Stability of Reeb graphs under function perturbations: the case of closed curves

Reeb graphs provide a method for studying the shape of a manifold by encoding the evolution and arrangement of level sets of a simple Morse function defined on the manifold. Since their introduction in computer graphics they have been gaining popularity as an effective tool for shape analysis and matching. In this context one question deserving attention is whether Reeb graphs are robust against function perturbations. Focusing on 1-dimensional manifolds, we define an editing distance between Reeb graphs of curves, in terms of the cost necessary to transform one graph into another. Our main result is that changes in Morse functions induce smaller changes in the editing distance between Reeb graphs of curves, implying stability of Reeb graphs under function perturbations.Comment: 23 pages, 12 figure

OPTIMAL HOMEOMORPHISMS BETWEEN CLOSED CURVES

The concept of natural pseudo-distance has proven to be a powerful tool for measuring the dissimilarity between topological spaces endowed with continuous real-valued functions. Roughly speaking, the natural pseudo-distance is defined as the infimum of the change of the functions' values, when moving from one space to the other through homeomorphisms, if possible. In this paper, we prove the first available result about the existence of optimal homeomorphisms between closed curves, i.e. inducing a change of the function that equals the natural pseudo-distance

Filtrations induced by continuous functions

In Persistent Homology and Topology, filtrations are usually given by introducing an ordered collection of sets or a continuous function from a topological space to $\R^n$. A natural question arises, whether these approaches are equivalent or not. In this paper we study this problem and prove that, while the answer to the previous question is negative in the general case, the approach by continuous functions is not restrictive with respect to the other, provided that some natural stability and completeness assumptions are made. In particular, we show that every compact and stable $1$-dimensional filtration of a compact metric space is induced by a continuous function. Moreover, we extend the previous result to the case of multidimensional filtrations, requiring that our filtration is also complete. Three examples show that we cannot drop the assumptions about stability and completeness. Consequences of our results on the definition of a distance between filtrations are finally discussed

Macroeconomic Modelling and the Effects of Policy Reforms: an Assessment for Italy using ITEM and

In this paper we compare the dynamic properties of the Italian Treasury Econometric Model (ITEM) with those of QUEST III, the endogenous growth model of the European Commission (DG ECFIN) in the version calibrated for Italy. We consider an array of shocks often examined in policy simulations and investigate their implications on macro variables. In doing so, we analyse the main transmission channels in the two models and provide a comparative assessment of the magnitude and the persistence of the effects, trying to ascertain whether the responses to shocks are consistent with the predictions of economic theory. We show that, despite substantial differences between the two models, the responses of the key variables are qualitatively similar when we consider competition enhancing policies and labour productivity improvements. On the other hand, we observe quantitative disparities between the two models, mainly due to the forward-looking behaviour and the endogenous growth mechanism incorporated into the QUEST model but not in ITEM. The simulation results show that Quest III is a powerful tool to capture the effects of structural economic reforms, like competitionenhancing policies or innovation-promoting policies. On the other hand, owing to the breakdown of fiscal variables in a large number of components, ITEM is arguably more suitable for the quantitative evaluation of fiscal policy and the study of the impact of reforms on the public sector balance sheet.Economic Modelling, DSGE, Structural Reforms, Italy

Multidimensional persistent homology is stable

Multidimensional persistence studies topological features of shapes by analyzing the lower level sets of vector-valued functions. The rank invariant completely determines the multidimensional analogue of persistent homology groups. We prove that multidimensional rank invariants are stable with respect to function perturbations. More precisely, we construct a distance between rank invariants such that small changes of the function imply only small changes of the rank invariant. This result can be obtained by assuming the function to be just continuous. Multidimensional stability opens the way to a stable shape comparison methodology based on multidimensional persistence.Comment: 14 pages, 3 figure

OPTIMAL HOMEOMORPHISMS BETWEEN CLOSED CURVES

The concept of natural pseudo-distance has proven to be a powerful tool for measuring the dissimilarity between topological spaces endowed with continuous real-valued functions. Roughly speaking, the natural pseudo-distance is defined as the infimum of the change of the functions' values, when moving from one space to the other through homeomorphisms, if possible. In this paper, we prove the first available result about the existence of optimal homeomorphisms between closed curves, i.e. inducing a change of the function that equals the natural pseudo-distance

An edit distance for Reeb graphs

We consider the problem of assessing the similarity of 3D shapes using Reeb graphs from the standpoint of robustness under perturbations. For this purpose, 3D objects are viewed as spaces endowed with real-valued functions, while the similarity between the resulting Reeb graphs is addressed through a graph edit distance. The cases of smooth functions on manifolds and piecewise linear functions on polyhedra stand out as the most interesting ones. The main contribution of this paper is the introduction of a general edit distance suitable for comparing Reeb graphs in these settings. This edit distance promises to be useful for applications in 3D object retrieval because of its stability properties in the presence of noise

The evolution of composite indices of well-being: An application to Italy

Abstract Prompted by the work of the Stiglitz's Commission, the growing attention to the beyond-GDP measures has led to the inclusion of well-being indicators in the policy agenda. This innovation asks for an improvement of the existing methodology to produce composite indices, in order to correctly address spatial and temporal comparisons as well as tackling for unbalances. Following a short review of the main international experiences, this paper will investigate these issues considering the methodology currently adopted to normalize and aggregate the selected individual indicators included in the Italian well-being. We study the properties of this methodology looking at different normalization and aggregation approaches and underlining some drawbacks, mostly due to the way in which time dimension, normalization, aggregation and unbalance adjustment interplay with each other. We illustrate our findings by means of examples related also to the ecological side. We argue that new efforts should be done to overcome these drawbacks extending the research agenda toward new non-compensatory approaches. Testing for time series methods, such as dynamic factor models could represent another important step forward. Meanwhile the introduction of a more traditional framework for the composite indicators for Italian well-being could be considered