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Yaroslav Shitov

Publications and source records attributed to Yaroslav Shitov.

At least 19 recordsLinked to original sources
arXiv2020

Sublinear extensions of polygons

Every convex polygon with $n$ vertices is a linear projection of a higher-dimensional polytope with at most $147\,n^{2/3}$ facets.

math.CO↗
arXiv2020

A sign pattern with non-zero elements on the diagonal whose minimal rank realizations are not diagonalizable over the complex numbers

The rank of the $9\times 9$ matrix $$ \left( \begin{array}{cccc|c|cccc} 1&1&0&0&1&0&0&0&0\\ 1&1&0&0&0&0&0&0&0\\ 0&0&1&1&1&0&0&0&0\\ 0&0&1&1&0&0&0&0&0\\\hline 0&0&0&0&1&0&1&0&1\\\hline 0&0&0&0&0&1&1&0&0\\ 0&0&0&0&0&1&1&0&0\\ 0&0&0&0&0&0&0&1&1\\ 0&0&0&0&0&0&0&1&1 \end{array} \right) $$ is $6$. If we replace the ones by arbitrary non-zero numbers, we get a matrix $B$ with $\operatorname{rank} B\geqslant6$, and if $\operatorname{rank} B=6$, the $6\times 6$ principal minors of $B$ vanish.

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arXiv2019

Linear mappings preserving the copositive cone

Let $\mathcal{S}_n$ be the set of all $n$-by-$n$ symmetric real matrices, and let $\mathcal{C}_n$ be the copositive cone, that is, the set of all matrices $a\in\mathcal{S}_n$ that fulfill the condition $u^\top a u\geqslant0$ for all $n$-vectors $u$ with nonnegative entries. We prove that a linear mapping $φ:\mathcal{S}_n\to \mathcal{S}_n$ satisfies $φ(\mathcal{C}_n)=\mathcal{C}_n$ if and only if $$φ(x)=m^\top xm$$ for a fixed monomial matrix $m$ with nonnegative entries.

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arXiv2019

Nonnegative rank depends on the field

We present an example of a subfield $\mathcal{F}\subset\mathbb{R}$ and a matrix $A$ whose conventional and nonnegative ranks equal five, but the nonnegative rank with respect to $\mathcal{F}$ equals six. In other words, $A$ can be represented as a sum of five rank-one matrices with nonnegative real entries but not as a sum of five rank-one matrices with nonnegative entries in $\mathcal{F}$.

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arXiv2019

One more counterexample on sign patterns

The \textit{sepr-sequence} of an $n\times n$ real matrix $A$ is $(s_1,\ldots,s_n)$, where $s_k$ is the subset of those signs of $+,-,0$ that appear in the values of the $k\times k$ principal minors of $A$. The $12\times 12$ matrix $$\left(\begin{array}{cccccc|ccc|ccc} 0&0&0&0&0&0&0&0&0&a_1&0&0\\ 0&0&0&0&0&0&0&0&0&0&a_2&0\\ 0&0&0&0&0&0&0&0&0&0&0&a_3\\ 0&0&0&0&0&0&0&0&0&0&0&a_4\\ 0&0&0&0&0&0&0&0&0&0&0&a_5\\ 0&0&0&0&0&0&0&0&0&0&0&a_6\\\hline b_1&b_2&0&0&0&0&0&0&0&0&0&0\\ b_3&b_4&0&0&b_5&-b_6&0&0&0&0&0&0\\ 0&b_7&b_8&-b_9&b_{10}&b_{11}&0&0&0&0&0&0\\\hline 0&0&0&0&0&0&c_1&0&0&0&0&0\\ 0&0&0&0&0&0&0&c_2&0&0&0&0\\ 0&0&0&0&0&0&0&0&c_3&0&0&0 \end{array}\right)$$ does always have $s_k=\{0,+,-\}$ if $k=3,6,9$ and $s_k=\{0\}$ otherwise, provided that the variables are positive. However, every principal $9\times 9$ minor that is not identically zero can take values of both signs.

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arXiv2019

An improvement to a recent upper bound for synchronizing words of finite automata

It has been known since the 60's that any complete discrete $n$-state automaton admits a reset word of length not exceeding $αn^3+o(n^3)$ for some absolute constant $α$. J.-E. Pin and P. Frankl proved this statement with $α=1/6=0.1666...$ in 1982, and this bound remained best known until 2017, when M. Szykuła decreased its value to $α\approx0.1664$. In this note, we present a modification to the latest approach and develop a different counting argument which leads to a more substantial improvement of $α\leqslant 0.1654$.

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arXiv2018

Quantum version of Wielandt's Inequality revisited

Consider a linear space L of complex D-dimensional linear operators, and assume that some power L^k of L is the whole space of DxD matrices. Perez-Garcia, Verstraete, Wolf and Cirac conjectured that the sequence L^1,L^2,... stablilizes after O(D^2) terms; we prove that this happens after O(D^2 log(D)) terms, improving the previously known bound of O(D^4).

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arXiv2018

An improved bound for the length of matrix algebras

Let $S$ be a set of $n\times n$ matrices over a field $\mathbb{F}$. We show that the $\mathbb{F}$-linear span of the words in $S$ of length at most $$2n\log_2n+4n$$ is the full $\mathbb{F}$-algebra generated by $S$. This improves on the $n^2/3+2/3$ bound by Paz (1984) and an $O\left(n^{1.5}\right)$ bound of Pappacena (1997).

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arXiv2018

A universality theorem for nonnegative matrix factorizations

Let $A$ be a matrix with nonnegative real entries. A nonnegative factorization of size $k$ is a representation of $A$ as a sum of $k$ nonnegative rank-one matrices. The space of all such factorizations is a bounded semialgebraic set, and we prove that spaces arising in this way are universal. More presicely, we show that every bounded semialgebraic set $U$ is rationally equivalent to the set of nonnegative size-$k$ factorizations of some matrix $A$ up to a permutation of matrices in the factorization. We prove that, if $U\subset\mathbb{R}^n$ is given as the zero locus of a polynomial with coefficients in $\mathbb{Q}$, then such a pair $(A,k)$ can be computed in polynomial time. This result gives a complete description of the algorithmic complexity of nonnegative rank, and it also allows one to solve the problem of Cohen and Rothblum on nonnegative factorizations restricted to matrices over different subfields of $\mathbb{R}$.

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arXiv2017

The nonnegative rank of a matrix: Hard problems, easy solutions

Using elementary linear algebra, we develop a technique that leads to solutions of two widely known problems on nonnegative matrices. First, we give a short proof of the result by Vavasis stating that the nonnegative rank of a matrix is NP-hard to compute. This proof is essentially contained in the paper by Jiang and Ravikumar, who discussed this topic in different terms fifteen years before the work of Vavasis. Secondly, we present a solution of the problem of Cohen and Rothblum on rational nonnegative factorizations, which was posed in 1993 and remained open.

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arXiv2017

A separation between tropical matrix ranks

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 prove the existence of a $01$-matrix with many ones and without large all-one submatrices, and we put our results together and construct an $n\times n$ matrix with tropical rank $o(n^{0.5+\varepsilon})$ and Kapranov rank $n-o(n)$.

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arXiv2017

On the complexity of graph coloring with additional local conditions

Let $G = (V,E)$ be a finite simple graph. Recall that a proper coloring of $G$ is a mapping $φ: V\to\{1,\ldots,k\}$ such that every color class induces an independent set. Such a $φ$ is called a semi-matching coloring if the union of any two consecutive color classes induces a matching. We show that the semi-matching coloring problem is NP-complete for any fixed $k\geqslant 3$, and we get the same result for another version of this problem in which any triangle of G is required to have vertices whose colors differ at least by three.

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arXiv2017

On tropical and nonnegative factorization ranks of band matrices

Matrix factorization problems over various semirings naturally arise in different contexts of modern pure and applied mathematics. These problems are very hard in general and cause computational difficulties in applications. We give a survey of what is known on the algorithmic complexity of Boolean, fuzzy, tropical, nonnegative, and positive semidefinite factorizations, and we examine the behavior of the corresponding rank functions on matrices of bounded bandwidth. We show that the Boolean, fuzzy, and tropical versions of matrix factorization become polynomial time solvable when restricted to this class of matrices, and we also show that the nonnegative rank of a tridiagonal matrix is easy to compute. We recall several open problems from earlier papers on the topic and formulate many new problems.

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arXiv2017

A counterexample to Comon's conjecture

We present an example of a symmetric tensor of size $800\times 800\times 800$ which can be written a sum of $903$ simple tensors with complex entries but not as a sum of $903$ symmetric simple tensors.

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