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The trace norm of a matrix is the sum of its singular values. This paper presents results on the minimum trace norm $psi_{n}left( mright) $ of $left( 0,1right) $-matrices of size $ntimes n$ with exactly $m$ ones. It is shown that: (1) if $ngeq2$ and $n<mleq2n,$ then $psi_{n}left( mright) leq sqrt{m+sqrt{2left( m-1right) }}$ , with equality if and only if $m$ is a prime; (2) if $ngeq4$ and $2n<mleq3n,$ then $psi_{n}left( mright) leq sqrt{m+2sqrt{2leftlfloor m/3rightrfloor }}$ , with equality if and only if $m$ is a prime or a double of a prime; (3) if $3n<mleq4n,$ then $psi_{n}left( mright) leqsqrt{m+2sqrt{m-2}}% $ , with equality if and only if there is an integer $kgeq1$ such that $m=12kpm2$ and $4kpm1,6kpm1,12kpm1$ are primes.
The permanent of a multidimensional matrix is the sum of products of entries over all diagonals. By Mincs conjecture, there exists a reachable upper bound on the permanent of 2-dimensional (0,1)-matrices. In this paper we obtain some generalizations
Using the $ell_1$-norm to regularize the estimation of the parameter vector of a linear model leads to an unstable estimator when covariates are highly correlated. In this paper, we introduce a new penalty function which takes into account the correl
The minimum rank of a simple graph $G$ is defined to be the smallest possible rank over all symmetric real matrices whose $ij$th entry (for $i eq j$) is nonzero whenever ${i,j}$ is an edge in $G$ and is zero otherwise. Minimum rank is a difficult par
Evaluating adversarial robustness amounts to finding the minimum perturbation needed to have an input sample misclassified. The inherent complexity of the underlying optimization requires current gradient-based attacks to be carefully tuned, initiali
We prove several results in the theory of fusion categories using the product (norm) and sum (trace) of Galois conjugates of formal codegrees. First, we prove that finitely-many fusion categories exist up to equivalence whose global dimension has a f