For any positive real number @math , the @math -frame potential of @math unit vectors @math is defined as @math . In this paper, we focus on the special case @math and establish the unique minimizer of @math for @math . Our results completely solve the minimization problem of @math -frame potential when @math , which confirms a conjecture posed by Chen, Gonzales, Goodman, Kang and Okoudjou.
The cases @math and @math for Conjecture are already solved in @cite_12 and @cite_9 , respectively. The first new result for Conjecture is obtained by Glazyrin in @cite_14 who shows that an orthonormal basis in @math plus a repeated vector minimizes @math for any @math . Combining Glazyrin's result with the previous ones, the minimizer of @math is only known for @math . Recently, Park extented Glazyrin's result to the case @math where @math , and showed that an orthonormal basis plus @math repeated vectors is the minimizer for any @math (see @cite_5 ). But the minimal @math -frame potential problem remains open for the case @math when @math .
@cite_5 @cite_9 @cite_14 @cite_12
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An extension is given of a recent result of Glazyrin, showing that an orthonormal basis @math joined with the vectors @math , where @math minimizes the @math -frame potential for @math over all collections of @math vectors @math in @math . We develop fast and memory efficient numerical methods for learning functions of many variables that admit sparse representations in terms of general bounded orthonormal tensor product bases. Such functions appear in many applications including, e.g., various Uncertainty Quantification(UQ) problems involving the solution of parametric PDE that are approximately sparse in Chebyshev or Legendre product bases. We expect that our results provide a starting point for a new line of research on sublinear-time solution techniques for UQ applications of the type above which will eventually be able to scale to significantly higher-dimensional problems than what are currently computationally feasible. More concretely, let @math be a finite Bounded Orthonormal Product Basis (BOPB) of cardinality @math . We will develop methods that approximate any function @math that is sparse in the BOPB, that is, @math of the form @math with @math of cardinality @math . Our method has a runtime of just @math , uses only @math function evaluations on a fixed and nonadaptive grid, and not more than @math bits of memory. For @math , the runtime @math will be less than what is required to simply enumerate the elements of the basis @math ; thus our method is the first approach applicable in a general BOPB framework that falls into the class referred to as "sublinear-time". This and the similarly reduced sample and memory requirements set our algorithm apart from previous works based on standard compressive sensing algorithms such as basis pursuit which typically store and utilize full intermediate basis representations of size @math . We study a statistical model for the tensor principal component analysis problem introduced by Montanari and Richard: Given a order- @math tensor @math of the form @math , where @math is a signal-to-noise ratio, @math is a unit vector, and @math is a random noise tensor, the goal is to recover the planted vector @math . For the case that @math has iid standard Gaussian entries, we give an efficient algorithm to recover @math whenever @math , and certify that the recovered vector is close to a maximum likelihood estimator, all with high probability over the random choice of @math . The previous best algorithms with provable guarantees required @math . In the regime @math , natural tensor-unfolding-based spectral relaxations for the underlying optimization problem break down (in the sense that their integrality gap is large). To go beyond this barrier, we use convex relaxations based on the sum-of-squares method. Our recovery algorithm proceeds by rounding a degree- @math sum-of-squares relaxations of the maximum-likelihood-estimation problem for the statistical model. To complement our algorithmic results, we show that degree- @math sum-of-squares relaxations break down for @math , which demonstrates that improving our current guarantees (by more than logarithmic factors) would require new techniques or might even be intractable. Finally, we show how to exploit additional problem structure in order to solve our sum-of-squares relaxations, up to some approximation, very efficiently. Our fastest algorithm runs in nearly-linear time using shifted (matrix) power iteration and has similar guarantees as above. The analysis of this algorithm also confirms a variant of a conjecture of Montanari and Richard about singular vectors of tensor unfoldings. This paper considers the problem of designing maximum distance separable (MDS) codes over small fields with constraints on the support of their generator matrices. For any given @math binary matrix @math , the GM-MDS conjecture, proposed by , states that if @math satisfies the so-called MDS condition, then for any field @math of size @math , there exists an @math MDS code whose generator matrix @math , with entries in @math , fits the matrix @math (i.e., @math is the support matrix of @math ). Despite all the attempts by the coding theory community, this conjecture remains still open in general. It was shown, independently by and , that the GM-MDS conjecture holds if the following conjecture, referred to as the TM-MDS conjecture, holds: if @math satisfies the MDS condition, then the determinant of a transform matrix @math , such that @math fits @math , is not identically zero, where @math is a Vandermonde matrix with distinct parameters. In this work, we first reformulate the TM-MDS conjecture in terms of the Wronskian determinant, and then present an algebraic-combinatorial approach based on polynomial-degree reduction for proving this conjecture. Our proof technique's strength is based primarily on reducing inherent combinatorics in the proof. We demonstrate the strength of our technique by proving the TM-MDS conjecture for the cases where the number of rows ( @math ) of @math is upper bounded by @math . For this class of special cases of @math where the only additional constraint is on @math , only cases with @math were previously proven theoretically, and the previously used proof techniques are not applicable to cases with @math .