One-dimensional space | EPFL Graph Search

In physics and mathematics, a sequence of n numbers can specify a location in n-dimensional space. When n = 1, the set of all such locations is called a one-dimensional space. An example of a one-dimensional space is the number line, where the position of each point on it can be described by a single number. In algebraic geometry there are several structures that are technically one-dimensional spaces but referred to in other terms. A field k is a one-dimensional vector space over itself. Similarly, the projective line over k is a one-dimensional space. In particular, if , the complex numbers, then the complex projective line is one-dimensional with respect to , even though it is also known as the Riemann sphere. More generally, a ring is a length-one module over itself. Similarly, the projective line over a ring is a one-dimensional space over the ring. In case the ring is an algebra over a field, these spaces are one-dimensional with respect to the algebra, even if the algebra is of higher dimensionality. The hypersphere in 1 dimension is a pair of points, sometimes called a 0-sphere as its surface is zero-dimensional. Its length is where is the radius. Coordinate system One dimensional coordinate systems include the number line. Image:Coord NumberLine.

Neural-network quantum states for periodic systems in continuous space

Giuseppe Carleo, Gabriel Maria Pescia

We introduce a family of neural quantum states for the simulation of strongly interacting systems in the presence of spatial periodicity. Our variational state is parametrized in terms of a permutationally invariant part described by the Deep Sets neural-network architecture. The input coordinates to the Deep Sets are periodically transformed such that they are suitable to directly describe periodic bosonic systems. We show example applications to both one- and two-dimensional interacting quantum gases with Gaussian interactions, as well as to He-4 confined in a one-dimensional geometry. For the one-dimensional systems we find very precise estimations of the ground-state energies and the radial distribution functions of the particles. In two dimensions we obtain good estimations of the ground-state energies, comparable to results obtained from more conventional methods.

AMER PHYSICAL SOC2022

Reduced representations of complexes, signals, and multifiltrations

Celia Camille Hacker

The field of computational topology has developed many powerful tools to describe the shape of data, offering an alternative point of view from classical statistics. This results in a variety of complex structures that are not always directly amenable for machine learning tasks. We develop theory and algorithms to produce computable representations of simplicial or cell complexes, potentially equipped with additional information such as signals and multifiltrations. The common goal of the topics discussed in this thesis is to find reduced representations of these often high dimensional and complex structures to better visualize, transform or formulate theoretical results about them. We extend the well known graph learning algorithm node2vec to simplicial complexes, a higher dimensional analogue of graphs. To this end we propose a way to define random walks on simplicial complexes, which we then use to design an extension of node2vec called k-simplex2vec, producing a representation of the simplices in a Euclidean space. Furthermore, the study of this method leads to interesting questions about robustness of graph and simplicial learning methods. In the case of graphs, we study node2vec embeddings arising from different parameter sets, analysing their quality and stability using various measures. In the topic of signal processing, we explore how discrete Morse theory can be used for compression and reconstruction of cell complexes equipped with signals. In particular we study the effect of the compression of a complex on the Hodge decomposition of its signals. We study how the signal changes through compression and reconstruction by introducing a topological reconstruction error, showing in particular that part of the Hodge decomposition is preserved. Moreover, we prove that any deformation retract over R can be expressed as a Morse deformation retract in a well-chosen basis, thus extending the reconstruction results to any deformation retract. In addition, we introduce an algorithm to minimize the loss induced by the reconstruction of a compressed signal. Finally, we use discrete Morse theory to compute an invariant of multi-parameter persistent homology, the rank invariant. We can restrict a multi-parameter persistence module to a one- dimensional persistence module along any line of positive slope and compute the one-dimensional analogue of the rank invariant, namely the barcode. Through a discrete Morse matching we can determine critical values in the multifiltration, which in turn allows us to identify equivalence classes of lines in the parameter space. In our main result, we explain how to compute the barcode along any given line of an equivalence class given the barcode along a representative line. This provides a way to fiber the rank invariant according to the critical values of a discrete Morse matching and to perform computations in the corresponding one-dimensional module, which is much better understood.

EPFL2022

The Pristine survey - XV. A CHIT ESPaDOnS view on the Milky Way halo and disc populations

Pascale Jablonka, Willem Suter, Carmela Lardo, Romain Ely Roland Lucchesi

We present a one-dimensional, local thermodynamic equilibrium homogeneous analysis of 132 stars observed at high resolution with ESPaDOnS. This represents the largest sample observed at high resolution (R similar to 40 000) from the Pristine survey. This sample is based on the first version of the Pristine catalogue and covers the full range of metallicities from [Fe/II] similar to-3 to similar to+0.25, with nearly half of our sample (58 stars) composed of very metal-poor (VMP) stars ([Fe/H]

OXFORD UNIV PRESS2022

Reduced representations of complexes, signals, and multifiltrations

Celia Camille Hacker

EPFL2022