Related papers: Verifying an algorithm computing Discrete Vector F…

The analysis of digital images using homological procedures is an outstanding topic in the area of Computational Algebraic Topology. In this paper, we describe a certified reduction strategy to deal with digital images, but preserving their…

This notes explains how a standard algorithm that constructs the discrete Fourier transform has been formalised and proved correct in the Coq proof assistant using the SSReflect extension.

We propose a new library to model and verify hardware circuits in the Coq proof assistant. This library allows one to easily build circuits by following the usual pen-and-paper diagrams. We define a deep-embedding: we use a (dependently…

Persistent homology is one of the most active branches of Computational Algebraic Topology with applications in several contexts such as optical character recognition or analysis of point cloud data. In this paper, we report on the formal…

This paper describes a formalization of discrete real closed fields in the Coq proof assistant. This abstract structure captures for instance the theory of real algebraic numbers, a decidable subset of real numbers with good algorithmic…

Computing homology and cohomology is at the heart of many recent works and a key issue for topological data analysis. Among homological objects, homology generators are useful to locate or understand holes (especially for geometric…

We present a fast algorithm for computing discrete cubical homology of graphs over finite fields with an appropriate characteristic. This algorithm improves on several computational steps compared to constructions in the existing…

In this paper, three Computational Topology methods (namely effective homology, persistent homology and discrete vector fields) are mixed together to produce algorithms for homological digital image processing. The algorithms have been…

In this paper, I present VQ-DRAW, an algorithm for learning compact discrete representations of data. VQ-DRAW leverages a vector quantization effect to adapt the sequential generation scheme of DRAW to discrete latent variables. I show that…

In mathematics, it is common practice to have several constructions for the same objects. Mathematicians will identify them modulo isomorphism and will not worry later on which construction they use, as theorems proved for one construction…

Vector Quantization (VQ) is a well-known technique in deep learning for extracting informative discrete latent representations. VQ-embedded models have shown impressive results in a range of applications including image and speech…

As the variety of commercially available quantum computers continues to increase so does the need for tools that can characterize, verify and validate these computers. This work explores using quantum state tomography for characterizing the…

This extended abstract is about an effort to build a formal description of a triangulation algorithm starting with a naive description of the algorithm where triangles, edges, and triangulations are simply given as sets and the most complex…

Topological abstractions offer a method to summarize the behavior of vector fields but computing them robustly can be challenging due to numerical precision issues. One alternative is to represent the vector field using a discrete approach,…

We describe a method for building composable and extensible verification procedures within the Coq proof assistant. Unlike traditional methods that rely on run-time generation and checking of proofs, we use verified-correct procedures with…

An invaluable feature of computer algebra systems is their ability to plot the graph of functions. Unfortunately, when one is trying to design a library of mathematical functions, this feature often falls short, producing incorrect and…

In this paper we introduce a new approach to computing hidden features of sampled vector fields. The basic idea is to convert the vector field data to a graph structure and use tools designed for automatic, unsupervised analysis of graphs.…

Oblique images are aerial photographs taken at oblique angles to the earth's surface. Projections of vector and other geospatial data in these images depend on camera parameters, positions of the geospatial entities, surface terrain,…

Many current watermarking algorithms insert data in the spatial or transform domains like the discrete cosine, the discrete Fourier, and the discrete wavelet transforms. In this paper, we present a data-hiding algorithm that exploits the…

In Compressed Sensing, a real-valued sparse vector has to be estimated from an underdetermined system of linear equations. In many applications, however, the elements of the sparse vector are drawn from a finite set. For the estimation of…