid:19)(cid:18) , , , , , (cid:19) . By choosing an orthonormal basis agreeing with log N p M we can plot this hexagon, thoughit lives in R . Figure 2 shows the log-normal polytope and our numerical approximationof the logarithmic Voronoi cell (which is not a polytope) surrounding the point p . Byrejection sampling, we computed points u , u , . . . , u ∈ log N p M ∩ ∆ n − in thelog-normal polytope. By a result in [22], we know that the ML degree of this model is .Using the formulation presented in Theorem 13, we successfully computed all complexcritical points for each (cid:96) u i , i ∈ { , , . . . , } restricted to M . We easily find each Φ( u i ) by comparing the values, choosing the maximum. If p = Φ( u i ) then u i ∈ log Vor M ( p ) and we color that point green in Figure 2, while if p (cid:54) = Φ( u i ) we color the point pink.The repeated computations of each set of critical points were accomplished using thesoftware HomotopyContinuation.jl [7], which can efficiently compute the isolated solutionsto systems of polynomial equations using homotopy continuation [6, 35]. A full descriptionof the
Julia code needed to compute this example can be found online at [1].
Consider running experiments with sample size d and choosing the model defined by M := Z n ∩ d · ∆ n − d . Philosophically, M is the chaotic universe model . Adopting this model is to abandon theidea that experiments tell us about some simpler underlying truth, since the experimentaldata will always lie exactly on the model. In this section we investigate the Euclidean andlogarithmic Voronoi cells for p ∈ M n,d . For convenience we work with the scaled set d · ∆ n − since all polytopes considered will be combinatorially equivalent to those we could define in ∆ n − . Then we define M n,d as the N := (cid:0) n + d − d (cid:1) nonnegative integer vectors summing to d .Thus ( p , p , . . . , p n ) = p ∈ M n,d has all coordinates p i ∈ N . These vectors can be used tocreate a projective toric variety, the d th Veronese embedding of P n − into P N − [30, Chapter12], but instead we treat them as the model itself. By Proposition 2, the logarithmic Voronoicells for p ∈ M n,d are polytopes. For any p ∈ M n,d such that all coordinates p i > , we willprovide a full characterization of the faces of the corresponding logarithmic root polytopesin Theorem 18. Theorem 20 shows that these logarithmic root polytopes are dual to thelogarithmic Voronoi cells. These are the main results of the section. Again using orthogonalprojection from R , Figure 5 shows all the logarithmic Voronoi cells for interior points of M , and M , .Figure 5: Logarithmic Voronoi cells (rhombic dodecahedra) of interior points for n = 4 , d = 9 (on the left) and d = 10 (on the right).The Euclidean Voronoi cells for p ∈ M n,d are the duals of root polytopes of type A n − , i.e.the facets are defined by inequalities whose normal vectors are { e i − e j : i (cid:54) = j } . Rootpolytope often refers to the convex hull of the origin and the positive roots { e i − e j : i < j } .These were studied in [19] in terms of their relationship to certain hypergeometric functions.However, we define root polytopes to be the convex hull of all roots, as studied in [8]. Wealso note that these polytopes are Young orbit polytopes for the partition ( n − , and findapplication in combinatorial optimization [31].Denote the ( n − -dimensional root polytope by P n ⊂ R n , so that the Euclidean Voronoicells of p ∈ M n,d are the dual P ∗ n . The volume of P n is equal to n ( n − C n − , where C n − isa Catalan number. Every nontrivial face of P n is a Cartesian product of two simplices, andcorresponds to a pair of nonempty, disjoint subsets I, J ⊂ [ n ] . Every m -dimensional faceof P n is the convex hull of the vectors { e i − e j : i ∈ I, j ∈ J } with | I | + | J | = m + 2 , sothere is a bijection between nontrivial faces and the set of ordered partitions of subsets of [ n ] with two blocks [8, Theorem 1]. This result is related to the face description of Π n − , thepermutahedron, since P n is a generalized permutahedron and can be obtained by collapsingcertain faces of Π n − .In the logarithmic setting, analogous polytopes log P n ( p ) exist, playing the same role as theroot polytopes in the Euclidean case. However, their details are more complicated. The13orrect modifications motivate the following definition. Definition 17.
The logarithmic root polytope for p ∈ M n,d is defined as the convex hull ofthe (cid:0) n (cid:1) vertices v ij for i (cid:54) = j ∈ [ n ] given by the formulas v ij := 1 b j p j − a i p i (cid:20) a i e i − b j e j − ( a i − b j ) n (cid:21) where a i := log( p i +1 p i ) b j := log( p j p j − ) and where := (cid:80) k ∈ [ n ] e k . Note that a i , b j > are always positive real numbers and allvectors v ij are orthogonal to . We denote the polytope by log P n ( p ) .The statement and proof of the following Theorem 18 was inspired by and closely follows[8, Theorem 1]. However, significant details needed to be modified. For example, the linearfunctional g = (1 , , − , , − , , − is replaced by − a a b b p − a a b b p − a a b b p − a b b b p + a b b b p + a b b b p + a b b b p + a b b b p a a b b p − a a b b p − a a b b p − a b b b p − a b b b p + a a b b p + a a b b p + a a b b p a b b b p + a b b b p − a a b b p − a a b b p − a a b b p − a b b b p + a b b b p + a b b b p a a b b p + a a b b p + a a b b p − a a b b p − a a b b p − a b b b p − a b b b p + a a b b p a a b b p + a a b b p + a a b b p + a a b b p − a a b b p − a a b b p − a b b b p − a b b b p . This linear functional plays the same role for the logarithmic root polytope of ( p , p , . . . , p ) ∈M ,d as g plays for the usual root polytope in the proof of [8, Theorem 1]. Theorem 18.
For m ∈ { , , . . . , n − } , every m -dimensional face of the logarithmic rootpolytope for p ∈ M n,d is given by the convex hull of the vertices v ij for i ∈ I, j ∈ J , where I, J are disjoint nonempty subsets of [ n ] such that | I | + | J | = m + 2 . Thus there is a bijectionbetween nontrivial faces and the set of ordered partitions of subsets of [ n ] with two blocks,where the dimension of the face corresponding to ( I, J ) is | I | + | J | − . Proof.
Each face of a polytope can be described as the subset of the polytope maximizing alinear functional. Recall that we have fixed some p ∈ M n,d with all p k > and that a i := log( p i +1 p i ) and b j := log( p j p j − ) . In our formula (3) we use a shorthand for writing square-free monomials in the a , a , . . . , a n and the b , b , . . . , b n . For example if I = { , , } then a I = a a a , while if J = { , } then b J = b b . For a pair of disjoint nonempty subsets I, J of [ n ] we define the linear functional g IJ = ( g , g , . . . , g n ) ∈ ( R n ) ∗ by the formulasIf (cid:96) ∈ I, g (cid:96) = (cid:80) i ∈ I \ (cid:96) a I \{ (cid:96),i } b J ( a i p i − a (cid:96) p (cid:96) ) + (cid:80) j ∈ J a I \ (cid:96) b J \ j ( b j p j − a (cid:96) p (cid:96) ) If (cid:96) ∈ J, g (cid:96) = (cid:80) i ∈ I a I \ i b J \ (cid:96) ( a i p i − b (cid:96) p (cid:96) ) + (cid:80) j ∈ J \ (cid:96) a I b J \{ (cid:96),j } ( b j p j − b (cid:96) p (cid:96) ) Else, g (cid:96) = 0 . (3)14hen the convex hull of the vectors { v ij : i ∈ I, j ∈ J } is the face maximizing g IJ . To seethis, first note that g IJ · = 0 . Because of this fact we can ignore the component of v ij inthe direction. Recall that v ij := 1 b j p j − a i p i (cid:20) a i e i − b j e j − ( a i − b j ) n (cid:21) , so that to evaluate g IJ on v ij it is enough to evaluate on b j p j − a i p i [ a i e i − b j e j ] . Recalling that the a i and b j are always positive and that the p k > , it can be seen that g IJ takes equal value on every vertex v rs for r ∈ I, s ∈ J , and strictly less on every othervertex. We omit the details of the admittedly lengthy calculation, but note that the commonmaximum value attained on all vertices v rs for r ∈ I, s ∈ J , is equal to (cid:88) i ∈ I a I \ i b J + (cid:88) j ∈ J a I b J \ j . Conversely, given an arbitrary linear functional f = ( f , f , . . . , f n ) determining a nontrivialface F , collect the indices where its components are nonnegative in a set I and the indiceswhere its components are negative in a set J . Then ( I, J ) is a partition of [ n ] and we referto the same formulas (3) as above in order to define the sets ( I (cid:48) , J (cid:48) ) as follows. If I (cid:54) = ∅ and J (cid:54) = ∅ then let I (cid:48) := { i : f i /g i = max ( f (cid:96) /g (cid:96) : (cid:96) ∈ I ) } J (cid:48) := { j : f j /g j = max ( f (cid:96) /g (cid:96) : (cid:96) ∈ J ) } . If I = ∅ then let I (cid:48) := { i : f i /g i = min ( f (cid:96) /g (cid:96) : (cid:96) ∈ J ) } J (cid:48) := { j : f j /g j = max ( f (cid:96) /g (cid:96) : (cid:96) ∈ J ) } , while if J = ∅ then let I (cid:48) := { i : f i /g i = max ( f (cid:96) /g (cid:96) : (cid:96) ∈ I ) } J (cid:48) := { j : f j /g j = min ( f (cid:96) /g (cid:96) : (cid:96) ∈ I ) } . Note that the face F is the convex hull of the vectors { v ij : i ∈ I (cid:48) , j ∈ J (cid:48) } and hence ( I (cid:48) , J (cid:48) ) aredetermined independently of the choice of linear functional which maximizes the given face.Now we show that the dimension of the face corresponding to disjoint nonempty sets I, J of [ n ] is | I | + | J | − . Let I = { i , . . . , i | I | } and J = { j , . . . , j | J | } . Then X = { v i ,j (cid:96) : (cid:96) = 1 , . . . , | J |} ∪ { v i (cid:96) ,j : (cid:96) = 2 , . . . , | I |}
15s a maximal linearly independent subset of | I | + | J | − of the vectors v ij , i ∈ I, j ∈ J . Inaddition, for any i ∈ I, j ∈ J either v ij ∈ X or we can write it as an affine combination(coefficients sum to 1) of vectors in X, namely v i,j = (cid:18) b j p j − a i p i b j p j − a i p i (cid:19) v i,j − (cid:18) b j p j − a i p i b j p j − a i p i (cid:19) v i ,j + (cid:18) b j p j − a i p i b j p j − a i p i (cid:19) v i ,j . Hence, X is an affine basis of the face corresponding to I, J , whose dimension is | X | − ,which is | I | + | J | − as desired. This completes the proof. Example 19.
Let n = 6 , I = { , } , J = { , , } and p = (2 , , , , , . We implementedthe formulas (3) in floating point arithmetic (due to the logarithms) and obtain (shown toonly three digits) g IJ = (0 . , − . , − . , . , − . , − . . We can evaluate this linear functional on the vertices v ij for i (cid:54) = j where i, j ∈ [6] and obtainthe following values, which are as expected. . v (1 ,
2) = (1 . , − . , − . , − . , − . , − . . v (1 ,
3) = (1 . , . , − . , . , . , . . v (1 ,
4) = (1 . , − . , − . , − . , − . , − . . v (1 ,
5) = (1 . , − . , − . , − . , − . , − . . v (1 ,
6) = (1 . , − . , − . , − . , − . , − . − . v (2 ,
1) = ( − . , . , . , . , . , . . v (2 ,
3) = (0 . , . , − . , . , . , . − . v (2 ,
4) = (0 . , . , . , − . , . , . . v (2 ,
5) = (0 . , . , . , . , − . , . − . v (2 ,
6) = (0 . , . , . , . , . , − . − . v (3 ,
1) = ( − . , . , . , . , . , . − . v (3 ,
2) = ( − . , − . , . , − . , − . , − . − . v (3 ,
4) = ( − . , − . , . , − . , − . , − . − . v (3 ,
5) = ( − . , − . , . , − . , − . , − . − . v (3 ,
6) = ( − . , − . , . , − . , − . , − . − . v (4 ,
1) = ( − . , . , . , . , . , . . v (4 ,
2) = ( − . , − . , − . , . , − . , − . . v (4 ,
3) = (0 . , . , − . , . , . , . . v (4 ,
5) = ( − . , − . , − . , . , − . , − . . v (4 ,
6) = (0 . , . , . , . , . , − . − . v (5 ,
1) = ( − . , . , . , . , . , . − . v (5 ,
2) = ( − . , − . , − . , − . , . , − . . v (5 ,
3) = (0 . , . , − . , . , . , . − . v (5 ,
4) = (0 . , . , . , − . , . , . − . v (5 ,
6) = (0 . , . , . , . , . , − . − . v (6 ,
1) = ( − . , . , . , . , . , . . v (6 ,
2) = ( − . , − . , − . , − . , − . , . . v (6 ,
3) = (0 . , . , − . , . , . , . − . v (6 ,
4) = (0 . , . , . , − . , . , . . v (6 ,
5) = ( − . , − . , − . , − . , − . , . Theorem 20.
The logarithmic Voronoi cells for p ∈ M n,d with all p i > are the dualpolytopes (log P n ( p )) ∗ of the logarithmic root polytopes log P n ( p ) . Proof.
Given a point p ∈ M n,d , the logarithmic Voronoi cell can be defined as the intersectionof d · ∆ n − with all the halfspaces H q ( u ) ≥ for all points q ∈ M n,d with q (cid:54) = p , where H q ( u ) := (cid:88) i ∈ [ n ] u i log (cid:18) p i q i (cid:19) .
16e say that this system of inequalities is sufficient to define the logarithmic Voronoi cell.However, not all of these inequalities are necessary. Lemma 21 shows that a certain setof (cid:0) n (cid:1) inequalities is sufficient for all n ∈ Z ≥ . These are the inequalities H q ( u ) ≥ for q = p + e i − e j for i (cid:54) = j . We avoid logarithms of zero since p k > and we are away from thesimplex boundary. In other words, we get one inequality from every point q reachable from p by moving along a root of type A n − .These H q ( u ) ≥ inequalities are linear, with constant term zero. However, projecting thenormal vectors of these hyperplanes along the all ones vector and viewing p as the originof a new coordinate system, we obtain inequalities with nonzero constant terms. Theseinequalities describe the same logarithmic Voronoi polytope on the hyperplane (cid:80) k u k = d .Dividing each inequality by the constant terms we obtain a system of inequalities which is ofthe form Au ≤ , following the notation of [41], where the rows of A are exactly the vectors v ij . By [41, Theorem 2.11], the dual polytope is given by the convex hull of these v ij . Lemma 21.
Let p ∈ M n,d with every entry p i > . A sufficient system of inequalitiesdefining the logarithmic Voronoi cell is given by the (cid:0) n (cid:1) halfspaces u ∈ R n such that H δ ( u ) ≥ for δ ∈ R := { e i − e j : i (cid:54) = j, i, j ∈ [ n ] } and the affine plane (cid:80) u i = d , where H δ ( u ) := (cid:88) i ∈ [ n ] u i log (cid:18) p i p i + δ i (cid:19) . Proof.
We prove that the (cid:0) n (cid:1) inequalities H δ ( u ) ≥ for δ ∈ R are sufficient. Fix p ∈ M with all p i > . Let u ∈ R n such that H δ ( u ) ≥ for all δ ∈ R . Fix some q = p + δ + δ (cid:48) where δ, δ (cid:48) ∈ R , and assume that δ + δ (cid:48) / ∈ R . We wish to show H q ( u ) = (cid:80) i u i log p i q i ≥ . Considerseveral cases. First, if δ = δ (cid:48) = e j − e k , it suffices to show that u j log p j p j + 2 + u k log p k p k − ≥ . We claim that u j log p j p j + 2 + u k log p k p k − ≥ u j log p j p j + 1 + 2 u k log p k p k − , (4)which would be sufficient, since the right-hand side of the above equation is ≥ by assump-tion. We show that u j log p j p j + 2 ≥ u j log p j p j + 1 and u k log p k p k − ≥ u k log p k p k − . (5)Observe: u j log p j p j + 2 ≥ u j log p j p j + 1 ⇐⇒ p j + 2 p j + 1 ≥ p j + 2 p j ,u k log p k p k − ≥ u k log p k p k − ⇐⇒ p k − p k + 1 ≥ p k − p k . δ (cid:54) = δ (cid:48) , but theyshare both indices, then p = q , and we’re done. If they do not share any indices, then wehave that H q ( u ) = H δ ( u ) + H δ (cid:48) ( u ) ≥ by assumption. Suppose δ (cid:54) = δ (cid:48) , and δ and δ (cid:48) shareone index, j . If δ = e i − e j and δ (cid:48) = e j − e k for i (cid:54) = j (cid:54) = k , then δ + δ (cid:48) = e i − e k , a contradictionto the assumption δ + δ (cid:48) / ∈ R . Similarly when δ = e j − e i and δ (cid:48) = e k − e j . Suppose thenthat δ = e i − e j and δ (cid:48) = e i − e k . We wish to show that u i log p i p i + 2 + u j log p j p j − u k log p k p k − ≥ . Note then that u i log p i p i + 2 ≥ u i log p i p i + 1 ⇐⇒ p i + 2 p i + 1 ≥ p i + 2 p i , and the last inequality always holds for positive p i , so the lemma is true for this case. Thecase when δ = e j − e i and δ (cid:48) = e k − e i is proved similarly. Since H q ( u ) ≥ in all of the caseswe considered, and the cases are exhaustive, we conclude that the lemma holds. A family of polytopes.
For n = 2 , , , , , we write below the f -vectors for the log-arithmic Voronoi cells of any point p ∈ M n,d with p i > in all coordinates. These werecomputed numerically and using the face characterization of Theorem 18. The logarithmicVoronoi cells for every M n,d are combinatorially isomorphic to the dual of the correspondingroot polytope, exactly as in the Euclidean case. n = 2 (1 , , n = 3 (1 , , , n = 4 (1 , , , , n = 5 (1 , , , , , n = 6 (1 , , , , , , n = 7 (1 , , , , , , , We have a family of Euclidean Voronoi polytopes that tile R n − and a family of logarithmicVoronoi polytopes that tile the open simplex ∆ n − . This family begins n − n − n − · · · line segment hexagon rhombic dodecahedron · · · Root polytopes of type A have connections to tropical geometry. The rhombic dodecahedronis a polytrope which has been called the -pyrope because of the mineral Mg Al ( SiO ) whose pure crystal can take the same shape. For more on root polytopes, tropical geometry,and polytropes, see [25].Georgy Voronoi devoted many years of his life to studying properties of 3-dimensional par-allelohedra, convex polyhedra that tessellate 3-dimensional Euclidean space. His paper onthe subject called Recherches sur les parallélloèdres primitifs [39] was a result of his twelve-year work. In a cover letter to the manuscript, he wrote: “I noticed already long ago that18he task of dividing the n -dimensional analytical space into convex congruent polyhedra isclosely related to the arithmetic theory of positive quadratic forms” [38]. Indeed, Voronoiwas interested in studying cells of lattices in Z n with the aim of applying them to the theoryof quadratic forms. This motivated us to study a lattice intersected with the probabilitysimplex, the topic of our current section. Today, Voronoi decomposition finds applicationsto the analysis of spatially distributed data in many fields of science, including mathematics,physics, biology, archaeology, and even cinematography. In [40], the author uses Voronoicells to optimize search paths in an attempt to improve the final 6-minute scene of AndreiTarkovsky’s Offret (the Sacrifice) . Voronoi diagrams are so versatile they even found theirway into baking: Ukrainian pastry chef Dinara Kasko uses Voronoi diagrams to 3D-printsilicone molds which she then uses to make cakes [26].
Acknowledgements:
Both authors would like to thank Bernd Sturmfels for suggesting thistopic during the Summer of 2019, including many helpful suggestions along the way. Wealso thank the Max Planck Institute of Mathematics in the Sciences for support during thesummers of 2019 and 2020, and also the library staff for their exceptional support duringthe difficult pandemic, allowing us access to the resources we need for research. The firstauthor was also supported by the Berkeley Chancellor’s fellowship.
References [1] Yulia Alexandr, Alexander Heaton, and Sascha Timme.
Computing a logarithmic Voronoicell . . Accessed: September 18, 2019.[2] Carlos Améndola, Nathan Bliss, Isaac Burke, Courtney R. Gibbons, Martin Helmer,Serkan Hoşten, Evan D. Nash, Jose Israel Rodriguez, and Daniel Smolkin. “The maxi-mum likelihood degree of toric varieties”. In: J. Symbolic Comput.
92 (2019), pp. 222–242.[3] Carlos Améndola, Kathlén Kohn, Philipp Reichenbach, and Anna Seigal.
Invarianttheory and scaling algorithms for maximum likelihood estimation . 2020.[4] T. W. Anderson. “Estimation of covariance matrices which are linear combinations orwhose inverses are linear combinations of given matrices”. In:
Essays in Probability andStatistics . Univ. of North Carolina Press, Chapel Hill, N.C., 1970, pp. 1–24.[5] Nihat Ay, Jürgen Jost, Hông Vân Lê, and Schwachhöfer.
Information Geometry . SpringerVerlag, New York, 2017.[6] Daniel J. Bates, Andrew J. Sommese, Jonathan D. Hauenstein, and Charles W. Wampler.
Numerically Solving Polynomial Systems with Bertini . Philadelphia, PA: Society forIndustrial and Applied Mathematics, 2013.197] Paul Breiding and Sascha Timme. “HomotopyContinuation.jl: A Package for Homo-topy Continuation in Julia”. In:
Mathematical Software – ICMS 2018 . Ed. by James H.Davenport, Manuel Kauers, George Labahn, and Josef Urban. Cham: Springer Inter-national Publishing, 2018, pp. 458–465.[8] Soojin Cho. “Polytopes of roots of type A n ”. In: Bull. Austral. Math. Soc.
VoronoiCells of Varieties . 2018.[10] Rodrigo A. Collazo, Christiane Görgen, and Jim Q. Smith.
Chain event graphs . Chap-man & Hall/CRC Computer Science and Data Analysis Series. CRC Press, Boca Ra-ton, FL, 2018.[11] Sandra Di Rocco, David Eklund, and Madeleine Weinstein. “The Bottleneck Degree ofAlgebraic Varieties”. In:
SIAM J. Appl. Algebra Geom.
Found. Comput. Math.
Lectures on algebraic statistics .Vol. 39. Oberwolfach Seminars. Birkhäuser Verlag, Basel, 2009.[14] Eliana Duarte, Orlando Marigliano, and Bernd Sturmfels.
Discrete Statistical Modelswith Rational Maximum Likelihood Estimator . 2019. to appear in Bernoulli.[15] Nicholas Eriksson, Stephen E. Fienberg, Alessandro Rinaldo, and Seth Sullivant. “Poly-hedral conditions for the nonexistence of the MLE for hierarchical log-linear models”.In:
J. Symbolic Comput.
J. Amer. Statist. Assoc.
65 (1970), pp. 1610–1616.[17] Luis David Garcia, Michael Stillman, and Bernd Sturmfels. “Algebraic geometry ofBayesian networks”. In:
J. Symbolic Comput.
Ann. Statist.
The Arnold-Gelfand math-ematical seminars . Birkhäuser Boston, Boston, MA, 1997, pp. 205–221.[20] Elizabeth Gross and Jose Israel Rodriguez. “Maximum likelihood geometry in the pres-ence of data zeros”. In:
ISSAC 2014—Proceedings of the 39th International Symposiumon Symbolic and Algebraic Computation . ACM, New York, 2014, pp. 232–239.[21] Emil Horobeţ and Madeleine Weinstein. “Offset hypersurfaces and persistent homologyof algebraic varieties”. In:
Comput. Aided Geom. Design
74 (2019), pp. 101767, 14.[22] Serkan Hoşten, Amit Khetan, and Bernd Sturmfels. “Solving the likelihood equations”.In:
Found. Comput. Math.
J. Algebr. Stat.
Combinatorial algebraicgeometry . Vol. 2108. Lecture Notes in Math. Springer, Cham, 2014, pp. 63–117.[25] Michael Joswig and Katja Kulas. “Tropical and ordinary convexity combined”. In:
Adv.Geom.
Pastry Art . Accessed: 2020-06-09. url : (visited on 06/10/2020).[27] Kaie Kubjas, Elina Robeva, and Bernd Sturmfels. “Fixed points EM algorithm andnonnegative rank boundaries”. In: Ann. Statist.
Graphical models . Vol. 17. Oxford Statistical Science Series. Ox-ford Science Publications. The Clarendon Press, Oxford University Press, New York,1996.[29] John B. Little. “The many lives of the twisted cubic”. In:
Amer. Math. Monthly
Invitation to Nonlinear Algebra . GraduateStudies in Mathematics. https : / / personal - homepages . mis . mpg . de / michalek /NonLinearAlgebra.pdf . American Mathematical Society, Providence, RI, 2021.[31] Shmuel Onn. “Geometry, complexity, and combinatorics of permutation polytopes”. In:
J. Combin. Theory Ser. A
Algebraic statistics for computational biology .Cambridge University Press, New York, 2005.[33] Igor R. Shafarevich.
Basic algebraic geometry. 1 . Russian. Varieties in projective space.Springer, Heidelberg, 2013.[34] Jim Q. Smith and Paul E. Anderson. “Conditional independence and chain eventgraphs”. In:
Artificial Intelligence
The numerical solution of systems ofpolynomials . Arising in engineering and science. World Scientific Publishing Co. Pte.Ltd., Hackensack, NJ, 2005.[36] Frank Sottile. “Toric ideals, real toric varieties, and the moment map”. In:
Topics inalgebraic geometry and geometric modeling . Vol. 334. Contemp. Math. Amer. Math.Soc., Providence, RI, 2003, pp. 225–240.[37] Seth Sullivant.
Algebraic statistics . Vol. 194. Graduate Studies in Mathematics. Amer-ican Mathematical Society, Providence, RI, 2018.[38] Halyna Syta and Rien van de Weygaert.
Life and Times of Georgy Voronoi . 2009.[39] Georges Voronoi. “Nouvelles applications des paramètres continus à la théorie desformes quadratiques. Deuxième mémoire. Recherches sur les parallélloèdres primitifs”.In:
J. Reine Angew. Math.
134 (1908), pp. 198–287.2140] Nico Zavallos.