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The nearest point map of a real algebraic variety with respect to Euclidean distance is an algebraic function. For instance, for varieties of low rank matrices, the Eckart-Young Theorem states that this map is given by the singular value decomposition. This article develops a theory of such nearest point maps from the perspective of computational algebraic… (More)

We consider a natural generalization of an abelian Hidden Subgroup Problem where the subgroups and their cosets correspond to graphs of linear functions over a finite field F with d elements. The hidden functions of the generalized problem are not restricted to be linear but can also be m-variate polynomial functions of total degree n ≥ 2. For fixed m and… (More)

We show that the kernel I of the ring homomorphism R[y ij | i, j ∈ N, i > j] → R[s i , t i | i ∈ N] determined by y ij → s i s j + t i t j is generated by two types of polynomials: off-diagonal 3 × 3-minors and pentads. This confirms a conjecture by Drton, Sturmfels, and Sullivant on the Gaussian two-factor model. Our proof is computational: inspired by… (More)

Consider the " Number in Hand " multiparty communication complexity model, where k players holding inputs x1,. .. , x k ∈ {0, 1} n communicate to compute the value f (x1,. .. , x k) of a function f known to all of them. The main lower bound technique for the communication complexity of such problems is that of partition arguments: partition the k players… (More)

abstract We present an algorithm for computing the dimensions of higher secant varieties of minimal orbits. Experiments with this algorithm lead to many conjectures on secant dimensions, especially of Grassmannians and Segre products. For these two classes of minimal orbits, we also point out a relation between the existence of certain codes and… (More)

We extend Guillemin and Sternberg's Realization Theorem for transitive Lie algebras of formal vector fields to certain Lie algebras of formal first order differential operators, and show that Blattner's proof of the Realization Theorem allows for a computer implementation that automatically reproduces many realizations derived in the existing literature,… (More)

The nearest point map of a real algebraic variety with respect to Euclidean distance is an algebraic function. For instance, for varieties of low rank matrices, the Eckart-Young Theorem states that this map is given by the singular value decomposition. This article develops a theory of such nearest point maps from the perspective of computational algebraic… (More)