Related papers: Separating the complexity classes NL and NP

In a paper by the first author it was shown that for certain arithmetical results on conjugacy class sizes it is enough to only consider the vanishing conjugacy class sizes. In this paper we further weaken the conditions to consider only…

The $P$ versus $NP$ problem is still unsolved. But there are several oracles with $P$ unequal $NP$ relative to them. Here we will prove, that $P\not=NP$ relative to a $P$-complete oracle. In this paper, we use padding arguments as the proof…

This paper establishes the separation of complexity classes $\mathbf{P}$ and $\mathbf{NP}$ through a novel homological algebraic approach grounded in category theory. We construct the computational category $\mathbf{Comp}$, embedding…

Complex-valued neural networks are not a new concept, however, the use of real-valued models has often been favoured over complex-valued models due to difficulties in training and performance. When comparing real-valued versus…

Second-order methods are of great importance for composite convex optimization problems due to their local super-linear convergence rates (under appropriate assumptions). However, the presence of even a simple nonsmooth function in the…

The second author previously discussed how classical complexity separation conjectures, we call them "axioms", have implications in three manifold topology: polynomial length stings of operations which preserve certain Jones polynomial…

In the Structure of Appearance and in Problems and Projects, Nelson Goodman has constructed a theory of complexity whose elements are the predicates of a system. One of his main results is a closed formula to evaluate v[n-pl], the maximum…

We analyse the complexity of learning first-order queries in a model-theoretic framework for supervised learning introduced by (Grohe and Tur\'an, TOCS 2004). Previous research on the complexity of learning in this framework focussed on the…

Learning complex programs through inductive logic programming (ILP) remains a formidable challenge. Existing higher-order enabled ILP systems show improved accuracy and learning performance, though remain hampered by the limitations of the…

We first highlight the main differences between second order and higher order linear parabolic equations. Then we survey existing results for the latter, in particular by analyzing the behavior of the convolution kernels. We illustrate the…

Deep neural networks use multiple layers of functions to map an object represented by an input vector progressively to different representations, and with sufficient training, eventually to a single score for each class that is the output…

We analyze the trade-off between model complexity and accuracy for random forests by breaking the trees up into individual classification rules and selecting a subset of them. We show experimentally that already a few rules are sufficient…

Given a neural network, training data, and a threshold, it was known that it is NP-hard to find weights for the neural network such that the total error is below the threshold. We determine the algorithmic complexity of this fundamental…

We show that unlike machine learning classifiers, there are no complex boundary structures in the decision boundaries for well-trained deep models. However, we found that the complicated structures do appear in training but they vanish…

Using only undergraduate-level methods, we classify all groups of order $p^4$, where $p$ is an odd prime.

First-order logic is known to have limited expressive power over finite structures. It enjoys in particular the locality property, which states that first-order formulae cannot have a global view of a structure. This limitation ensures on…

In the last decade, Convolutional Neural Network with a multi-layer architecture has advanced rapidly. However, training its complex network is very space-consuming, since a lot of intermediate data are preserved across layers, especially…

This paper presents a novel and straight formulation, and gives a complete insight towards the understanding of the complexity of the problems of the so called NP-Class. In particular, this paper focuses in the Searching of the Optimal…

Nonuniformity is a central concept in computational complexity with powerful connections to circuit complexity and randomness. Nonuniform reductions have been used to study the isomorphism conjecture for NP and completeness for larger…

This paper has been withdrawn by the author due to a mistake in the proof of the main theorem.