Related papers: The complexity of MinRank
We define the topological complexity sequence of a group as the sequence of topological complexities of its Milnor constructions. This sequence may be regarded as an intrinsic refinement of the topological complexity of a group and, unlike…
We provide new approximation guarantees for greedy low rank matrix estimation under standard assumptions of restricted strong convexity and smoothness. Our novel analysis also uncovers previously unknown connections between the low rank…
we derive new, improved lower bounds for the block complexity of an irrational algebraic number and for the number of digit changes in the b-ary expansion of an irrational algebraic number. To this end, we apply a quantitative version of…
In this paper, we define an ordering relation for a set of complex numbers, and research the properties and theorems of the ordering, solve some simple complex inequalities with the ordering.
Low-rank approximation of a matrix by means of random sampling has been consistently efficient in its empirical studies by many scientists who applied it with various sparse and structured multipliers, but adequate formal support for this…
Matrix Completion is the problem of recovering an unknown real-valued low-rank matrix from a subsample of its entries. Important recent results show that the problem can be solved efficiently under the assumption that the unknown matrix is…
We solve a long-standing problem by enumerating the number of non-degenerate Desargues configurations. We extend the result to the more difficult case involving Desargues blockline structures in Section 8. A transparent proof of Desargues…
The application of generalized inverses is usually neglected in pure mathematical research. However, it is very effective for this paper. We expand the famous matrix rank theorem due to R. Penrose to operators between Banach paces.…
This is a survey on the use of low-degree polynomials to predict and explain the apparent statistical-computational tradeoffs in a variety of average-case computational problems. In a nutshell, this framework measures the complexity of a…
We revisit Haagerup's enigmatic reduction theorem \cite[Theorems 2.1 \& 3.1]{HJX} showing how that theorem may be extended to general von Neumann algebras $\M$ equipped with an arbitrary faithful normal semifinite weight in a manner which…
The most developed aspect of the theory of finite semigroups is their classification in pseudovarieties. The main motivation for investigating such entities comes from their connection with the classification of regular languages via…
Machine learning algorithms generally suffer from a problem of explainability. Given a classification result from a model, it is typically hard to determine what caused the decision to be made, and to give an informative explanation. We…
We study the relative complexity of equivalence relations and preorders from computability theory and complexity theory. Given binary relations $R, S$, a componentwise reducibility is defined by $ R\le S \iff \ex f \, \forall x, y \, [xRy…
In this note we investigate the complexity of the Minimum Label Alignment problem and we show that such a problem is APX-hard.
The permanent vs. determinant problem is one of the most important problems in theoretical computer science, and is the main target of geometric complexity theory proposed by Mulmuley and Sohoni. The current best lower bound for the…
This document gives a list of finite semigroups that are interesting from the point of view of Krohn-Rhodes complexity theory. The list will be expanded and updates as "time goes by".
We give here a simple proof of the centrality of the congruence subgroup kernel in the higher rank isotropic case.
Minimizing the rank of a matrix subject to constraints is a challenging problem that arises in many applications in control theory, machine learning, and discrete geometry. This class of optimization problems, known as rank minimization, is…
We construct the first examples of an algorithmically complex finitely presented residually finite groups and first examples of finitely presented residually finite groups with arbitrarily large (recursive) Dehn function and depth function.…
Some examples are easier for humans to classify than others. The same should be true for deep neural networks (DNNs). We use the term example perplexity to refer to the level of difficulty of classifying an example. In this paper, we…