Related papers: Rank-Based Norms, Capra-Conjugacies and the Rank F…
We determine the rank of a random matrix over an arbitrary field with prescribed numbers of non-zero entries in each row and column. As an application we obtain a formula for the rate of low-density parity check codes. This formula…
We use techniques from (tracial noncommutative) polynomial optimization to formulate hierarchies of semidefinite programming lower bounds on matrix factorization ranks. In particular, we consider the nonnegative rank, the positive…
We consider change-point tests based on rank statistics to test for structural changes in long-range dependent observations. Under the hypothesis of stationary time series and under the assumption of a change with decreasing change-point…
We prove that for two-component maps in dimension two, rank-one convexity is equivalent to quasiconvexity. The essential tool for the proof is a fixed-point argument for a suitable set-valued map going from one component to the other that…
We consider the problem of recovering low-rank matrices from random rank-one measurements, which spans numerous applications including covariance sketching, phase retrieval, quantum state tomography, and learning shallow polynomial neural…
Let $k$ be a field of characteristic zero and let $K=k(t)$ be the rational function field over $k$. In this paper we combine a formula of Ulmer for ranks of certain Jacobians over $K$ with strong upper bounds on endomorphisms of Jacobians…
This paper delves into vector and matrix norms of Fibonacci numbers. Two classes of Fibonacci vectors and a parametric p-norm are defined. From this definition, several properties of Fibonacci vector and matrix p-norms are described by…
Multiclass classification (MCC) is a fundamental machine learning problem of classifying each instance into one of a predefined set of classes. In the deep learning era, extensive efforts have been spent on developing more powerful neural…
Conjugation, or Legendre transformation, is a basic tool in convex analysis, rational mechanics, economics and optimization. It maps a function on a linear topological space into another one, defined in the dual of the linear space by…
This work adresses the question of density of piecewise constant (resp. rigid) functions in the space of vector valued functions with bounded variation (resp. deformation) with respect to the strict convergence. Such an approximation…
The link prediction task on knowledge graphs without explicit negative triples in the training data motivates the usage of rank-based metrics. Here, we review existing rank-based metrics and propose desiderata for improved metrics to…
A duality of $\kappa$-normed topological vector spaces is defined and investigated. For such spaces the analog of the Mackey-Arens theorem is proved. There are investigated cases, when $\kappa$-normability of a topological vector space…
We consider the problem of minimizing a linear function over an affine section of the cone of positive semidefinite matrices, with the additional constraint that the feasible matrix has prescribed rank. When the rank constraint is active,…
We determine the rank of a random matrix A over a finite field with prescribed numbers of non-zero entries in each row and column. As an application we obtain a formula for the rate of low-density parity check codes. This formula verifies a…
Adaptive nuclear-norm penalization is proposed for low-rank matrix approximation, by which we develop a new reduced-rank estimation method for the general high-dimensional multivariate regression problems. The adaptive nuclear norm of a…
We continue to study the rank functions of tropical matrices. In this paper, we explain how to reduce the computation of ranks for matrices over the `supertropical semifield' to the standard tropical case. Using a counting approach, we…
We propose a multivariate functional responses low rank regression model with possible high dimensional functional responses and scalar covariates. By expanding the slope functions on a set of sieve basis, we reconstruct the basis…
In this paper we further develop the theory of matrices over the extended tropical semiring. Introducing a notion of tropical linear dependence allows for a natural definition of matrix rank in a sense that coincides with the notions of…
A matrix always has a full rank submatrix such that the rank of this matrix is equal to the rank of that submatrix. This property is one of the corner stones of the matrix rank theory. We call this property the max-full-rank-submatrix…
A criterion is given for a type in a finite rank stable theory to be (almost) internal to a given nonmodular minimal type. The motivation comes from results of Campana which give criteria for compact complex analytic spaces to be algebraic…