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Papers by Maria Angelica Cueto

In this paper we present an effective method for linearizing rational varieties of codimension at... more In this paper we present an effective method for linearizing rational varieties of codimension at least two under Cremona transformations, starting from a given parametrization. Using these linearizing Cremonas, we simplify the equations of secant and tangential varieties of some classical examples, including Veronese, Segre and Grassmann varieties. We end the paper by treating the special case of the Segre embedding of the n-fold product of projective spaces, where cumulant Cremonas, arising from algebraic statistics, appear as specific cases of our general construction.

The mixed discriminant of n Laurent polynomials in n variables is the irreducible polynomial in t... more The mixed discriminant of n Laurent polynomials in n variables is the irreducible polynomial in the coefficients which vanishes whenever two of the roots coincide. The Cayley trick expresses the mixed discriminant as an A-discriminant. We show that the degree of the mixed discriminant is a piecewise linear function in the Plucker coordinates of a mixed Grassmannian. An explicit degree formula is given for the case of plane curves.

We construct and study an embedded weighted balanced graph in R n+1 parameterized by a strictly i... more We construct and study an embedded weighted balanced graph in R n+1 parameterized by a strictly increasing sequence of n coprime numbers {i1, . . . , in}, called the tropical secant surface graph. We identify it with the tropicalization of a surface in C n+1 parameterized by binomials. Using this graph, we construct the tropicalization of the first secant variety of a monomial projective curve with exponent vector (0, i1, . . . , in), which can be described by a balanced graph called the tropical secant graph. The combinatorics involved in computing the degree of this classical secant variety is non-trivial. One earlier approach to this is due to K. Ranestad. Using techniques from tropical geometry, we give algorithms to effectively compute this degree (as well as its multidegree) and the Newton polytope of the first secant variety of any given monomial curve in P 4 .

The nodes of the de Bruijn graph B(d,3) consist of all strings of length 3, taken from an alphabe... more The nodes of the de Bruijn graph B(d,3) consist of all strings of length 3, taken from an alphabet of size d, with edges between words which are distinct substrings of a word of length 4. We give an inductive characterization of the maximum independent sets of the de Bruijn graphs B(d,3) and for the de Bruijn graph of diameter three with loops removed, for arbitrary alphabet size. We derive a recurrence relation and an exponential generating function for their number. This recurrence allows us to construct exponentially many comma-free codes of length 3 with maximal cardinality.

We study tree metrics that can be realized as a mixture of two star tree metrics. We prove that t... more We study tree metrics that can be realized as a mixture of two star tree metrics. We prove that the only trees admitting such a decomposition are the ones coming from a tree with at most one internal edge, and whose weight satisfies certain linear inequalities. We also characterize the fibers of the corresponding mixture map. In addition, we discuss the general framework of tropical secant varieties and we interpret our results within this setting. Finally, we show that the set of tree metric ranks of metrics on nnn taxa is unbounded.

The restricted Boltzmann machine is a graphical model for binary random variables. Based on a com... more The restricted Boltzmann machine is a graphical model for binary random variables. Based on a complete bipartite graph separating hidden and observed variables, it is the binary analog to the factor analysis model. We study this graphical model from the perspectives of algebraic statistics and tropical geometry, starting with the observation that its Zariski closure is a Hadamard power of the first secant variety of the Segre variety of projective lines. We derive a dimension formula for the tropicalized model, and we use it to show that the restricted Boltzmann machine is identifiable in many cases. Our methods include coding theory and geometry of linear threshold functions.

In this paper we further develop the theory of geometric tropicalization due to Hacking, Keel and... more In this paper we further develop the theory of geometric tropicalization due to Hacking, Keel and Tevelev and we describe tropical methods for implicitization of surfaces. More precisely, we enrich this theory with a combinatorial formula for tropical multiplicities of regular points in arbitrary dimension and we prove a conjecture of Sturmfels and Tevelev regarding sufficient combinatorial conditions to compute tropical varieties via geometric tropicalization. Using these two results, we extend previous work of Sturmfels, Tevelev and Yu for tropical implicitization of generic surfaces, and we provide methods for approaching the non-generic cases.

Journal of Symbolic Computation, 2010

We use tropical geometry to compute the multidegree and Newton polytope of the hypersurface of a ... more We use tropical geometry to compute the multidegree and Newton polytope of the hypersurface of a statistical model with two hidden and four observed binary random variables, solving an open question stated by Drton, Sturmfels and Sullivant in (Drton et al., 2009, Ch. VI, Problem 7.7). The model is obtained from the undirected graphical model of the complete bipartite graph K2,4K2,4 by marginalizing two of the six binary random variables. We present algorithms for computing the Newton polytope of its defining equation by parallel walks along the polytope and its normal fan. In this way we compute vertices of the polytope. Finally, we also compute and certify its facets by studying tangent cones of the polytope at the symmetry classes of vertices. The Newton polytope has 17 214 912 vertices in 44 938 symmetry classes and 70 646 facets in 246 symmetry classes.

We study generalized Horn-Kapranov rational parametrizations of inhomogeneous sparse discriminant... more We study generalized Horn-Kapranov rational parametrizations of inhomogeneous sparse discriminants from both a theoretical and an algorithmic perspective. We show that all these parametrizations are birational and prove some results on the corresponding implicit equations. We also propose a combinatorial algorithm to compute the degree of inhomogeneous discriminantal surfaces associated to uniform matrices.

Bulletin of Mathematical Biology, 2010

It is well known among phylogeneticists that adding an extra taxon (e.g. species) to a data set c... more It is well known among phylogeneticists that adding an extra taxon (e.g. species) to a data set can alter the structure of the optimal phylogenetic tree in surprising ways. However, little is known about this "rogue taxon" effect. In this paper we characterize the behavior of balanced minimum evolution (BME) phylogenetics on data sets of this type using tools from polyhedral geometry. First we show that for any distance matrix there exist distances to a "rogue taxon" such that the BME-optimal tree for the data set with the new taxon does not contain any nontrivial splits (bipartitions) of the optimal tree for the original data. Second, we prove a theorem which restricts the topology of BME-optimal trees for data sets of this type, thus showing that a rogue taxon cannot have an arbitrary effect on the optimal tree. Third, we construct polyhedral cones computationally which give complete answers for BME rogue taxon behavior when our original data fits a tree on four, five, and six taxa. We use these cones to derive sufficient conditions for rogue taxon behavior for four taxa, and to understand the frequency of the rogue taxon effect via simulation.

In this paper we present an effective method for linearizing rational varieties of codimension at... more In this paper we present an effective method for linearizing rational varieties of codimension at least two under Cremona transformations, starting from a given parametrization. Using these linearizing Cremonas, we simplify the equations of secant and tangential varieties of some classical examples, including Veronese, Segre and Grassmann varieties. We end the paper by treating the special case of the Segre embedding of the n-fold product of projective spaces, where cumulant Cremonas, arising from algebraic statistics, appear as specific cases of our general construction.

The mixed discriminant of n Laurent polynomials in n variables is the irreducible polynomial in t... more The mixed discriminant of n Laurent polynomials in n variables is the irreducible polynomial in the coefficients which vanishes whenever two of the roots coincide. The Cayley trick expresses the mixed discriminant as an A-discriminant. We show that the degree of the mixed discriminant is a piecewise linear function in the Plucker coordinates of a mixed Grassmannian. An explicit degree formula is given for the case of plane curves.

We construct and study an embedded weighted balanced graph in R n+1 parameterized by a strictly i... more We construct and study an embedded weighted balanced graph in R n+1 parameterized by a strictly increasing sequence of n coprime numbers {i1, . . . , in}, called the tropical secant surface graph. We identify it with the tropicalization of a surface in C n+1 parameterized by binomials. Using this graph, we construct the tropicalization of the first secant variety of a monomial projective curve with exponent vector (0, i1, . . . , in), which can be described by a balanced graph called the tropical secant graph. The combinatorics involved in computing the degree of this classical secant variety is non-trivial. One earlier approach to this is due to K. Ranestad. Using techniques from tropical geometry, we give algorithms to effectively compute this degree (as well as its multidegree) and the Newton polytope of the first secant variety of any given monomial curve in P 4 .

The nodes of the de Bruijn graph B(d,3) consist of all strings of length 3, taken from an alphabe... more The nodes of the de Bruijn graph B(d,3) consist of all strings of length 3, taken from an alphabet of size d, with edges between words which are distinct substrings of a word of length 4. We give an inductive characterization of the maximum independent sets of the de Bruijn graphs B(d,3) and for the de Bruijn graph of diameter three with loops removed, for arbitrary alphabet size. We derive a recurrence relation and an exponential generating function for their number. This recurrence allows us to construct exponentially many comma-free codes of length 3 with maximal cardinality.

We study tree metrics that can be realized as a mixture of two star tree metrics. We prove that t... more We study tree metrics that can be realized as a mixture of two star tree metrics. We prove that the only trees admitting such a decomposition are the ones coming from a tree with at most one internal edge, and whose weight satisfies certain linear inequalities. We also characterize the fibers of the corresponding mixture map. In addition, we discuss the general framework of tropical secant varieties and we interpret our results within this setting. Finally, we show that the set of tree metric ranks of metrics on nnn taxa is unbounded.

The restricted Boltzmann machine is a graphical model for binary random variables. Based on a com... more The restricted Boltzmann machine is a graphical model for binary random variables. Based on a complete bipartite graph separating hidden and observed variables, it is the binary analog to the factor analysis model. We study this graphical model from the perspectives of algebraic statistics and tropical geometry, starting with the observation that its Zariski closure is a Hadamard power of the first secant variety of the Segre variety of projective lines. We derive a dimension formula for the tropicalized model, and we use it to show that the restricted Boltzmann machine is identifiable in many cases. Our methods include coding theory and geometry of linear threshold functions.

In this paper we further develop the theory of geometric tropicalization due to Hacking, Keel and... more In this paper we further develop the theory of geometric tropicalization due to Hacking, Keel and Tevelev and we describe tropical methods for implicitization of surfaces. More precisely, we enrich this theory with a combinatorial formula for tropical multiplicities of regular points in arbitrary dimension and we prove a conjecture of Sturmfels and Tevelev regarding sufficient combinatorial conditions to compute tropical varieties via geometric tropicalization. Using these two results, we extend previous work of Sturmfels, Tevelev and Yu for tropical implicitization of generic surfaces, and we provide methods for approaching the non-generic cases.

Journal of Symbolic Computation, 2010

We use tropical geometry to compute the multidegree and Newton polytope of the hypersurface of a ... more We use tropical geometry to compute the multidegree and Newton polytope of the hypersurface of a statistical model with two hidden and four observed binary random variables, solving an open question stated by Drton, Sturmfels and Sullivant in (Drton et al., 2009, Ch. VI, Problem 7.7). The model is obtained from the undirected graphical model of the complete bipartite graph K2,4K2,4 by marginalizing two of the six binary random variables. We present algorithms for computing the Newton polytope of its defining equation by parallel walks along the polytope and its normal fan. In this way we compute vertices of the polytope. Finally, we also compute and certify its facets by studying tangent cones of the polytope at the symmetry classes of vertices. The Newton polytope has 17 214 912 vertices in 44 938 symmetry classes and 70 646 facets in 246 symmetry classes.

We study generalized Horn-Kapranov rational parametrizations of inhomogeneous sparse discriminant... more We study generalized Horn-Kapranov rational parametrizations of inhomogeneous sparse discriminants from both a theoretical and an algorithmic perspective. We show that all these parametrizations are birational and prove some results on the corresponding implicit equations. We also propose a combinatorial algorithm to compute the degree of inhomogeneous discriminantal surfaces associated to uniform matrices.

Bulletin of Mathematical Biology, 2010

It is well known among phylogeneticists that adding an extra taxon (e.g. species) to a data set c... more It is well known among phylogeneticists that adding an extra taxon (e.g. species) to a data set can alter the structure of the optimal phylogenetic tree in surprising ways. However, little is known about this "rogue taxon" effect. In this paper we characterize the behavior of balanced minimum evolution (BME) phylogenetics on data sets of this type using tools from polyhedral geometry. First we show that for any distance matrix there exist distances to a "rogue taxon" such that the BME-optimal tree for the data set with the new taxon does not contain any nontrivial splits (bipartitions) of the optimal tree for the original data. Second, we prove a theorem which restricts the topology of BME-optimal trees for data sets of this type, thus showing that a rogue taxon cannot have an arbitrary effect on the optimal tree. Third, we construct polyhedral cones computationally which give complete answers for BME rogue taxon behavior when our original data fits a tree on four, five, and six taxa. We use these cones to derive sufficient conditions for rogue taxon behavior for four taxa, and to understand the frequency of the rogue taxon effect via simulation.