Related papers: Pebble-Depth
The theory of tunnel number 1 knots detailed in our previous paper, The tree of knot tunnels, provides a non-negative integer invariant called the depth of the tunnel. We give various results related to the depth invariant. Noting that it…
For a finite binary string $x$ its logical depth $d$ for significance $b$ is the shortest running time of a program for $x$ of length $K(x)+b$. There is another definition of logical depth. We give a new proof that the two versions are…
Phantom depth, phantom nonzerodivisors, and phantom exact sequences are analogues of the non-"phantom" notions which have been useful in tackling the (very difficult) localization problem in tight closure theory. In the present paper, these…
Deep learning has arguably achieved tremendous success in recent years. In simple words, deep learning uses the composition of many nonlinear functions to model the complex dependency between input features and labels. While neural networks…
Finite-state dimension (Dai, Lathrop, Lutz, and Mayordomo (2004)) quantifies the information rate in an infinite sequence as measured by finite-state automata. In this paper, we define a relative version of finite-state dimension. The…
We provide several new depth-based separation results for feed-forward neural networks, proving that various types of simple and natural functions can be better approximated using deeper networks than shallower ones, even if the shallower…
In their seminal paper Artemov and Protopopescu provide Hilbert formal systems, Brower-Heyting-Kolmogorov and Kripke semantics for the logics of intuitionistic belief and knowledge. Subsequently Krupski has proved that the logic of…
In the context of incompressible fluids, the observation that turbulent singular structures fail to be space filling is known as ``intermittency'' and it has strong experimental foundations. Consequently, as first pointed out by Landau,…
In this paper, we explore different ways to extend a recurrent neural network (RNN) to a \textit{deep} RNN. We start by arguing that the concept of depth in an RNN is not as clear as it is in feedforward neural networks. By carefully…
Based on its great successes in inference and denosing tasks, Dictionary Learning (DL) and its related sparse optimization formulations have garnered a lot of research interest. While most solutions have focused on single layer…
We study neural network compressibility by using singular learning theory to extend the minimum description length (MDL) principle to singular models like neural networks. Through extensive experiments on the Pythia suite with quantization,…
Membranes are of great technological and biological as well as theoretical interest. Two main classes of membranes can be distinguished: Fluid membranes and polymerized, tethered membranes. Here, we review progress in the theoretical…
We obtain in arbitrary codimension a removability result on the order of singularity of weak limits and bubbles of Willmore immersions measured by the second residue. This permits to reduce significantly the number of possible bubbling…
In studying the expressiveness of neural networks, an important question is whether there are functions which can only be approximated by sufficiently deep networks, assuming their size is bounded. However, for constant depths, existing…
Quasi-saddles or inherent saddles of the potential energy surface, $U$, of a liquid are defined as configurations which correspond to absolute minima of the pseudo-potential surface, $W =\wf$, as identified by a multi-dimensional…
There is no single universally accepted definition of "Complexity". There are several perspectives on complexity and what constitutes complex behaviour or complex systems, as opposed to regular, predictable behaviour and simple systems. In…
A non-distributive two-sorted hypersequent calculus \textbf{PDBL} and its modal extension \textbf{MPDBL} are proposed for the classes of pure double Boolean algebras and pure double Boolean algebras with operators respectively. A relational…
We propose a formal model of reasoning limitations in large neural net models for language, grounded in the depth of their neural architecture. By treating neural networks as linear operators over logic predicate space we show that each…
We propose a general framework for neural network compression that is motivated by the Minimum Description Length (MDL) principle. For that we first derive an expression for the entropy of a neural network, which measures its complexity…
We define and study a variant of the \emph{Stanley depth} which we call \emph{total depth} for partially ordered sets (posets). This total depth is the most natural variant of Stanley depth from $\llbracket S_k\rrbracket$ -- the poset of…